Mireille Bousquet-Mélou

Rational and algebraic series in combinatorial enumeration

Keywords: Mathematics Subject Classification (2000). Primary 05A15; Secondary 68Q45 Enumerative combinatorics, generating functions, rational and algebraic power series, formal languages

Let A be a class of objects, equipped with an integer size such that for all n the number an of objects of size n is finite. We are interested in the case where the generating function n ant n is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers an, but also a combinatorial one: the rational or algebraic nature of the generating function suggests that the objects have a (possibly hidden) structure, similar to the linear structure of words in the rational case, and to the branching structure of trees in the algebraic case. We describe and illustrate this combinatorial intuition, and discuss its validity. While it seems to be satisfactory in the rational case, it is probably incomplete in the algebraic one. We conclude with open questions.

Introduction

The general topic of this paper is the enumeration of discrete objects (words, trees, graphs...) and more specifically the rational or algebraic nature of the associated generating functions. Let A be a class of discrete objects equipped with a size:

size : A → N A → |A|.
Assume that for all n, the number a n of objects of size n is finite. The generating function of the objects of A, counted by their size, is the following formal power series in the indeterminate t:

A(t) := n≥0 a n t n = A∈A t |A| . (1)
To take a very simple example, if A is the set of words on the alphabet {a, b} and the size of a word is its number of letters, then the generating function is n≥0 2 n t n = 1/(1 -2t).

Generating functions provide both a tool for solving counting problems, and a concise way to encode their solution. Ideally, one would probably dream of finding a closed formula for the numbers a n . But the world of mathematical objects would be extremely poor if this was always possible. In practise, one is usually happy with an expression of the generating function A(t), or even with a recurrence relation defining the sequence a n , or a functional equation defining A(t).

Enumerative problems arise spontaneously in various fields of mathematics, computer science, and physics. Among the most generous suppliers of such problems, let us cite discrete probability theory, the analysis of the complexity of algorithms [START_REF] Knuth | The art of computer programming[END_REF][START_REF] Flajolet | An Introduction to the Analysis of Algorithms[END_REF], and the discrete models of statistical physics, like the famous Ising model [START_REF] Baxter | Exactly solved models in statistical mechanics[END_REF]. More generally, counting the objects that occur in one's work seems to answer a natural curiosity. It helps to understand the objects, for instance to appreciate how restrictive are the conditions that define them. It also forces us to get some understanding of the structure of the objects: an enumerative result never comes for free, but only after one has elucidated, at least partly, what the objects really are.

We focus in this survey on objects having a rational, or, more generally, algebraic generating function. Rational and algebraic formal power series are wellbehaved objects with many interesting properties. This is one of the reasons why several classical textbooks on enumeration devote one or several chapters to these series [START_REF] Flajolet | Analytic Combinatorics[END_REF][START_REF] Stanley | Enumerative combinatorics[END_REF][START_REF] Stanley | Enumerative combinatorics[END_REF]. These chapters give typical examples of objects with a rational [resp. algebraic] generating function (GF). After a while, the collection of these examples builds up a general picture: one starts thinking that yes, all these objects have something in common in their structure. At the same time arises the following question: do all objects with a rational [algebraic] GF look like that? In other words, what does it mean, what does it suggest about the objects when they are counted by a rational [algebraic] GF ?

This question is at the heart of this survey. For each of the two classes of series under consideration, we first present a general family of enumerative problems whose solution falls invariably in this class. These problems are simple to describe: the first one deals with walks in a directed graph, the other with plane trees. Interestingly, these families of objects admit alternative descriptions in language theoretic terms: they correspond to regular languages, and to unambiguous contextfree languages, respectively. The words of these languages have a clear recursive structure, which explains directly the rationality [algebraicity] of their GF.

The series counting words of a regular [unambiguous context-free] language are called N-rational [N-algebraic]. It is worth noting that a rational [algebraic] series with non-negative coefficients is not necessarily N-rational [N-algebraic]. Since we want to appreciate whether our two generic classes of objects are good representatives of objects with a rational [algebraic] GF, the first question to address is the following: do we always fall in the class of N-rational [N-algebraic] series when we count objects with a rational [algebraic] GF? More informally, do these objects exhibit a structure similar to the structure of regular [context-free] languages? Is such a structure usually clearly visible? That is to say, is it easy to feel, to predict rationality [algebraicity]?

We shall see that the answer to all these questions tends to be yes in the rational case (with a few warnings...) but is probably no in the algebraic case. In particular, the rich world of planar maps (planar graphs embedded in the sphere) abounds in candidates for non-N-algebraicity. The algebraicity of the associated GFs has been known for more than 40 years (at least for some families of maps), but it is only in the past 10 years that a general combinatorial explanation of this algebraicity has emerged. Moreover, the underlying constructions are more general that those allowed in context-free descriptions, as they involve taking complements.

Each of the main two sections ends with a list of questions. In particular, we present at the end of Section 3 several counting problems that are simple to state and have an algebraic GF, but for reasons that remain mysterious.

The paper is sometimes written in an informal style. We hope that this will not stop the reader. We have tried to give precise references where he/she will find more details and more material on the topics we discuss. In particular, this survey borrows a lot to two books that we warmly recommend: Stanley's Enumerative Combinatorics [START_REF] Stanley | Enumerative combinatorics[END_REF][START_REF] Stanley | Enumerative combinatorics[END_REF], and Flajolet & Sedgewick's Analytic Combinatorics [START_REF] Flajolet | Analytic Combinatorics[END_REF].

Notation and definitions. Given a (commutative) ring R, we denote by R[t] the ring of polynomials in t having coefficients in R. A Laurent series in t is a series of the form A(t) = n≥n0 a n t n , with n 0 ∈ Z and a n ∈ R for all n. If n 0 ≥ 0, we say that A(t) is a formal power series. The coefficient of t n is denoted a n := [t n ]A(t). The set of Laurent series forms a ring, and even a field if R is a field. The quasi-inverse of A(t) is the series A * (t) := 1/(1 -A(t)). If A(t) is a formal power series with constant term 0, then A * (t) is a formal power series too.

In most occasions, the series we consider are GFs of the form (1) and thus have rational coefficients. However, we sometimes consider refined enumeration problems, in which every object A is weighted, usually by a monomial w(A) in some additional indeterminates x 1 , . . . , x m . The weighted GF is then A∈A w(A)t |A| , so that the coefficient ring is Q[x 1 , . . . , x m ] rather than Q.

We denote k = {1, 2, . . . , k}. We use the standard notation N, Z, Q, and P := {1, 2, 3, . . .}.

Rational generating functions

2.1. Definitions and properties. The Laurent series A(t) with coefficients in the field R is said to be rational if it can be written in the form

A(t) = P (t) Q(t)
where P (t) and

Q(t) belong to R[t].
There is probably no need to spend a lot of time explaining why such series are simple and well-behaved. We refer to [START_REF] Stanley | Enumerative combinatorics[END_REF]Ch. 4] and [START_REF] Flajolet | Analytic Combinatorics[END_REF]Ch. IV] for a survey of their properties. Let us review briefly some of them, in the case where R = Q. The set of (Laurent) rational series is closed under sum, product, derivation, reciprocals -but not under integration as shown by A(t) = 1/(1 -t). The coefficients a n of a rational series A(t) satisfy a linear recurrence relation with constant coefficients:

for n large enough,

a n = c 1 a n-1 + c 2 a n-2 + • • • + c k a n-k .
The partial fraction expansion of A(t) provides a closed form expression of these coefficients of the form:

a n = k i=0 P i (n)µ n i (2)
where the µ i are the reciprocals of the roots of the denominator Q(t), and the P i are polynomials. In particular, if A(t) has non-negative integer coefficients, its radius of convergence ρ is one its the poles (Pringsheim) and the "typical" asymptotic behaviour of a n is

a n ∼ κρ -n n d (3)
where d ∈ N and κ is an algebraic number. The above statement has to be taken with a grain of salt: all poles of minimal modulus may actually contribute to the dominant term in the asymptotic expansion of a n , as indicated by [START_REF] Banderier | Generating functions for generating trees[END_REF]. Let us add that Padé approximants allow us to guess whether a generating function whose first coefficients are known is likely to be rational. For instance, given the 10 first coefficients of the series

A(t) = t+2 t 2 +6 t 3 +19 t 4 +61 t 5 +196 t 6 +629 t 7 +2017 t 8 +6466 t 9 +20727 t 10 +O t 11 ,
it is easy to conjecture that actually

A(t) = t (1 -t) 3 1 -5 t + 7 t 2 -4 t 3 .
Padé approximants are implemented in most computer algebra packages. For instance, the relevant Maple command is convert/ratpoly.

2.2.

Walks on a digraph. We now introduce our typical "rational" objects.

Let G = (V, E) be a directed graph with (finite) vertex set V = p and (directed) edge set E ⊂ V × V . A walk of length n on G is a sequence of vertices w = (v 0 , v 1 , . . . , v n ) such that for all i, the pair (v i , v i+1 ) is an edge. Such a walk goes from v 0 to v n . We denote |w| = n. Now assign to each directed edge e a weight (an indeterminate) x e . Define the weight x w of the walk w as the product of the weights of the edges it visits: more precisely,

x w = n-1 i=0 x (vi,vi+1) .
See Fig. 1(a) for an example. Let X denote the (weighted) adjacency matrix of G: for i and j in p , the entry X i,j is x e if (i, j) = e is an edge of G and 0 otherwise. Let W i,j (t) be the weighted generating function of walks going from i to j: W i,j (t) = w:i;j x w t |w| . It is well-known, and easy to prove, that W i,j is a rational function in t with coefficicients in Q[x e , e ∈ E] (see [START_REF] Stanley | Enumerative combinatorics[END_REF]Thm. 4.7.1]).

Theorem 2.1. The series W i,j (t) is the (i, j)-entry in the matrix (1 -tX) -1 .

This theorem reduces the enumeration of walks on a digraph to the calculation of the inverse of a matrix with polynomial coefficients. It seems to be little known in the combinatorics community that this inverse matrix can be computed by studying the elementary cycles of the digraph G. This practical tool relies on Viennot's theory of heaps of pieces [START_REF] Viennot | Heaps of pieces. I. Basic definitions and combinatorial lemmas[END_REF]. Since it is little known, and often convenient, let us advertise it here. It will be illustrated further down.

An elementary cycle of G is a closed walk w = (v 0 , v 1 , . . . , v n-1 , v 0 ) such that v 0 , . . . , v n-1 are distinct. It is defined up to a cyclic permutation of the v i . That is, (v 1 , v 2 , . . . , v n-1 , v 0 , v 1 ) is the same cycle as w. A collection γ = {γ 1 , . . . , γ r } of (elementary) cycles is non-intersecting if the γ i are pairwise disjoint. The weight x γ of γ is the product of the weights of the γ i . We denote |γ| = |γ i |.

Proposition 2.2 ([81]

). The generating function of walks going from i to j reads

W i,j (t) = N i,j D , where D = γ={γ1,...,γr} (-1) r x γ t |γ| and N i,j = w;γ={γ1,...,γr} (-1) r x w x γ t |w|+|γ| .
The polynomial D is the alternating generating function of non-intersecting collections of cycles. In the expression of N , γ a non-intersecting collection of cycles and w a self-avoiding walk going from i to j, disjoint from the cycles of γ.

To illustrate this result, let us determine the generating function of walks going from 1 to 2 and from 1 to 3 on the digraph of Fig. 1(a). This graph contains 4 cycles of length 1, 2 cycles of length 2, 2 cycles of length 3 and 1 cycle of length 4. By forming all non-intersecting collections of cycles, one finds:

D(t) = 1 -(3 + x)t+ (3 + 3x-2)t 2 + (-1 -3x+ 3 + x-2)t 3 + (x-1 -x+ 1 + x-y)t 4 = 1 -(3 + x)t + (1 + 3x)t 2 -2xt 3 + (x -y)t 4 .
There is only one self-avoiding walk (SAW) going from 1 to 2, and one SAW going from 1 to 3 (via the vertex 2). The collections of cycles that do not intersect these walks are formed of loops, which gives

N 1,2 = t(1 -t) 2 (1 -xt) and N 1,3 = t 2 (1 -t) 2 .
Hence the generating function of walks that start from 1 and end at 2 or 3 is:

W 1,2 + W 1,3 = N 1,2 + N 1,3 D = t(1 -t) 2 (1 + t -xt) 1 -(3 + x)t + (1 + 3x)t 2 -2xt 3 + (x -y)t 4 . (4)

Regular languages and automata.

There is a very close connection between the collection of walks on a digraph and the words of regular languages. Let A be an alphabet, that is, a finite set of symbols (called letters). A word on A is a sequence u = u 1 u 2 • • • u n of letters. The number of occurrences of the letter a in the word u is denoted |u| a . The product of two words

u 1 u 2 • • • u n and v 1 v 2 • • • v m is the concatenation u 1 u 2 • • • u n v 1 v 2 • • • v m .
The empty word is denoted ǫ. A language on A is a set of words. We define two operations on languages:

-the product LK of two languages L and K is the set of words uv, with u ∈ L and v ∈ K; this product is easily seen to be associative, -the star L * of the language L is the union of all languages L k , for k ≥ 0. By convention, L 0 is reduced to the empty word ǫ.

A finite state automaton on A is a digraph (V, E) with possibly multiple edges, together with: -a labelling of the edges by letters of A, that is to say, a function L : E → A, -an initial vertex i, -a set V f ⊂ V of final vertices. The vertices are usually called the states of the automaton. The automaton is deterministic if for every state v and every letter a, there is at most one edge labelled a starting from v.

To every walk on the underlying multigraph, one associates a word on the alphabet A by reading the letters met along the walk. The language L recognized by the automaton is the set of words associated with walks going from the initial state i to one of the states of V f . For j ∈ V , let L j denote the set of words associated with walks going from i to j. These sets admit a recursive description. For the automaton of Fig. 1

(b), one has L = L 2 ∪ L 3 with L 1 = {ǫ}, L 2 = L 1 c ∪ L 2 a ∪ L 3 a ∪ L 4 c, L 4 = L 2 ā ∪ L 3 ā ∪ L 4 a ∪ L 5 b, L 3 = L 2 c ∪ L 3 b ∪ L 3 c, L 5 = L 2 c ∪ L 3 b ∪ L 3 c ∪ L 5 b.
Remarkably, there also exists a non-recursive combinatorial description of the languages that are recognized by an automaton [START_REF] Hopcroft | Formal languages and their relation to automata[END_REF]Thms. 3.3 and 3.10].

Theorem 2.3. Let L be a language on the alphabet A. There exists a finite state automaton that recognizes L if and only if L can be expressed in terms of finite languages on A, using a finite number of unions, products and stars of languages.

If these conditions hold, L is said to be regular. Moreover, there exists a deterministic automaton that recognizes L.

Regular languages and walks on digraphs. Take a deterministic automaton, and associate with it a weighted digraph as follows: the vertices are those of the automaton, and for all vertices j and k, if m edges go from j to k in the automaton, they are replaced by a single edge labelled m in the digraph. For instance, the automaton of Fig. 1(b) gives the digraph to its left, with x = y = 2. Clearly, the length GF of words of L is the GF of (weighted) walks of this digraph going from the initial vertex i to one of the final vertices of V f . For instance, according to (4), the length GF of the language recognized by the automaton of Fig. 1(b) is

A(t) = t (1 -t) 3 1 -5 t + 7 t 2 -4 t 3 . (5)
Take a regular language L recognized by a deterministic automaton A. There exists another deterministic automaton that recognizes L and does not contain multiple edges. The key is to create a state (j, a) for every edge labelled a ending at j in the automaton A. The digraph associated with this new automaton has all its edges labelled 1, so that there exists a length preserving bijection between the words of L and the walks on the digraph going from a specified initial vertex v 0 to one of the vertices of a given subset V f of vertices.

Conversely, starting from a digraph with all edges labelled 1, together with a specified vertex v 0 and a set V f of final vertices, it is easy to construct a regular language that is in bijection with the walks of the graph going from v 0 to V f (consider the automaton obtained by labelling all edges with distinct letters). This shows that counting words of regular languages is completely equivalent to counting walks in digraphs. In particular, the set of rational series obtained in both types of problems coincide, and have even been given a name: Definition 2.4. A series A(t) = n≥0 a n t n with coefficients in N is said to be N-rational if there exists a regular language having generating function A(t) -a 0 .

The description of regular languages given by Theorem 2.3 implies that the set of N-rational series contains the smallest set of series containing N[t] and closed under sum, product and quasi-inverse. The converse is true [71, Thm. II.5.1]. There exists a simple way to decide whether a given rational series with coefficients in N is N-rational [71, Thms. II.10.2 and II.10.5].

Theorem 2.5. A series A(t) = n≥0 a n t n with coefficients in N is N-rational if and only if there exists a positive integer p such that for all r ∈ {0, . . . , p}, the series A r,p (t) := n≥0 a np+r t n has a unique singularity of minimal modulus (called dominant).

There exist rational series with non-negative integer coefficients that are not N-rational. For instance, let α be such that cos α = 3/5 and sin α = 4/5, and define a n = 25 n cos(nα) 2 . It is not hard to see that a n is a non-negative integer. The associated series A(t) reads

A(t) = 1 -2t + 225t 2 (1 -25t)(625t 2 + 14t + 1)
.

It has 3 distinct dominant poles. As α is not a rational multiple of π, the same holds for all series A 0,p (t), for all values of p. Thus A(t) is not N-rational.

2.4. The combinatorial intuition of rational generating functions. We have described two families of combinatorial objects that naturally yield rational generating functions: walks in a digraph and words of regular languages. We have, moreover, shown that the enumeration of these objects are equivalent problems. It seems that these families convey the "right" intuition about objects with a rational GF. By this, we mean informally that:

(i) "every" family of objects with a rational GF has actually an N-rational GF, (ii) for almost all families of combinatorial objects with a rational GF, it is easy to foresee that there will be a bijection between these objects and words of a regular language.

Point (ii) means that most of these families F have a clear automatic structure, similar to the automatic structure of regular languages: roughly speaking, the objects of F can be constructed recursively using unions of sets and concatenation of cells (replacing letters). A more formal definition would simply paraphrase the definition of automata. Point (i) means simply that I have never met a counting problem that would yield a rational, but not N-rational GF. This includes problems coming from algebra, like growth functions of groups. On the contrary, Point (ii) only concerns purely combinatorial problems (but I do not want to be asked about the border between combinatorics and algebra). It admits very few counter-examples. Some will be discussed in Section 2.5. For the moment, let us illustrate the two above statements by describing the automatic structure of certain classes of objects (some being rather general), borrowed from [START_REF] Stanley | Enumerative combinatorics[END_REF]Ch. 4].

Column-convex polyominoes.

A polyomino is a finite union of cells of the square lattice, whose interior is connected. Polyominoes are considered up to a translation. A polyomino is column-convex (cc) if its intersection with every vertical line is connected. Let a n be the number of cc-polyominoes having n cells, and let A(t) be the associated generating function. We claim that these polyominoes have an automatic structure.

Consider a cc-polyomino P having n cells. Let us number these cells from 1 to n as illustrated in Fig. column, cells are numbered from bottom to top. In each of the other columns, the lowest cell that has a left neighbour gets the smallest number; then the cells lying below it are numbered from top to bottom, and finally the cells lying above it are numbered from bottom to top. Note that for all i, the cells labelled 1, 2, . . . , i form a cc-polyomino. This will be essential in our description of the automatic structure of these objects. Associate with P the word u = u 1 • • • u n on the alphabet {a, b, c} defined by -u i = c (like Column) if the ith cell is the first to be visited in its column, -u i = b (like Below) if the ith cell lies below the first visited cell of its column, -u i = a (like Above) if the ith cell lies above the first visited cell of its column. Then, add a bar on the letter u i if the ith cell of P has a South neighbour, an East neighbour, but no South-East neighbour. (In other words, the barred letters indicate where to start a new column, when the bottommost cell of this new column lies above the bottommost cell of the previous column.) This gives a word v on the alphabet {a, b, c, ā, b, c}. It is not hard to see that the map that sends P on the word v is a size-preserving bijection between cc-polyominoes and words recognized by the automaton of Fig. 1(b). Hence by [START_REF] Baxter | Exactly solved models in statistical mechanics[END_REF], the generating function of column-convex polyominoes is [START_REF] Temperley | Combinatorial problems suggested by the statistical mechanics of domains and of rubber-like molecules[END_REF]:

A(t) = t (1 -t) 3 1 -5 t + 7 t 2 -4 t 3 .

P-partitions.

A partition of the integer n into at most k parts is a nondecreasing k-tuple λ = (λ 1 , . . . , λ k ) of nonnegative integers that sum to n. This classical number-theoretic notion is generalized by the notion of P-partitions. Let P be a natural partial order on k (by natural we mean that if i < j in P , then i < j in N). A P -partition of n is a k-tuple λ = (λ 1 , . . . , λ k ) of nonnegative integers that sum to n and satisfy λ i ≤ λ j if i ≤ j in P . Thus when P is the natural total order on k , a P -partition is simply a partition1 .

We are interested in the following series:

F P (t) = λ t |λ| ,
where the sum runs over all P -partitions and

|λ| = λ 1 + • • • + λ k is the weight of λ.
The case of ordinary partitions is easy to analyze: every partition can be written in a unique way as a linear combination

c 1 λ (1) + • • • + c k λ (k) (6)
where λ (i) = (0, 0, . . . , 0, 1, 1, . . . , 1) has exactly i parts equal to 1 and c i ∈ N. The weight of λ (i) is i, and one obtains:

F P (t) = 1 (1 -t)(1 -t 2 ) • • • (1 -t k ) . (7)
The automatic structure of (ordinary) partitions is transparent: since they are constructed by adding a number of copies of λ (1) , then a number of copies of λ (2) , and so on, there is a size preserving bijection between these partitions and walks starting from 1 and ending anywhere in the following digraph:

1 4 2 3 [1] [2] [2] [3] [3] [4] [4] [4] [3] [4]
Note that this graph corresponds to k = 4, and that an edge labelled [ℓ] must be understood as a sequence of ℓ edges. These labels do not correspond to multiplicities. Observe that the only cycles in this digraph are loops. This, combined with Proposition 2.2, explains the factored form of the denominator of [START_REF] Bernardi | Bijective counting of Kreweras walks and loopless triangulations[END_REF].

Consider now the partial order on 4 defined by 1 < 3, 2 < 3 and 2 < 4. The partitions of weight at most 2 are (0, 0, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (1, 0, 1, 0), (0, 0, 1, 1), (0, 0, 2, 0), (0, 0, 0, 2), so that F P (t) = 1 + 2t + 4t 2 + O(t 3 ). If one is brave enough to list P -partitions of weight at most 20, the Padé approximant of the truncated series thus obtained is remarkably simple:

F P (t) = 1 + t + t 2 + t 3 + t 4 (1 -t)(1 -t 2 )(1 -t 3 )(1 -t 4 ) + O(t 21 ),
and allows one to make a (correct) conjecture. It turns out that the generating function of P -partitions is always a rational series of denominator (1 -t)(1 -t 2 ) . . . (1 -t k ). Moreover, P -partitions obey our general intuition about objects with a rational GF. The following proposition, illustrated below by an example, describes their automatic structure: the set of P -partitions can be partitioned into a finite number of subsets; in each of these subsets, partitions have a structure similar to [START_REF] Bender | The number of degree-restricted rooted maps on the sphere[END_REF]. Recall that a linear extension of P is a bijection σ on k such that σ(i) < σ(j) if i < j in P .

Proposition 2.6 ([74], Section 4.5). Let P be a natural order on k .

For every P -partition λ, there exists a unique linear extension σ of P such that for all i, λ σ(i) ≤ λ σ(i+1) , the inequality being strict if σ(i) > σ(i + 1). We say that λ is compatible with σ.

Given a linear extension σ, the P -partitions that are compatible with σ can be written in a unique way as a linear combination with coefficients in N:

λ (σ,0) + c 1 λ (σ,1) + • • • + c k λ (σ,k) (8)
where λ (σ,0) is the smallest P -partition compatible with σ:

λ (σ,0) σ(j) = |{i < j : σ(i) > σ(i + 1)}| for 1 ≤ j ≤ k,
and for

1 ≤ i ≤ k, (λ (σ,i) σ(1) , . . . , λ (σ,i) σ(k) ) = (0, 0, . . . , 0, 1, 1, . . . , 1)
has exactly i parts equal to 1. Thus the GF of these P -partitions is

F P,σ (t) = t e(σ) (1 -t)(1 -t 2 ) . . . (1 -t k )
where e(σ) is a variant of the Major index of σ:

e(σ) = i:σ(i)>σ(i+1) (k -i).
Example. Let us return to the order 1 < 3, 2 < 3 and 2 < 4. The 5 linear extensions are 1234, 2134, 1243, 2143 and 2413. Take σ = 2143. The P -partitions λ that are compatible with σ are those that satisfy λ 2 < λ 1 ≤ λ 4 < λ 3 . The smallest of them is thus λ (σ,0) = (1, 0, 2, 1). Then λ (σ,1) = (0, 0, 1, 0), λ (σ,2) = (0, 0, 1, 1), λ (σ,3) = (1, 0, 1, 1) and λ (σ,4) = (1, 1, 1, 1).

Integer points in a convex polyhedral cone (

[74], Sec. 4.6). Let H be a finite collection of linear half-spaces of R m of the form c 1 α 1 + • • • + c m α m ≥ 0, with c i ∈ Z.
We are interested in the set E of non-negative integer points α = (α 1 , . . . , α m ) lying in the intersection of those half-spaces. For instance, we could have the following set E, illustrated in Fig. 3(a):

E = {(α 1 , α 2 ) ∈ N 2 : 2α 1 ≥ α 2 and 2α 2 ≥ α 1 }. (9)
Numerous enumerative problems (including P -partitions) can be formulated in terms of linear inequalities as above. The generating function of E is The set E is a monoid (it is closed under summation). In general, it is not a free monoid. Geometrically, the set C of non-negative real points in the intersection of the half-spaces of H forms a pointed convex polyhedral cone (the term pointed means that it does not contain a line), and E is the set of integer points in C.

E(t) = α∈E t |α| , where |α| = α 1 + • • • + α m . In the above example, E(t) = 1 + t 2 + 2t 3 + t 4 + 2t 5 + 3t 6 + 2t 7 + O(t 8 ).
The simplicial case. In the simplest case, the cone C is simplicial. This implies that the monoid E is simplicial, meaning that there exists linearly independent vectors α (1) , . . . , α (k) such that

E = α ∈ N m : α = q 1 α (1) + • • • + q k α (k) with q i ∈ Q, q i ≥ 0 .
This is the case in Example ( 9), with α (1) = (1, 2) and α (2) = (2, 1). The interior of E (the set of points of E that are not on the boundary of C) is then

E = α ∈ N m : α = q 1 α (1) + • • • + q k α (k) with q i ∈ Q, q i > 0 . ( 10
)
Then there exists a finite subset D of E [resp. D of E] such that every element of E [resp. E] can be written uniquely in the form

α = β + c 1 α (1) + • • • + c k α (k) , (11)
with β ∈ D [resp. β ∈ D] and c i ∈ N [74, Lemma 4.6.7]. In our running example (9), taken with α (1) = (1, 2) and α (2) = (2, 1), one has D = {(0, 0), (1, 1), (2, 2)} while D = {(1, 1), (2, 2), (3, 3)}. Compare [START_REF] Beukers | Monodromy for the hypergeometric function nFn-1[END_REF] with the structure found for Ppartitions [START_REF] Bernardi | On triangulations with high vertex degree[END_REF]. Thus E and E have an automatic structure and their GFs read

E(t) = β∈D t |β| k i=1 (1 -t |α (i) | ) resp. Ē(t) = β∈D t |β| k i=1 (1 -t |α (i) | )
.

In Example ( 9), one thus obtains

E(t) = 1 + t 2 + t 4 (1 -t 3 ) 2 = 1 -t + t 2 (1 -t)(1 -t 3 ) and Ē(t) = t 2 E(t).
The general case. The set E can always be partitioned into a finite number of sets F of the form [START_REF] Bétréma | Modèles avec particules dures, animaux dirigés et séries en variables partiellement commutatives[END_REF], where F is a simplicial monoid [74, Ch. 4, Eq. ( 24)]. Thus E, as a finite union of sets with an automatic structure, has an automatic structure as well. The associated generating function E(t) is N-rational, with a denominator which is a product of cyclotomic polynomials. Consider, for example, the set

E = {(α 1 , α 2 , α 3 ) ∈ N 3 : α 3 ≤ α 1 + α 2 }.
The cone C of non-negative real points α satisfying α 3 ≤ α 1 + α 2 is not simplicial, as it has 4 faces of dimension 2, lying respectively in the hyperplanes α i = 0 for i = 1, 2, 3 and

α 3 = α 1 + α 2 (Fig. 3(b)). But
E(t) = E 1 (t) + E 2 (t) -E 12 (t) (12)
where E 12 (t) counts integer points in the intersection of C 1 and C 2 (that is, in the plane α 1 = α 3 ). Since E 1 , E 2 and E 1 ∩ E 2 are simplicial cones (of dimension 3, 3 and 2 respectively), the method presented above for simplicial cones applies.

Indeed, E 1 [resp. E 2 ; E 12 ] is the set of linear combinations (with coefficients in N) of (1, 0, 1), (0, 1, 0) and (1, 0, 0) [resp. (1, 0, 1), (0, 1, 0) and (0, 1, 1); (1, 0, 1) and (0, 1, 0)]. This implies:

E(t) = 1 (1 -t) 2 (1 -t 2 ) + 1 (1 -t)(1 -t 2 ) 2 - 1 (1 -t)(1 -t 2 ) = 1 + t + t 2 (1 -t)(1 -t 2 ) 2 .
However, the "minus" sign in [START_REF] Bousquet-Mélou | Limit results for embedded trees. Applications to the integrated super-Brownian excursion[END_REF] prevents us from seeing directly the automatic nature of E (the difference of N-rational series is not always N-rational). This structure only becomes clear when we write E as the disjoint union of the interiors of all simplicial monoids induced by the triangulation of C into C 1 and C 2 . These monoids are the integer points of the faces (of all possible dimensions) of C 1 and C 2 . As there are 12 such faces (more precisely, 1 [resp. 4, 5, 2] faces of dimension 0 [resp. 1, 2, 3]), this gives E as the disjoint union of 12 sets having an automatic structure of the form (10).

2.5. Rational generating functions: more difficult questions.

2.5.1. Predicting rationality. We wrote in Section 2.4 that it is usually easy to foresee, to predict when a class of combinatorial objects has a rational GF. There are a few exceptions. Here is one of the most remarkable ones.

Example 2.7 (Directed animals). A directed animal with a compact source of size k is a finite set of points A on the square lattice Z 2 such that:

-the points (-i, i) for 0 ≤ i < k belong to A; they are called the source points, -all the other points in A can be reached from one of the source points by a path made of North and East steps, having all its vertices in A. See Fig. 4 for an illustration. A similar notion exists for the triangular lattice. It turns out that these animals have extremely simple generating functions [START_REF] Gouyou-Beauchamps | Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem[END_REF][START_REF] Bétréma | Modèles avec particules dures, animaux dirigés et séries en variables partiellement commutatives[END_REF].

Theorem 2.8. The number of compact-source directed animals of cardinality n is 3 n-1 on the square lattice, and 4 n-1 on the triangular lattice.

The corresponding GFs are respectively t/(1 -3t) and t/(1 -4t), and are as rational as a series can be. There is at the moment no simple combinatorial intuition as to why these animals have rational GFs. A bijection between square lattice animals and words on a 3-letter alphabet was described in [START_REF] Gouyou-Beauchamps | Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem[END_REF], but it does not shed a clear light on the structure of these objects. Still, there is now a convincing explanation of the algebraicity of these series (see Section 3.4.2).

Example 2.9 (The area under Dyck paths). Another family of (slightly less natural) examples is provided by the enumeration of points lying below certain lattice paths. For instance, let us call Dyck path of length 2n any path P on Z 2 formed of steps (1, 1) and (1, -1), that starts from (0, 0) and ends at (2n, 0) without ever hitting a point with a negative ordinate. The area below P is the number of non-negative integer points (i, j), with i ≤ 2n, lying weakly below P (Fig. 5). It turns out that the sum of the areas of Dyck paths of length 2n is simply

P :|P |=2n a(P ) = 4 n .
Again, the rationality of the associated generating function does not seem easy to predict, but there are good combinatorial reasons explaining why it is algebraic. See [START_REF] Chottin | Une preuve combinatoire de la rationalité d'une série génératrice associée aux arbres[END_REF][START_REF] Pergola | A bijective approach to the area of generalized Motzkin paths[END_REF] for a direct explanation of this result, references, and a few variations on this phenomenon, first spotted by Kreweras [START_REF] Kreweras | Aires des chemins surdiagonaux et application un problme conomique[END_REF]. Finally, let us mention that our optimistic statement about how easy it is to predict the rationality of a generating function becomes less and less true as we move from purely combinatorial problems to more algebraic ones. For instance, it is not especially easy to foresee that a group has an automatic structure [START_REF] Epstein | Word processing in groups[END_REF]. Let us give also an example coming from number theory. Let P (x) ≡ P (x 1 , . . . , x r ) be a polynomial with integer coefficients, and take p a prime. For n ≥ 0, let a n be the number of x ∈ (Z/p n Z) r such that P (x) = 0 mod p n . Then the generating function n a n t n is rational. A related result holds with p-adic solutions [START_REF] Denef | The rationality of the Poincaré series associated to the p-adic points on a variety[END_REF][START_REF] Igusa | Complex powers and asymptotic expansions. II. Asymptotic expansions[END_REF].

2.5.2. Computing a rational generating function. Let us start with an elementary, but important observation. Many enumerative problems, including some very hard, can be approximated by problems having a rational GF. To take one example, consider the notoriously difficult problem of counting self-avoiding polygons (elementary cycles) on the square lattice. It is easy to convince oneself that the generating function of SAP lying in a horizontal strip of height k is rational for all k. This does not mean that it will be easy (or even possible, in the current state of affairs) to compute the corresponding generating function when k = 100. Needless to say, there is at the moment no hope to express this GF for a generic value of k. The generating function of SAP having 2k horizontal steps can also be seen to be rational. Moreover, these SAP can be described in terms of linear inequalities (as in Section 2.4.3), which implies that the denominator of the corresponding series G k is a product of cyclotomic polynomials. But again, no one knows what this series is for a generic value of k, or even for k = 100. Still, some progress have been made recently, since it has been proved that the series G k have more and more poles as k increases, which means that their denominators involve infinitely many cyclotomic polynomials [START_REF] Rechnitzer | Haruspicy 2: The anisotropic generating function of self-avoiding polygons is not D-finite[END_REF]. This may be considered as a proof of the difficulty of this enumerative problem [START_REF] Guttmann | Indicators of solvability for lattice models[END_REF].

In general, computing the (rational) generating function of a family of objects depending on a parameter k may be non-obvious, if not difficult, even if the objects are clearly regular, and even if the final result turns out to be nice. A classical example is provided by the growth functions of Coxeter groups [START_REF] Macdonald | The Poincaré series of a Coxeter group[END_REF]. Here is a more combinatorial example. A partition λ = (λ 1 , . . . , λ k ) is said to be a k-Lecture Hall partition (k-LHP) if

0 ≤ λ 1 1 ≤ λ 2 2 ≤ • • • ≤ λ k k .
Since these partitions are defined by linear inequalities, it follows from Section 2.4.3 that their weight generating function is rational, with a denominator formed of cyclotomic polynomials. Still, there is no clear reason to expect that [START_REF] Bousquet-Mélou | Lecture hall partitions[END_REF]:

λ k-LHP t |λ| = 1 (1 -t)(1 -t 3 ) • • • (1 -t 2k-1 )
.

Several proofs have been given for this result and variations on it. See for instance [START_REF] Bousquet-Mélou | Lecture hall partitions[END_REF][START_REF] Corteel | Enumeration of sequences constrained by the ratio of consecutive parts[END_REF] and references in the latter paper. Some of these proofs are based on a bijection between lecture hall partitions and partitions into parts taken in {1, 3, . . . , 2k -1}, but these bijections are never really simple [START_REF] Yee | On the combinatorics of lecture hall partitions[END_REF][START_REF] Eriksen | A simple bijection between lecture hall partitions and partitions into odd integers[END_REF].

2.5.3. N-rationality. As we wrote in Section 2.4, we do not know of a counting problem that would yield a rational, but not N-rational series. It would certainly be interesting to find one (even if it ruins some parts of this paper).

Let us return to Soittola's criterion for N-rationality (Theorem 2.5). It is not always easy to prove that a rational series has non-negative coefficients. For instance, it was conjectured in [START_REF] Friedman | More monotonicity theorems for partitions[END_REF] that for any odd k, the number of partitions of n into parts taken in {k, k + 1, . . . , 2k -1} is a non-decreasing function of n, for n ≥ 1. In terms of generating functions, this means that the series

q + 1 -q (1 -q k )(1 -q k+1 ) • • • (1 -q 2k-1 )
has non-negative coefficients. This was only proved recently [START_REF] Prellberg | Proof of a monotonicity conjecture[END_REF]. When k is even, a similar result holds for the series

q + 1 -q (1 -q k )(1 -q k+1 ) • • • (1 -q 2k )(1 -q 2k+1 )
.

Once the non-negativity of the coefficients has been established, it is not hard to prove that these series are N-rational. This raises the question of finding a family of combinatorial objects that they count.

Algebraic generating functions

3.1. Definitions and properties. The Laurent series A(t) with coefficients in the field R is said to be algebraic (over R(t)) if it satisfies a non-trivial algebraic equation:

P (t, A(t)) = 0
where P is a bivariate polynomial with coefficients in R.

We assume below that R = Q. Again, the set of algebraic Laurent series possesses numerous interesting properties [START_REF] Stanley | Enumerative combinatorics[END_REF]Ch. 6], [43, Ch. VII]. It is closed under sum, product, derivation, reciprocals, but not under integration. These closure properties become effective using either the theory of elimination or Gröbner bases, which are implemented in most computer algebra packages. The coefficients a n of an algebraic series A(t) satisfy a linear recurrence relation with polynomial coefficients: for n large enough,

p 0 (n)a n + p 1 (n)a n-1 + p 2 (n)a n-2 + • • • + p k (n)a n-k = 0.
Thus the first n coefficients can be computed using a linear number of operations.

There is no systematic way to express the coefficients of an algebraic series in closed form. Still, one can sometimes apply the Lagrange inversion formula: Proposition 3.1. Let Φ and Ψ be two formal power series and let U ≡ U (t) be the unique formal power series with no constant term satisfying

U = tΦ(U ).
Then for n > 0, the coefficient of t n in Ψ(U ) is:

[t n ]Ψ(U ) = 1 n [t n-1 ] (Ψ ′ (t)Φ(t) n ) .
Given an algebraic equation P (t, A(t)) = 0, one can decide whether there exists a series U (t) and two rational series Φ and Ψ satisfying U = tΦ(U ) and A = Ψ(U ).

(

) 13
Indeed, such series exist if and only if the genus of the curve P (t, a) is zero [START_REF] Abhyankar | Algebraic geometry for scientists and engineers[END_REF]Ch. 15]. Moreover, both the genus and a parametrization of the curve in the form (13) can be determined algorithmically.

Example 3.2 (Finding a rational parametrization). The following algebraic equation was recently obtained [START_REF] Bousquet-Mlou | Planar maps and algebraic series: a generalization of the quadratic method[END_REF], after a highly non-combinatorial derivation, for the GF of certain planar graphs carrying a hard-particle configuration:

0 =
The package algcurves of Maple, and more precisely the commands genus and parametrization, reveal that a rational parametrization is obtained by setting

t = -3 (3 U + 7) 9 U 2 + 33 U + 37 (3 U + 1) 4 .
Of course, this is just the net result of Maple, which is not necessarily very meaningful for combinatorics. Still, starting from this parametrization, one obtains after a few attempts an alternative parametrizing series V with positive coefficients:

V = t (1 -2V )(1 -3V + 3V 2 ) . ( 15
)
The main interest of such a parametrization for this problem does not lie in the possibility of applying the Lagrange inversion formula. Rather, it suggests that a more combinatorial approach exists, based on the enumeration of certain trees, in the vein of [START_REF] Bousquet-Mélou | The degree distribution of bipartite planar maps: applications to the Ising model[END_REF][START_REF] Bouttier | Combinatorics of bicubic maps with hard particles[END_REF]. It also gives a hint of what these trees may look like.

Another convenient tool borrowed from the theory of algebraic curves is the possibility to explore all branches of the curve P (t, A(t)) = 0 in the neighbourhood of a given point t 0 . This is based on Newton's polygon method. All branches have a Puiseux expansion, that is, an expansion of the form:

A(t) = n≥n0 a n (t -t 0 ) n/d
with n 0 ∈ Z, d ∈ P. The coefficients a n belong to C (in general, to an algebraic closure of the ground field). These expansions can be computed automatically using standard software. For instance, the Maple command puiseux of the algcurves package tells us that (14) has a unique solution that is a formal power series, the other three solutions starting with a term t -2 . Such Puiseux expansions are crucial for studying the asymptotic behaviour of the coefficients of an algebraic series A(t). As in the rational case, one has first to locate the singularities of A(t), considered as a function of a complex variable t. These singularities are found among the roots of the discriminant and of the leading coefficient of P (t, a) (seen as a polynomial in a). The singular expansion of A(t) near its singularities of smallest modulus can then be converted, using certain transfer theorems, into an asymptotic expansion of the coefficients [START_REF] Flajolet | Singularity analysis of generating functions[END_REF][START_REF] Flajolet | Analytic Combinatorics[END_REF]VII.4].

Example 3.3 (Asymptotics of the coefficients of an algebraic series).

Consider the series V (t) defined by [START_REF] Bousquet-Mélou | Lecture hall partitions[END_REF]. Its singularities lie among the roots of the discriminant

∆(t) = -3 + 114t -4635t 2 + 55296t 3 .
Only one root is real. Denote it t 0 ∼ 0.065. The modulus of the other two roots is smaller than t 0 , so they could, in theory, be candidates for singularities. However, V (t) has non-negative coefficients, and this implies, by Pringsheim's theorem, that one of the roots of minimal modulus is real and positive. Hence V (t) has a unique singularity, lying at t 0 . A Puiseux expansion at this point gives

V (t) = c 0 -c 1 1 -t/t 0 + O(t -t 0 ),
for some explicit (positive) algebraic numbers c 0 and c 1 , which translates into

[t n ]V (t) = c 1 2 √ π t -n 0 n -3/2 (1 + o(1)) .
The determination of asymptotic expansions for the coefficients of algebraic series is probably not far from being completely automated, at least in the case of series with non-negative coefficients [START_REF] Chabaud | Sries gnratrices algbriques ; asymptotique et applications combinatoires[END_REF][START_REF] Flajolet | Analytic Combinatorics[END_REF]. The "typical" behaviour is

a n ∼ κ Γ(d + 1) ρ -n n d , ( 16
)
where κ is an algebraic number and d ∈ Q \ {-1, -2, -3, . . .}. Compare with the result (3) obtained for rational series. Again, the above statement is not exact, as the contribution of all dominant singularities must be taken into account. See [43, Thm. VII.6] for a complete statement.

Let us add that, again, one can guess if a series A(t) given by its first coefficients satisfies an algebraic equation P (t, A(t)) = 0 of a given bi-degree (d, e). The guessing procedure requires to know at least (d+1)(e+1) coefficients, and amounts to solving a system of linear equations. It is implemented in the package Gfun of Maple [START_REF] Salvy | Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable[END_REF]. For instance, given the 10 first coefficients of the series V (t) satisfying V (0) = 0 and (15), one automatically conjectures (15).

Plane trees.

Our typical "algebraic" objects will be (plane) trees. Let us begin with their usual intuitive recursive definition. A tree is a graph formed of a distinguished vertex (called the root) to which are attached a certain number (possibly zero) of trees, ordered from left to right. The number of these trees is the degree of the root. The roots of these trees are the children of the root. A more rigorous definition describes a tree as a finite set of words on the alphabet P satisfying certain conditions [START_REF] Neveu | Arbres et processus de Galton-Watson[END_REF]. We hope that our less formal definition and Fig. 6(a) suffice to understand what we mean. The vertices of a tree are often called nodes. Nodes of degree 0 are called leaves, the others are called inner nodes.

The enumeration of classes of trees yields very often algebraic equations. Let us consider for instance the complete binary trees, that is, the trees in which all vertices have degree 0 or 2 (Fig. 12). Let a n be the number of such trees having n leaves. Then, by looking at the two (sub)trees of the root, one gets, for n > 1:

a n = n-1 k=1 a k a n-k .
The initial condition is a 1 = 1. In terms of GFs, this gives A(t) = t + A(t) 2 , which is easily solved:

A(t) = 1 - √ 1 -4t 2 = n≥0 1 n + 1 2n n t n+1 . ( 17
)
More generally, many algebraic series obtained in enumeration are given as the first component of the solution of a system of the form

A i = P i (t, A 1 , . . . , A k ), (18)
for some polynomials P i (t, x 1 , . . . , x k ) having coefficients in Z. This system is said to be proper if P i has no constant term (P i (0, . . . , 0) = 0) and does not contain any linear term x i . It is positive if the coefficients of the P i are non-negative. For instance,

A 1 = t 2 + A 1 A 2 and A 2 = 2tA 3 1
is a proper positive system. The system is quadratic if every P i (t, x 1 , . . . , x k ) is a linear combination of the monomials t and x ℓ x m , for 1 ≤ ℓ ≤ m ≤ k.

Theorem 3.4 ([75], Thm. 6.6.10 and [START_REF] Salomaa | Automata-theoretic aspects of formal power series[END_REF], Thm. IV.2.2). A proper algebraic system has a unique solution (A 1 , . . . , A k ) in the set of formal power series in t with no constant term. This solution is called the canonical solution of the system. The series A 1 is also the first component of the solution of -a proper quadratic system, -a proper system of the form B i = tQ i (t, B 1 , . . . , B ℓ ), for 1 ≤ i ≤ ℓ. These two systems can be chosen to be positive if the original system is positive.

Proof. Let us prove the last property, which we have not found in the above references. Assume A 1 satisfies [START_REF] Bousquet-Mélou | Enumeration of planar constellations[END_REF] and that this system is quadratic. The ith equation reads A i = m i t + n i A σ(i) A τ (i) . Rewrite each monomial A i A j as tU ij and add the equations

U ij = t m i m j + m i n j U σ(j)τ (j) + m j n i U σ(i)τ (i) + n i n j U σ(i)τ (i) U σ(j)τ (j) .
The new system has the required properties. 0) is the first component of the solution of a proper positive system.

Definition 3.5. A series A(t) is N-algebraic if it has coefficients in N and if A(t) - A(
Proper positive systems like [START_REF] Bousquet-Mélou | Enumeration of planar constellations[END_REF] can always be given a combinatorial interpretation in terms of trees. Every vertex of these trees carries a label (i, c) where i ∈ k and c ∈ P. We say that i is the type of the vertex and that c is its colour. The type of a tree is the type of its root. Write A 0 = t, so that A i = P i (A 0 , A 1 , . . . , A k ). Let A 0 be the set reduced to the tree with one node, labelled (0, 1). For i ∈ k , let A i be the set of trees such that -the root has type i, -the types of the subtrees of the root, visited from left to right, are 0, . . . , 0, 1, . . . , 1, . . . , k, . . . , k, in this order, -if exactly e j children of the root have type j, the colour of the root is any integer in the interval [1, m], where m is the coefficient of x e0 0 • • • x e k k in P i (x 0 , . . . , x k ). Then it is not hard to see that A i (t) is the generating function of trees of type i, counted by the number of leaves. This explains why trees will be, in the rest of this paper, our typical "algebraic" objects.

Context-free languages.

As in the case of rational (and, more precisely, N-rational) series, there exists a family of languages that is closely related to algebraic series. A context-free grammar G consists of -a set S = {S 1 , . . . , S k } of symbols, with one distinguished symbol, say, S 1 , -a finite alphabet A of letters, disjoint from S, -a set of rewriting rules of the form S i → w where w is a non-empty word on the alphabet S ∪ A.

The grammar is proper if there is no rule S i → S j . The language L(G) generated by G is the set of words on the alphabet A that can be obtained from S 1 by applying iteratively the rewriting rules. A language is context-free is there exists a context-free grammar that generates it. In this case there exists also a proper context-free grammar that generates it. A derivation tree associated with G is a plane tree in which all inner nodes are labelled by symbols, and all leaves by letters, in such a way that if a node is labelled S i and its children w 1 , . . . , w k (from left to right), then the rewriting rule S i → w 1 • • • w k is in the grammar. If the root is labelled S 1 , then the word obtained by reading the labels of the leaves in prefix order (i.e., from left to right) belongs to the language generated by G. Conversely, for every word w in L(G), there exists at least one derivation tree with root labelled S 1 that gives w. The grammar is said to be unambiguous if every word of L(G) admits a unique derivation tree.

Assume G is proper. For 1 ≤ i ≤ k, let A i (t) be the generating function of derivation trees rooted at S i , counted by the number of leaves. With each rule r, associate the monomial

M (r) = x e0 0 • • • x e k
k where e 0 [resp. e i , with i > 0] is the number of letters of A [resp. occurrences of S i ] in the right-hand side of r. Then the series A 1 , . . . , A k form the canonical solution of the proper positive system [START_REF] Bousquet-Mélou | Enumeration of planar constellations[END_REF], with

P i (x 0 , x 1 , . . . , x k ) = r M (r),
where the sum runs over all rules r with left-hand side S i .

Conversely, starting from a positive system B i = tQ i (t, B 1 , . . . , B k ) and its canonical solution, it is always possible to construct an unambiguous grammar with symbols S 1 , . . . , S k such that B i is the generating function of derivation trees rooted at S i (the idea is to introduce a new letter a i for each occurrence of t). In view of Theorem 3.4 and Definition 3.5, this gives the following alternative characterization of N-algebraic series: Proposition 3.7. A series A(t) is N-algebraic if and only if only A(0) ∈ N and there exists an unambiguous context-free language having generating function A(t) -A(0).

The combinatorial intuition of algebraic generating functions.

We have described two families of combinatorial objects that naturally yield algebraic GFs: plane trees and words of unambiguous context-free languages. We have, moreover, shown a close relationship between these two types of objects. These two families convey the standard intuition of what a family with an algebraic generating function looks like: the algebraicity suggests that it may (or should...) be possible to give a recursive description of the objects based on disjoint union of sets and concatenation of objects. Underlying such a description is a context-free grammar. This intuition is the basis of the so-called Schützenberger methodology, according to which the "right" combinatorial way of proving algebraicity is to describe a bijection between the objects one counts and the words of an unambiguous context-free language. This approach has led in the 80's and 90's to numerous satisfactory explanations of the algebraicity of certain series, and we describe some of them in this subsection. Let us, however, warn the reader that the similarities with the rational case will stop here. Indeed, it seems that the "context-free" intuition is far from explaining all algebraicity phenomena in enumerative combinatorics. In particular, (i) it is very likely that many families of objects have an algebraic, but not N-algebraic generating function, (ii) there are many families of combinatorial objects with an algebraic GF that do not exhibit a clear "context-free" structure, based on union and concatenation. For several of these families, there is just no explanation of this type, be it clear or not.

This will be discussed in the next subsections. For the moment, let us illustrate the "context-free" intuition.

3.4.1. Walks on a line. Let S be a finite subset of Z. Let W be the set of walks on the line Z that start from 0 and take their steps in S. The length of a walk is its number of steps. Let W k be the set of walks ending at position k. For k ≥ 0, let M k be the subset of W k consisting of walks that never visit a negative position, and let M be the union of the sets M k . In probabilistic terms, the walks in M would be called meanders and the walks of M 0 excursions. Of course, a walk is simply a sequence of steps, hence a word on the alphabet S. Thus the sets of walks we have defined can be considered as languages on this alphabet.

Theorem 3.8. The language W is simply S * and is thus regular. The languages M, W k and M k are unambiguous context-free for all k.

Proof. We only describe the (very simple) case S = {+1, -1}, to illustrate the ideas that are involved in the construction of the grammar. We encode the steps +1 by the letter a, the steps -1 by b, and introduce some auxiliary languages:

• M - 0 , the subset of W 0 formed of walks that never visit a positive position,

• W + 0 [resp. W - 0 ]
, the subset of W 0 formed of walks that start with a [resp. b]. The language M 0 will be generated from the symbol M 0 , and similarly for the other languages. By looking at the first time a walk of M 0 [resp. M - 0 ] reaches position 0 after its first step, one obtains

M 0 → a(1 + M 0 )b(1 + M 0 ) and M - 0 → b(1 + M - 0 )a(1 + M - 0 ).
By considering the last visit to 0 of a walk of M k , one obtains, for k > 0:

M k → (1 + M 0 )a (1 k=1 + M k-1 ) .
This is easily adapted to general meanders:

M → M 0 + (1 + M 0 )a (1 + M ) .
Considering the first step of a walk of W 0 gives

W 0 → W + 0 + W - 0 with W + 0 → M 0 (1 + W - 0 ) and W - 0 → M - 0 (1 + W + 0 ).
Finally, for k > 0, looking at the first visit at 1

[resp. -1] of a walk of W k [resp. W -k ] yields W k → (1 + M - 0 )a (1 k=1 + W k-1 ) resp. W -k → (1 + M 0 )b 1 k=1 + W -(k-1) .
For a general set of steps S, various grammars have been described for the languages M k of meanders [START_REF] Duchon | On the enumeration and generation of generalized Dyck words[END_REF][START_REF] Labelle | Generalized Dyck paths[END_REF][START_REF] Labelle | Langages de Dyck généralisés[END_REF]. For W k , we refer to [START_REF] Labelle | Langages de Dyck généralisés[END_REF]Section 4] where the (representative) case S = {-2, -1, 0, 1, 2} is treated. Theorem 3.8 is often described in terms of walks in Z 2 starting from (0, 0) and taking their steps in {(1, j), j ∈ S}. The conditions on the positions of the walks that lead to the definition of M k and W k are restated in terms of conditions on the ordinates of the vertices visited by the walk. A harmless generalization is obtained by taking steps in a finite subset S of P × Z. A walk is still encoded by a word on the alphabet S. The languages W k remain unambiguous context-free. If each step (i, j) is, moreover, weighted by a rational number w i,j , then the generating function of walks of W, counted by the coordinates of their endpoint, is

W (t, s) = 1 1 -(i,j)∈S w i,j t i s j .
The generating function W k (t) that counts (weighted) walks ending at ordinate k is the coefficient of s k in W (t, s). Since W k is unambiguous context-free, the series W k (t) is algebraic. This gives a combinatorial explanation of the following result [START_REF] Stanley | Enumerative combinatorics[END_REF]Thm. 6.3.3]. Theorem 3.9 (Diagonals of rational series). Let A(x, y) = m,n≥0 a m,n x m y n be a series in two variables x and y, with coefficients in Q, that is rational. Then the diagonal of A, that is, the series ∆A(t) = n≥0 a n,n t n , is algebraic. Proof. By linearity, it suffices to consider the case A(x, y) =

x a y b 1 -0≤m,n≤d c m,n x m y n , with c 0,0 = 0. Set x = ts and y = t/s. The diagonal of A satisfies

∆A(t 2 ) = [s 0 ]A(ts, t/s) = t a+b [s b-a ] 1 1 -0≤m,n≤d c m,n t m+n s m-n ,
which is algebraic as it counts weighted paths in W b-a , for a certain set of steps. Hence ∆A(t) is algebraic too.

The converse of Theorem 3.9 holds: every series B(t) that is algebraic over Q(t) is the diagonal of a bivariate rational series A(x, t) [START_REF] Safonov | On conditions for the sum of a power series to be algebraic and rational[END_REF].

Note. If one is simply interested in obtaining a set of algebraic equations defining the GFs of the sets M k and W k , a more straightforward approach is to use a partial fraction decomposition (for W k ) and the kernel method (for M k ). See [75, 6.3], and [17, Example 3].

Directed animals.

Let us move to an example where a neat context-free exists, but is uneasy to discover. We return to the directed animals defined in Section 2.5.1. As discussed there, there is no simple explanation as to why the number of compact-source animals is so simple (Theorem 2.8). Still, there is a convincing explanation for the algebraicity of the corresponding series: directed animals have, indeed, a context-free structure. This structure was discovered a few years after the proof of Theorem 2.8, with the development by Viennot of the theory of heaps [START_REF] Viennot | Heaps of pieces. I. Basic definitions and combinatorial lemmas[END_REF], a geometric version of partially commutative monoids [START_REF] Cartier | Problèmes combinatoires de commutation et réarrangements[END_REF]. Intuitively, a heap is obtained by dropping vertically some solid pieces, the one after the other. Thus, a piece lies either on the "floor" (then it is said to be minimal), or covers, at least partially, another piece.

Directed animals are, in essence, heaps. To see this, replace every point of the animal by a dimer (Fig. 7). Note that if the animal has a unique source, the associated heap has a unique minimal piece. Such heaps are named pyramids. What makes heaps interesting here is that there exists a monoid structure on the set of heaps: The product of two heaps is obtained by putting one heap above the other and dropping its pieces. This product is the key in our context-free description of directed animals.

Let us begin with the description of pyramids (one-source animals). A pyramid is either a half-pyramid (Fig. 7(b)), or the product of a half-pyramid and a pyramid (Fig. 8, top). Let P (t) denote the GF of pyramids counted by the number of dimers, and H(t) denote the GF of half-pyramids. Then P (t) = H(t)(1 + P (t)). Now, a half-pyramid may be reduced to a single dimer. If it has several dimers, it is the product of a single dimer and of one or two half-pyramids (Fig. 8, bottom), which implies H(t) = t + tH(t) + tH 2 (t). A trivial computation finally provides the GF of directed (single-source) animals:

P (t) = 1 2 1 + t 1 -3t -1 while H(t) = 1 -t -(1 + t)(1 -3t) 2t .
The enumeration of compact-source directed animals is equivalent to the enumeration of heaps having a compact basis (the minimal dimers are adjacent). The generating function of heaps having a compact basis formed with k dimers is P (t)H(t) k-1 (Fig. 9), which implies that the generating function of compactsource animals is

P (t) 1 -H(t) = t 1 -3t
. 3.5. The world of planar maps. We have seen in Section 3.2 that plane trees are the paradigm for objects with an algebraic generating function. A more general family of plane objects seems to be just as deeply associated with algebraic series, but for reasons that are far more mysterious: planar maps. A (planar) map is a proper embedding of a planar graph in the sphere (Fig. 6(b)). In order to avoid symmetries, all the maps we consider are rooted: this means that one edge is distinguished and oriented. Maps are only considered up to a continuous deformation of the sphere. A map induces a 2-cell decomposition of the sphere: the cells of dimension 0 [resp. [START_REF] Abhyankar | Algebraic geometry for scientists and engineers[END_REF][START_REF] Banderier | Generating functions for generating trees[END_REF] are called vertices [resp. edges, faces]. Hence plane trees are maps with a single face.

The interest for the enumeration of planar maps dates back to the early 60's, in connection with the 4-colour theorem. The first results are due to Tutte [START_REF] Tutte | A census of planar triangulations[END_REF][START_REF] Tutte | A census of slicings[END_REF][START_REF] Tutte | A census of planar maps[END_REF]. Ten to fifteen years later, maps started to be investigated independently in theoretical physics, as a model for 2-dimensional quantum gravity [START_REF] Brézin | Planar diagrams[END_REF][START_REF] Bessis | Quantum field theory techniques in graphical enumeration[END_REF]. However, neither the recursive approach used by Tutte and his disciples, nor the physics approach based on matrix integrals were able to explain in a combinatorially satisfactory way the following observations:

-the generating functions of many classes of planar maps are algebraic, -the associated numbers are often irritatingly simple. Let us illustrate this with three examples: 1. General maps. The number of planar maps having n edges is [START_REF] Tutte | On the enumeration of planar maps[END_REF]:

g n = 2.3 n (n + 1)(n + 2) 2n n . (19)
The associated generating function G ≡ G(t) = n≥0 g n t n satisfies:

-1 + 16t + (1 -18t)G + 27t 2 G 2 = 0. ( 20
)
2. Loopless triangulations. The number of loopless triangulations (maps in which all faces have degree 3) having 2n + 2 faces is [START_REF] Mullin | On counting rooted triangular maps[END_REF]:

t n = 2 n (n + 1)(2n + 1) 3n n .
The associated generating function T ≡ T (t) = n t n t n satisfies

1 -27t + (-1 + 36t)T -8tT 2 -16t 2 T 3 = 0.
3. Three-connected triangulations. The number of 3-connected triangulations having 2n + 2 faces is [START_REF] Tutte | A census of planar triangulations[END_REF]:

m n = 2 (n + 1)(3n + 2) 4n + 1 n .
The associated generating function M ≡ M (t) = n t n t n satisfies

-1 + 16t + (1 -20t)M + (3t + 8t 2 )M 2 + 3t 2 M 3 + t 3 M 4 = 0.
These maps are in bijection with rooted maximal planar simple graphs (graphs with no loop nor multiple edge that lose planarity as soon as one adds an edge). At last, in the past ten years, a general combinatorial picture has emerged, suggesting that maps are, in essence, unrooted plane trees. In what follows, we illustrate on the example of general maps the main three approaches that now exist, and give references for further developments of these methods.

3.5.1. The recursive approach. We leave to the reader to experience personally that maps do not have an obvious context-free structure. Still, maps do have a simple recursive structure, based on the deletion of the root-edge. However, in order to exploit this structure, one is forced to keep track of the degree of the rootface (the face lying to the right of the root edge). The decomposition illustrated in Fig. 10 leads in a few lines to the following equation:

G(u, t) = 1 + tu 2 G(u, t) 2 + tu uG(u, t) -G(1, t) u -1 , (21)
where G(u, t) counts planar maps by the number of edges (t) and the degree of the root-face (u).

It can be checked that the above equation defines G(u, t) uniquely as a formal power series in t (with polynomial coefficients in u). However, it is not clear on the equation why G(1, t) (and hence G(u, t)) are algebraic. In his original paper, Tutte first guessed the value of G 1 (t) := G(1, t), and then proved the existence of a series G(u, t) that fits with G 1 (t) when u = 1, and satisfies the above equation. Still, a bit later, Brown came with a method for solving [START_REF] Bousquet-Mlou | The density of the ISE and local limit laws for embedded trees[END_REF]: the so-called quadratic method [START_REF] Brown | On the existence of square roots in certain rings of power series[END_REF][START_REF] Goulden | Combinatorial enumeration[END_REF]Sec. 2.9]. Write [START_REF] Bousquet-Mlou | The density of the ISE and local limit laws for embedded trees[END_REF] in the form (2aG(u, t) + b) 2 = δ, where a, b and δ are polynomials in t, u and G 1 (t). That is,

+ + =
2tu 2 (u -1)G(u, t) + tu 2 -u + 1 2 = 4t 2 u 3 (u-1)G 1 +(1-u) 2 -4tu 4 +6tu 3 +u 4 t 2 -2tu 2 .
It is not hard to see, even without knowing the value of G(u, t), that there exists a (unique) formal power series in t, say U ≡ U (t), that cancels the left-hand side of this equation. That is,

U = 1 + tU 2 + 2tU 2 (U -1)G(U, t).
This implies that the series U is a double root of the polynomial δ that lies on the right-hand side. The discriminant of this polynomial (in u) thus vanishes: this gives the algebraic equation ( 20) satisfied by G(1, t).

The enumeration of many other families of planar maps can also be attacked by a recursive description based on the deletion of an edge (or vertex, or face...). See for instance [START_REF] Mullin | On counting rooted triangular maps[END_REF] for 2-connected triangulations, or [START_REF] Bender | The number of degree-restricted rooted maps on the sphere[END_REF] for maps with prescribed face degrees. (For maps with high connectivity, like 3-connected triangulations, an additional composition formula is often required [START_REF] Tutte | A census of planar triangulations[END_REF][START_REF] Banderier | Random maps, coalescing saddles, singularity analysis, and Airy phenomena[END_REF].) The resulting equations are usually of the form P (F (u), F 1 , . . . , F k , t, u) = 0, [START_REF] Bousquet-Mlou | Planar maps and algebraic series: a generalization of the quadratic method[END_REF] where F (u) ≡ F (t, u), the main generating function, is a series in t with polynomial coefficients in u, and F 1 , . . . , F k are series in t only, independent of u. Brown's quadratic method applies as long as the degree in F (u) is 2 (for the linear case, see the kernel method in [START_REF] Bousquet-Mélou | Linear recurrences with constant coefficients: the multivariate case[END_REF][START_REF] Banderier | Generating functions for generating trees[END_REF]). Recently, it was understood how these equations could be solved in full generality [START_REF] Bousquet-Mlou | Planar maps and algebraic series: a generalization of the quadratic method[END_REF]. Moreover, the solution of any (well-founded) equation of the above type was shown to be algebraic. This provides two types of enumerative results:

-the proof that many map generating functions are algebraic: it now suffices to exhibit an equation of the form [START_REF] Bousquet-Mlou | Planar maps and algebraic series: a generalization of the quadratic method[END_REF], or to explain why such an equation exists,

-the solution of previously unsolved map problems (like the enumeration of hard-particle configurations on maps, which led to [START_REF] Bousquet-Mélou | Walks in the quarter plane: Kreweras' algebraic model[END_REF], or that of triangulations with high vertex degrees [START_REF] Bernardi | On triangulations with high vertex degree[END_REF]).

Matrix integrals.

In the late 70's, it was understood by a group of physicists that certain matrix integral techniques coming from quantum field theory could be used to attack enumerative problems on maps [START_REF] Brézin | Planar diagrams[END_REF][START_REF] Bessis | Quantum field theory techniques in graphical enumeration[END_REF]. This approach proved to be extremely efficient (even if it is usually not fully rigorous). The first step is fairly automatized, and consists in converting the description of maps into a certain integral. For instance, the relevant integral for the enumeration of 4-valent maps (maps in which all vertices have degree 4) is

Z(t, N ) = 2 N (N -1)/2 (2π) N 2 /2 dHe tr -H 2 /2 + tH 4 /N ,
where the integration space is that of hermitian matrices H of size N , equipped with the Lebesgue measure dH = dx kk k<ℓ dx kℓ dy kℓ with h kℓ = x kℓ + iy kℓ . As there is a classical bijection between 4-valent maps with n vertices and planar maps with n edges (Fig. 6(c)), we are still dealing with our reference problem: the enumeration of general planar maps. The connection between the above integral and maps is

G(t) = tE ′ (t) with E(t) = lim N →∞ 1 N 2 log Z(t, N ).
Other map problems lead to integrals involving several hermitian matrices [START_REF] Kazakov | Ising model on a dynamical planar random lattice: exact solution[END_REF]. We refer to [START_REF] Zvonkin | Matrix integrals and map enumeration: an accessible introduction[END_REF] for a neat explanation of the encoding of map problems by integrals, and to [START_REF] Francesco | 2D Quantum gravity, matrix models and graph combinatorics[END_REF][START_REF] Eynard | An introduction to random matrix theory[END_REF] (and references therein) for the evaluation of integrals.

3.5.3. Planar maps and trees. We finally come to a combinatorial explanation of the formula/equation for g n and G(t). Take a plane binary tree with n (inner) nodes, planted at a leaf, and add to every inner node a new distinguished child, called a bud. At each node, we have three choices for the position of the bud (Fig. 11(a)). The new tree, called budding tree, has now n buds and n + 2 leaves. Now start from the root and walk around the tree in counterclockwise order, paying attention to the sequence of buds and leaves you meet. Each time a bud is immediately followed by a leaf in this sequence, match them by forming a new edge (Fig. 11(b)) and then go on walking around the plane figure thus obtained. At the end, exactly two leaves remain unmatched. Match them together and orient this final edge in one of the two possible ways. Also, mark the face to the left of the matching edge that ends at the root-leaf.

Theorem 3.10 ([73]). The above correspondence is a bijection between pairs (T, ǫ)

where T is a budding tree having n inner nodes and ǫ ∈ {0, 1}, and 4-valent maps with n vertices and a marked face.

The value of ǫ tells how to orient the final matching edge. Schaeffer first used this bijection to explain combinatorially the formula [START_REF] Bousquet-Mélou | The degree distribution of bipartite planar maps: applications to the Ising model[END_REF]. Indeed, the number of budding trees with n inner nodes is clearly 3 n 2n n /(n + 1) (see ( 17)), while the number of 4-valent maps with n vertices and a marked face is (n + 2)g n . Eq. ( 19) follows.

Later, it was realized that this construction could also be used to explain the algebraicity of the series G(t) [START_REF] Bouttier | Census of planar maps: from the onematrix model solution to a combinatorial proof[END_REF]. Say that a budding tree is balanced if the rootleaf is not matched by a bud. Take such a tree, match all buds, and orient the final edge from the root-leaf to the other unmatched leaf. This gives a bijection between balanced budding trees and 4-valent maps. We thus have to count balanced trees, or, equivalently, the unbalanced ones. By re-rooting them at the bud that matches the root-leaf, one sees that they are in bijection with a node attached to three budding trees. This gives

G(t) = B(t) -tB(t) 3 ,
where

B(t) = 3t(1 + B(t)) 2
counts budding trees by (inner) nodes. The above construction involves taking a difference of N-algebraic series, which needs not be N-algebraic. We actually conjecture that the series G(t) is not N-algebraic (see Section 3.6.4).

There is little doubt that the above construction (once described in greater detail...) explains in a very satisfactory way both the simplicity of the formula giving g n and the algebraicity of G(t). Moreover, this is not an ad hoc, isolated magic trick: over the past ten years, it was realized that this construction is one in a family of constructions of the same type, which apply to numerous families of maps (Eulerian maps [START_REF] Schaeffer | Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees[END_REF], maps with prescribed vertex degrees [START_REF] Bouttier | Census of planar maps: from the onematrix model solution to a combinatorial proof[END_REF], constellations [START_REF] Bousquet-Mélou | Enumeration of planar constellations[END_REF], bipartite maps with prescribed degrees [START_REF] Bousquet-Mélou | The degree distribution of bipartite planar maps: applications to the Ising model[END_REF], maps with higher connectivity [START_REF] Poulalhon | Optimal coding and sampling of triangulations[END_REF][START_REF] Fusy | Dissections and trees: applications to optimal mesh encoding and random sampling[END_REF]). Definitely, these constructions reveal a lot about the combinatorial nature of planar maps.

To conclude this section, let us mention that a different combinatorial construction for general planar maps, discovered in the early 80's [START_REF] Cori | Planar maps are well labeled trees[END_REF], has recently been simplified [START_REF] Chassaing | Random planar lattices and integrated superBrownian excursion[END_REF] and adapted to other families of maps [START_REF] Del Lungo | Left ternary trees and non-separable rooted planar maps[END_REF][START_REF] Jacquard | A bijective census of nonseparable planar maps[END_REF][START_REF] Bouttier | Statistics of planar graphs viewed from a vertex: a study via labeled trees[END_REF][START_REF] Bouttier | Planar maps as labeled mobiles[END_REF]. It is a bit less easy to handle than the one based on trees with buds, but it allows one to keep track of the distances between some vertices of the map. This has led to remarkable connections with a random probability distribution called the Integrated SuperBrownian Excursion [START_REF] Chassaing | Random planar lattices and integrated superBrownian excursion[END_REF]. A third type of construction has emerged even more recently [START_REF] Bernardi | Bijective counting of Kreweras walks and loopless triangulations[END_REF] for 2-connected triangulations, but no ones knows at the moment whether it will remain isolated or is just the tip of another iceberg.

3.6. Algebraic series: some questions. We begin with three simple classes of objects that have an algebraic GF, but for reasons that remain mysterious. We then discuss a possible criterion (or necessary condition) for N-algebraicity, and finally the algebraicity of certain hypergeometric series.

3.6.1. Kreweras' words and walks on the quarter plane. Let L be the set of words u on the alphabet {a, b, c} such that for every prefix v of u, |v| a ≥ |v| b and |v| a ≥ |v| c . These words encode certain walks on the plane: these walks start at (0, 0), are made of three types of steps, a = (1, 1), b = (-1, 0) and c = (0, -1), and never leave the first quadrant of the plane, defined by x, y ≥ 0. The pumping lemma [START_REF] Hopcroft | Formal languages and their relation to automata[END_REF]Thm. 4.7], applied to the word a n b n c n , shows that the language L is not context-free. However, its generating function is algebraic. More precisely, let us denote by ℓ i,j (n) the number of words u of L of length n such that |u| a -|u| b = i and |u| a -|u| c = j. They correspond to walks of length n ending at position (i, j). Then the associated three-variable generating function is

L(u, v; t) = i,j,n ℓ i,j (n)u i v j t n = (1/W -ū) √ 1 -uW 2 + (1/W -v) √ 1 -vW 2 uv -t(u + v + u 2 v 2 ) - 1 uvt
where ū = 1/u, v = 1/v and W ≡ W (t) is the unique power series in t satisfying W = t(2 + W 3 ). Moreover, the number of walks ending at (i, 0) is remarkably simple:

ℓ i,0 (3n + 2i) = 4 n (2i + 1) (n + i + 1)(2n + 2i + 1) 2i i 3n + 2i
n .

The latter formula was proved in 1965 by Kreweras, in a fairly complicated way [START_REF] Kreweras | Sur une classe de problèmes liés au treillis des partitions d'entiers[END_REF]. This rather mysterious result has attracted the attention of several combinatorialists since its publication [START_REF] Bousquet-Mélou | Walks in the quarter plane: Kreweras' algebraic model[END_REF][START_REF] Gessel | A probabilistic method for lattice path enumeration[END_REF][START_REF] Niederhausen | The ballot problem with three candidates[END_REF]. The first combinatorial explanation of the above formula (in the case i = 0) has just been found by Bernardi [START_REF] Bernardi | Bijective counting of Kreweras walks and loopless triangulations[END_REF].

Walks in the quarter plane do not always have an algebraic GF: for instance, the number of square lattice walks (with North, South, East and West steps) of size 2n that start and end at (0, 0) and remain in the quarter plane is 1 (2n + 1)(2n + 4) 2n + 2 n + 1 2 ∼ 4 2n+1 πn 3 , and this asymptotic behaviour prevents the corresponding generating function from being algebraic (see ( 16)). The above formula is easily proved by looking at the projections of the walk onto the horizontal and vertical axes.

3.6.2. Walks on the slit plane. Take now any finite set of steps S ⊂ Z × {-1, 0, 1} (we say that these steps have small height variations). Let s i,j (n) be the number of walks of length n that start from the origin, consist of steps of S, never return to the non-positive horizontal axis {(-k, 0), k ≥ 0}, and end at (i, j). Let S(u, v; t) be the associated generating function:

S(u, v; t) = i,j∈Z,n≥0 s i,j (n)u i v j t n .
Then this series is always algebraic, as well as the series S i,j (t) := n s i,j (n)t n that counts walks ending at (i, j) [START_REF] Bousquet-Mélou | Walks on the slit plane: other approaches[END_REF][START_REF] Bousquet-Mélou | Walks on the slit plane[END_REF]. For instance, when S is formed of the usual square lattice steps (North, South, West and East), then

S(u, v; t) = 1 -2t(1 + ū) + √ 1 -4t 1/2 1 + 2t(1 -ū) + √ 1 + 4t 1/2 1 -t(u + ū + v + v)
with ū = 1/u and v = 1/v. Moreover, the number of walks ending at certain specific points is remarkably simple. For instance: where C n = 2n n /(n+1) is the nth Catalan number, which counts binary trees [START_REF] Bousquet-Mélou | Linear recurrences with constant coefficients: the multivariate case[END_REF], Dyck words, and numerous other combinatorial objects [START_REF] Stanley | Enumerative combinatorics[END_REF]Ch. 6]. The first of these three identities has been proved combinatorially [START_REF] Barcucci | A bijection for some paths on the slit plane[END_REF]. The others still defeat our understanding.

3.6.3. Embedded binary trees. We consider again the complete binary trees met at the beginning of Section 3.2. Let us associate with each (inner) node of such a tree a label, equal to the difference between the number of right steps and the number of left steps one takes when going from the root to the node. In other words, the label of the node is its abscissa in the natural integer embedding of the tree (Fig. 12). Let S j ≡ S j (t, u) be the generating function of binary trees counted by the number of nodes (variable t) and the number of nodes at abscissa j (variable u). Then for all j ∈ Z, this series is algebraic of degree (at most) 8 (while S j (t, 1) is quadratic) [START_REF] Bousquet-Mélou | Limit results for embedded trees. Applications to the integrated super-Brownian excursion[END_REF]. Moreover, for j ≥ 0, S j = T

(1 + µZ j )(1 + µZ j+5 ) (1 + µZ j+2 )(1 + µZ j+3 ) ,

where

T = 1 + tT 2 , Z = t 1 + Z 2 2 (1 -Z + Z 2 )
, and µ ≡ µ(t, u) is the unique formal power series in t satisfying µ = (u -1)

Z(1 + µZ) 2 (1 + µZ 2 )(1 + µZ 6 ) (1 + Z) 2 (1 + Z + Z 2 )(1 -Z) 3 (1 -µ 2 Z 5 )
.

Why is that so? This algebraicity property holds as well for other families of labelled trees [START_REF] Bousquet-Mélou | Limit results for embedded trees. Applications to the integrated super-Brownian excursion[END_REF][START_REF] Bouttier | Random trees between two walls: exact partition function[END_REF]. From these series, one can derive certain limit results on the distribution of the number of nodes at abscissa ⌊λn 1/4 ⌋ in a random tree with n nodes [START_REF] Bousquet-Mélou | Limit results for embedded trees. Applications to the integrated super-Brownian excursion[END_REF]. These results provide some information about the law of the integrated super-Brownian excursion [START_REF] Bousquet-Mélou | Limit results for embedded trees. Applications to the integrated super-Brownian excursion[END_REF][START_REF] Bousquet-Mlou | The density of the ISE and local limit laws for embedded trees[END_REF].

3.6.4. N-algebraicity. N-algebraic series have been defined in Section 3.2 in terms of positive proper algebraic systems. The author has been unable to find in the literature a criterion, or even a necessary condition for an algebraic series with coefficients in N to be N-algebraic. Nor even an algebraic series with coefficients in N that would not be N-algebraic (together with a proof of this statement...).

A partial answer could be provided by the study of the possible asymptotic behaviour of coefficients of N-algebraic series. It is very likely that not all behaviours of the form (16) are possible. An important result in this direction states that, if a proper positive system ( 18) is strongly connected, the nth coefficient of, say, A 1 follows the general pattern ( 16), but with d = -3/2 [43, Thm. VII.7]. The system is strongly connected if, roughly speaking, the expression of every series A i involves (possibly after a few iterations of the system) every other series A j . For instance, the system defining the walks ending at 0 in Section 3.4.1 reads M 0 = t 2 (1 + M 0 ) 2 and W 0 = M 0 (2 + W 0 ). This system is not strongly connected, as M 0 does not involve W 0 . Accordingly, the number of 2n-step walks returning to 0 is 2n n ∼ κ4 n n -1/2 . If one can rule out the possibility that d = -5/2 for N-algebraic series, then this will prove that most map generating functions are not N-algebraic (see the examples in Section 3.5).

3.6.5. Some algebraic hypergeometric series. Consider the following series: Can we describe all algebraic cases? Well, one can easily obtain some necessary conditions on the sequences a and b by looking at the asymptotics of f n . First, an algebraic power series has a finite, positive radius of convergence (unless it is a polynomial). This, combined with Stirling's formula, gives at once Here are some algebraic instances: The degree of these series is rather big: 12 [resp. 30] for the first [second] series above. This theorem provides a collection of algebraic series with nice integer coefficients: are these series N-algebraic? Do they count some interesting objects?

F (t) =
a 1 + • • • + a d = b 1 + • • • + b e . (23
f n = (

Figure 1 .

 1 Figure 1. (a) A weighted digraph. The default value of the weight is 1. (b) A deterministic automaton on the alphabet {a, b, c, ā, b, c}. The initial state is 1 and and the final states are 2 and 3.

2 .Figure 2 .

 22 Figure 2. A column-convex polyomino, with the numbering and encoding of the cells.

Figure 3 .

 3 Figure 3. Integer points in a polyhedral cone.

Figure 4 .

 4 Figure 4. Compact-source directed animals on the square and triangular lattices.

Figure 5 .

 5 Figure 5. The 5 Dyck paths of length 6 and the 4 3 = 64 points lying below.

Figure 6 .

 6 Figure 6. (a) A plane tree. (b) A rooted planar map. (c) The corresponding 4-valent map (thick lines).

Example 3 . 6 (

 36 Dyck words). Consider the grammar G having only one symbol, S, alphabet {a, b}, and rules S → ab + abS + aSb + aSbS (which is short for S → ab, S → abS, S → aSb, S → aSbS). It is easy to see that L(G) is the set of non-empty words u on {a, b} such that |u| a = |u| b and for every prefix v of u, |v| a ≥ |v| b . These words, called Dyck words, provide a simple encoding of the Dyck paths met in Example 2.9.

Figure 7 .

 7 Figure 7. (a) A directed animal and the associated pyramid. (b) A half-pyramid.

Figure 8 .

 8 Figure 8. Decomposition of pyramids (top) and half-pyramids (bottom).

Figure 9 .

 9 Figure 9. Decomposition of heaps having a compact basis.

Figure 10 .

 10 Figure 10. Tutte's decomposition of rooted planar maps.

Figure 11 .

 11 Figure 11. (a). A budding tree. (b) An intermediate step in the matching procedure. (c) The resulting 4-valent map, with its marked face.

s 1 ,

 1 0 (2n + 1) = C 2n+1 , s 0,1 (2n + 1) = 4 n C n , s -1,1 (2n) = C 2n ,

Figure 12 .

 12 Figure 12. The integer embedding of a binary tree.

  j n)! t n ,where a 1 , . . . , a d , b 1 , . . . , b e are positive integers. This series is algebraic for some values of the a i 's and b j 's, as shown by the casen≥0 (2n)! n! 2 t n = 1 √ 1 -4t .

  )Moreover, by looking at the dominant term in the asymptotic behaviour of f n , and comparing with[START_REF] Bousquet-Mélou | Lecture hall partitions[END_REF], one obtains that either e = d, or e = d + 1. The case d = e only gives the trivial solution F (t) = 1/(1 -t), and the complete answer to this problem is as follows[START_REF] Beukers | Monodromy for the hypergeometric function nFn-1[END_REF][START_REF] Rodriguez-Villegas | Integral ratios of factorials and algebraic hypergeometric functions[END_REF]: Theorem 3.11. Assume (23) holds, and F (t) = 1/(1 -t). The series F (t) is algebraic if and only if f n ∈ N for all n and e = d + 1.

  6n)!(n)! (3n)!(2n)! 2 , f n = (10n)!(n)! (5n)!(4n)!(2n)! , f n = (20n)!(n)! (10n)!(7n)!(4n)!.

  it is the union of two simplicial cones C 1 and C 2 , obtained by intersecting C with the half-spaces α 1 ≥ α 3 and α 1 ≤ α 3 , respectively. Let E 1 [resp. E 2 ] denote the set of integer points of C 1 [resp. C 2 ].

The fastest way to obtain the generating function E(t) is to write

  23328 t 6 A 4 + 27 t 4 (91 -2088 t) A 3 + t 2 86 -3951 t + 46710 t 2 + 3456 t 3 A 2

+ 1 -69 t + 1598 t 2 -11743 t 3 -14544 t 4 A-1+66 t-1495 t 2 +11485 t 3 +128 t 4 .

A P -partition is usually defined as an order-reversing map from k to N [74, Section 4.5]. Both notions are of course completely equivalent.

Mireille Bousquet-Mélou

Acknowledgements. The parts of this survey that do not deal exactly with the enumeration of combinatorial objects have often be influenced by discussions with some of my colleagues, including Frédérique Bassino, Henri Cohen, Philippe Flajolet, François Loeser, Géraud Sénizergues. Still, they should not be hold responsible for any of the flaws of this paper...