Mireille Bousquet-Mélou
email: mireille.bousquet@labri.fr

Guillaume Chapuy
email: guillaume.chapuy@liafa.jussieu.fr

Louis-François Préville-Ratelle

TAMARI LATTICES AND PARKING FUNCTIONS: PROOF OF A CONJECTURE OF F. BERGERON

Keywords: 2000 Mathematics Subject Classification, 05A15, Key words and phrases, Enumeration -Lattice paths -Tamari lattices -Parking functions

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0, 0), ending at (mn, n), and never going below the line {x = my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T (m) n , which generalizes the usual Tamari lattice Tn obtained when m = 1. This lattice was introduced by F. Bergeron in connection with the study of coinvariant spaces. He conjectured several intriguing formulas dealing with the enumeration of intervals in this lattice. One of them states that the number of intervals in

This conjecture was proved recently, but in a non-bijective way, while its form strongly suggests a connection with plane trees.

Here, we prove another conjecture of Bergeron, which deals with the number of labelled intervals. An interval [P, Q] of T (m) n is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. We prove that the number of labelled intervals in

The form of these numbers suggests a connection with parking functions, but our proof is nonbijective. It is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. This equation involves a derivative and a divided difference, taken with respect to two additional variables. Solving this equation is the hardest part of the paper.

Finding a bijective proof remains an open problem.

Introduction and main results

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0, 0), ending at (mn, n), and never going below the line {x = my}. It is well-known that there are 1 mn + 1 (m + 1)n n such paths [START_REF] Dvoretzky | A problem of arrangements[END_REF], and that they are in bijection with (m+1)-ary trees with n inner nodes. Bergeron recently defined on the set T (m) n of m-ballot paths of size n a partial order. It is convenient to describe it via the associated covering relation, exemplified in Figure 1.

Definition 1. Let P and Q be two m-ballot paths of size n. Then Q covers P if there exists in P an east step a, followed by a north step b, such that Q is obtained from P by swapping a and S, where S is the shortest factor of P that begins with b and is a (translated) m-ballot path. It was shown in [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF] that this order endows T (m) n with a lattice structure, which is called the m-Tamari lattice of size n. When m = 1, it coincides with the classical Tamari lattice [START_REF] Bernardi | Intervals in Catalan lattices and realizers of triangulations[END_REF][START_REF] Friedman | Problèmes d'associativité: Une structure de treillis finis induite par une loi demi-associative[END_REF][START_REF] Huang | Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law[END_REF][START_REF] Knuth | The art of computer programming[END_REF]. Figure 2 shows two of the lattices T (m) n . These lattices are conjectured to have deep connections with the ring DR 3,n of polynomials in three sets of variables X = {x 1 , . . . , x n }, Y = {y 1 , . . . , y n }, Z = {z 1 , . . . , z n }, quotiented by the ideal generated by (trivariate) diagonal invariants. By diagonal invariants, one means constant term free polynomials that are invariant under the following action of the symmetric group S n : for σ ∈ S n and f a polynomial, σ(f (X, Y, Z)) = f (x σ(1) , . . . , x σ(n) , y σ(1) , . . . , y σ(n) , z σ(1) , . . . , z σ(n) ).

We refer to [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF][START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF] for details about these conjectures, which have striking analogies with the much studied case of two sets of variables [START_REF] Haglund | The q,t-Catalan numbers and the space of diagonal harmonics[END_REF][START_REF] Haglund | A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants[END_REF][START_REF] Haiman | Vanishing theorems and character formulas for the Hilbert scheme of points in the plane[END_REF][START_REF] Haiman | Conjectures on the quotient ring by diagonal invariants[END_REF][START_REF] Loehr | Multivariate analogues of Catalan numbers, parking functions, and their extensions[END_REF]. In particular, it seems that the role played by ballot paths for two sets of variables (see, e.g., [START_REF] Garsia | A proof of the q, t-Catalan positivity conjecture[END_REF][START_REF] Haglund | Conjectured statistics for the q, t-Catalan numbers[END_REF][START_REF] Haglund | A conjectured combinatorial formula for the Hilbert series for diagonal harmonics[END_REF]) is played for three sets of variables by intervals of ballot paths in the Tamari order. form a lattice that is isomorphic to T (see Proposition 4).

For instance, it is conjectured in [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF] that the the dimension of a certain polynomial ring related to DR 3,n , but involving one more parameter m, is m + 1 n(mn + 1)

(m + 1) 2 n + m n -1 , (1)
and that this number counts intervals in the Tamari lattice T (m) n

. The latter statement was proved in [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF] (the special case m = 1 had been proved earlier [START_REF] Chapoton | Sur le nombre d'intervalles dans les treillis de Tamari[END_REF]). The former one is presumably extremely difficult, given the complexity of the corresponding result for two sets of variables [START_REF] Haiman | Vanishing theorems and character formulas for the Hilbert scheme of points in the plane[END_REF]. The dimension related result was observed earlier for small values of n by Haiman [START_REF] Haiman | Conjectures on the quotient ring by diagonal invariants[END_REF] in the case m = 1. The aim of this paper is to prove another conjecture of [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF], dealing with labelled Tamari intervals. Let us say that an m-ballot path of size n is labelled if the north steps are labelled from 1 to n, in such a way the labels increase along any sequence of consecutive north steps (Figure 3). The number of labelled m-ballot paths of size n is

(mn + 1) n-1 .
Indeed, these paths are in bijection with (1, m, . . . , m)-parking functions of size n, in the sense of [START_REF] Stanley | A polytope related to empirical distributions, plane trees, parking functions, and the associahedron[END_REF][START_REF] Yan | Generalized parking functions, tree inversions, and multicolored graphs[END_REF]: the function f associated with a path Q satisfies f (i) = k if the north step of Q labelled i lies at abscissa k -1. Now, we say that an m-Tamari interval [P, Q] is labelled, if the upper path Q is labelled. It is conjectured in [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF] that the number of labelled m-Tamari intervals of size n is (m + 1) n (mn + 1) n-2 , (2) and this is what we prove in this paper. It is also conjectured in [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF] that this number is the dimension of a certain polynomial ring generalizing DR 3,n (which corresponds to the case m = 1).

Our proof is, at first blush, analogous to the proof of (1) presented in [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF]: we introduce a generating function F (m) (t; x, y) counting labelled intervals according to three parameters; we describe a recursive construction of intervals and translate it into a functional equation defining F (m) (t; x, y) ; we finally solve this equation, after having partially guessed its solution. However, the labelled case turns out to be significantly more difficult than the unlabelled one. It is not hard to explain the origin of this increased difficulty: for m fixed, the generating function of the numbers ( 1) is an algebraic series, and can be expressed in terms of the series Z ≡ Z(t) satisfying

Z = t (1 -Z) m(m+2) .
There exists a wealth of tools, both modern or ancient, to handle algebraic series (e.g., factorisation, elimination, Gröbner bases, rational parametrizations when the genus is zero, efficient guessing techniques, all tools made effective in Maple and its packages, like algcurves and gfun). Such tools play a key role in the proof of [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF]. But the generating function of the numbers (2) is related to the series Z satisfying Z = te m(m+1)Z , which lives in the far less polished world of differentially algebraic series, for which much fewer tools are available. Our main result is actually more general that [START_REF] Bernardi | Intervals in Catalan lattices and realizers of triangulations[END_REF]. Indeed, we refine the enumeration by taking into account two more parameters, which we now define. A contact of an m-ballot path P is a vertex of P lying on the line {x = my}. The initial rise of a ballot path Q is the length of the initial run of up steps in Q. A contact of a Tamari interval [P, Q] is a contact of the lower path P , while the initial rise of this interval is the initial rise of the upper path Q (see Figure 4). We consider the exponential generating function of labelled m-Tamari intervals, counted by the size, the number of contacts and the initial rise. More precisely,

F (m) (t; x, y) = I=[P,Q] t |I| |I|! x c(P ) y r(Q) , (3)
where the sum runs over all labelled m-Tamari intervals I, |I| denotes the size of I (that is, the number of up steps in P ), c(P ) the number of contacts of P and r(Q) the initial rise of Q. The main result of this paper is a complicated closed form expression of F (m) (t; x, y), which becomes simple when y = 1. In particular, extracting the n th coefficient in F (m) (t; 1, 1) proves Bergeron's conjecture (2).

Theorem 2. Let F (m) (t; x, y) ≡ F (t; x, y) be the exponential generating function of labelled m-Tamari intervals, defined by (3). Let z and u be two indeterminates, and write

t = ze -m(m+1)z and x = (1 + u)e -mzu . (4)
Then F (t; x, 1) becomes a series in z with polynomial coefficients in u, and this series has a simple expression:

F (t; x, 1) = (1 + u)e (m+1)z-(m-1)zu 1 + 1 -e mzu u . (5)
In particular,

F (t; 1, 1) = (1 -mz)e (m+1)z ,
and the number of labelled m-Tamari intervals of size n is

n![t n ]F (t; 1, 1) = (m + 1) n (mn + 1) n-2 .
Our expression of F (m) (t; x, y) is given in Theorem 17. When m = 1, it takes a reasonably simple form, which we now present (the case m = 2 is also detailed at the end of the paper). Given a Laurent polynomial P (u) in u, we denote by [u ≥ ]P (u) the non-negative part of P (u) in u, defined by

[u ≥ ]P (u) = i≥0 P i u i if P (u) = i∈Z P i u i .
The definition is then extended by linearity to power series whose coefficients are Laurent polynomials in u.

Theorem 3. Let F (1) (t; x, y) ≡ F (t; x, y) be the generating function of labelled 1-Tamari intervals, defined by (3). Let z and u be two indeterminates, and set

t = ze -2z and x = (1 + u)e -zu . (6)
Then F (t; x, y) becomes a formal power series in z with polynomial coefficients in u and y, which is given by F (t; x, y) = (1 + u) e 2yz [u ≥ ] e zu(y-1)+zy ū -ūe z ū(y-1)+zyu ,

with ū = 1/u. Equivalently,

F (t; x, y) (1 + u)e 2yz = 0≤i≤j u j-i z i+j y i (y -1) j i!j! - 0≤j<i u i-j-1 z i+j y i (y -1) j i!j! .
It is easily seen that the case y = 1 of the above formula reduces to the case m = 1 of (5). When x = 1, that is, u = 0, the double sums in the expression of F (t; x, y) reduce to simple sums, and the generating function of labelled Tamari intervals is expressed in terms of Bessel functions:

F (t; 1, y) e 2yz = i≥0 z 2i y i (y -1) i i! 2 - j≥0 z 2j+1 y j+1 (y -1) j (j + 1)!j! .
The outline of the paper goes as follows: in Section 2 we derive from a recursive description of labelled Tamari intervals a functional equation satisfied by their generating function. This equation involves a derivative (with respect to y) and a divided difference (with respect to x). We present in Section 3 the principle of the proof, and exemplify it on the case m = 1, thus obtaining Theorem 3 above. Section 4 deals with the general case, and proves Theorem 2.

We conclude this introduction with some notation and a few definitions. Let K be a commutative ring and t an indeterminate. We denote by K[t] (resp. K[[t]]) the ring of polynomials (resp. formal power series) in t with coefficients in K. If K is a field, then K(t) denotes the field of rational functions in t, and K((t)) the field of Laurent series in t (that is, series of the form n≥n0 a n t n , with n 0 ∈ Z). These notations are generalized to polynomials, fractions and series in several indeterminates. We denote by bars the reciprocals of variables: for instance, ū = 1/u, so that K[u, ū] is the ring of Laurent polynomials in u with coefficients in K. The coefficient of u n in a Laurent series F (u) is denoted by [u n ]F (x).

We have defined the non-negative part of a Laurent polynomial P (u) above Theorem 3. We define similarly the positive part of P (u), denoted by [u > ]P (u).

The series we handle in this paper involve a main variable t, or z after the change of variables (4), and then additional variables x and y. So they should in principle be denoted F (t; x, y), but we often omit the variable t (or z), to avoid heavy notation and enhance role of the additional variables x and y.

A functional equation

The aim of this section is to describe a recursive decomposition of labelled m-Tamari intervals, and to translate it into a functional equation satisfied by the associated generating function. Our description of the decomposition is self-contained, but we refer to [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF] for several proofs and details.

Recursive decomposition of Tamari intervals

We start by modifying the appearance of 1-ballot paths. We apply a 45 degree rotation to 1ballot paths to transform them into Dyck paths. A Dyck path of size n consists of steps u = (1, 1) (up steps) and steps d = (1, -1) (down steps), starts at (0, 0), ends at (2n, 0) and never goes below the x-axis. We say that an up step has rank i if it is the i th up step of the path. We often represent Dyck paths by words on the alphabet {u, d}.

Consider now an m-ballot path of size n, and replace each north step by a sequence of m north steps. This gives a 1-ballot path of size mn, and thus, after a rotation, a Dyck path. In this path, for each i ∈ 0, n -1 , the up steps of ranks mi + 1, . . . , m(i + 1) are consecutive. We call the Dyck paths satisfying this property m-Dyck paths, and say that the up steps of ranks mi + 1, . . . , m(i + 1) form a block. Clearly, m-Dyck paths of size mn (i.e., having n blocks) are in one-to-one correspondence with m-ballot paths of size n. We often denote by T n , rather than

I 1 Q 1 P ℓ 1 P r 1 I 2 Q 2 P 2 P I Q Figure 5
. The recursive construction of Tamari intervals.

T (1)
n , the usual Tamari lattice of size n. Similarly, the intervals of this lattice are called Tamari intervals, rather than 1-Tamari intervals. As proved in [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF], the transformation of m-ballot paths into m-Dyck paths maps T .

We now describe a recursive decomposition of (unlabelled) Tamari intervals, again borrowed from [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF]. Thanks to the embedding of T (m) n into T nm , it will also enable us to decompose m-Tamari intervals, for any value of m, in the next subsection.

A Tamari interval I = [P, Q] is pointed if its lower path P has a distinguished contact (we refer to the introduction for the definition of contacts). Such a contact splits P into two Dyck paths P ℓ and P r , respectively located to the left and to the right of the contact. The pointed interval I is proper is P ℓ is not empty, i.e., if the distinguished contact is not (0, 0). We often use the notation I = [P ℓ P r , Q] to denote a pointed Tamari interval.

Proposition 5. Let I 1 = [P ℓ 1 P r 1 , Q 1 ]
be a pointed Tamari interval, and let I 2 = [P 2 , Q 2 ] be a Tamari interval. Construct the Dyck paths P = uP ℓ 1 dP r 1 P 2 and Q = uQ 1 dQ 2 as shown in Figure 5. Then I = [P, Q] is a Tamari interval. Moreover, the mapping (I 1 , I 2 ) → I is a bijection between pairs (I 1 , I 2 ) formed of a pointed Tamari interval and a Tamari interval, and Tamari intervals I of positive size. Note that I 1 is proper if and only if the initial rise of P is not 1. Remarks 1. To recover P ℓ 1 , P r 1 , Q 1 , P 2 and Q 2 from P and Q, one proceeds as follows: Q 2 is the part of Q that follows the first return of Q to the x-axis; this defines Q 1 unambiguously. The path P 2 is the suffix of P having the same size as Q 2 . This defines P 1 := uP ℓ 1 dP r 1 unambiguously. Finally P r 1 is the part of P 1 that follows the first return of P 1 to the x-axis, and this defines P ℓ 1 unambiguously.

2. This proposition is obtained by combining Proposition 5 in [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF] and the case m = 1 of Lemma 9 in [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF]. With the notation (P ′ ; p 1 ) and (Q ′ , q 1 ) used therein, the above paths P 2 and Q 2 are respectively the parts of P ′ and Q ′ that lie to the right of q 1 , while P ℓ 1 P r 1 and Q 1 are the parts of P ′ and Q ′ that lie to the left of q 1 . The pointed vertex p 1 is the endpoint of P ℓ 1 . Proposition 5 in [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF] guarantees that, if P Q in the Tamari order, then P ℓ 1 P r 1 Q 1 and P 2 Q 2 . 3. One can keep track of several parameters in the construction of Proposition 5. For instance, the initial rise of Q equals the initial rise of Q 1 plus one. Also, the number of contacts of P is

c(P ) = c(P r 1 ) + c(P 2 ). (8)
2.2. From the decomposition to a functional equation

We will now establish the following functional equation.

Proposition 6. For m ≥ 1, let F (m) (t; x, y) ≡ F (x, y) be the exponential generating function of labelled m-Tamari intervals, defined by (3). Then F (x, 0) = x and ∂F ∂y (x, y) = tx (F (x, 1) • ∆) (m) (F (x, y)), ( 9
)
where ∆ is the following divided difference operator

∆S(x) = S(x) -S(1) x -1 ,
and the power m means that the operator G(x, y) → F (x, 1) • ∆G(x, y) is applied m times.

Proof. We constantly use the inclusion T (m) n ⊂ T nm given by Proposition 4. That is, we identify elements of T (m) n with m-Dyck paths having n blocks. It is obvious that F (x, 0) = x, since the interval of size 0 is the only interval of initial rise 0, and has one contact. The functional equation ( 9) relies on the decomposition of Tamari intervals described in Proposition 5.

We will actually apply this decomposition to a slight generalization of m-Tamari intervals. For k ≥ 0, a k-augmented m-Dyck path is a Dyck path of size k + mn for some integer n, where the first k steps are up steps, and all the other up steps can be partitioned into blocks of m consecutive up steps. The k first steps are not considered to be part of a block, even if k is a multiple of m. We denote by T (m,k) the set of k-augmented m-Dyck paths.

A Tamari interval I = [P, Q] is a k-augmented m-Tamari interval if both P and Q belong to T (m,k) . Assume that P and Q contain n blocks. Then I is labelled if the blocks of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive blocks. Note that labelled 0-augmented m-Tamari intervals coincide with labelled m-Tamari intervals. Generalizing (3), we denote by F k (t; x, y) ≡ F k (x, y) the exponential generating function of labelled k-augmented m-Tamari intervals, counted by the number of blocks (variable t), the number of non-initial contacts (that is, contacts distinct from (0, 0) -variable x), and the number of blocks contained in the first ascent (variable y).

In what follows, we first obtain an expression of F k (x, y) in terms of F (x, y):

F k (x, y) = F (x, y)/x if k = 0, (F (x, 1) • ∆) (k) F (x, y) otherwise. ( 10
)
We then relate m-augmented m-Tamari intervals to m-Tamari intervals, proving that

txF m (x, y) = ∂F ∂y (x, y). (11)
This identity, combined with the case k = m of (10), gives [START_REF] Haglund | Conjectured statistics for the q, t-Catalan numbers[END_REF].

We thus need to prove [START_REF] Haglund | The q,t-Catalan numbers and the space of diagonal harmonics[END_REF] and [START_REF] Haglund | A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants[END_REF]. The case k = 0 of (10) is clear, since 0-augmented m-Tamari intervals are just m-Tamari intervals. The factor x arises from the fact that F 0 (x, y) only keeps track of non-initial contacts, while F (x, y) counts all of them.

Let us now address the case k ≥ 1 of (10). Let I = [P, Q] be a labelled k-augmented m-Tamari interval. By Proposition 5, one can decompose I into a pair (I 1 , I 2 ) of Tamari intervals, with 5). Since the up steps of P 2 and Q 2 are not in the first ascent of P and Q, the paths P 2 and Q 2 are actually m-Dyck paths, so that I 2 is an m-Tamari interval. Similarly, I 1 is a pointed (k -1)-augmented m-Tamari interval, which is proper if k > 1 (Proposition 5). The blocks of Q 1 and Q 2 inherit a labelling from Q. We normalise these labellings in the usual way: if P 1 has n 1 blocks and Q 2 has n 2 blocks, we relabel the blocks of Q 1 (resp. Q 2 ) with 1, . . . , n 1 (resp. 1, . . . , n 2 ) while preserving the relative order of the labels occurring in Q

I 1 = [P ℓ 1 P r 1 , Q 1 ] and I 2 = [P 2 , Q 2 ] (see Figure
1 (resp. Q 2 ).
Conversely, consider a pair (I 1 , I 2 ), where [START_REF] Garsia | A proof of the q, t-Catalan positivity conjecture[END_REF], the number of non-initial contacts in I is the number of non-initial contacts in P r 1 , plus the number of contacts in P 2 . The number of blocks in the first ascent of Q is the number of blocks in the first ascent of Q 1 .

I 1 = [P ℓ 1 P r 1 , Q 1 ] is a labelled pointed (k -1)- augmented m-Tamari interval and I 2 is a labelled m-Tamari interval. If k > 1, assume moreover that I 1 is proper. If Q 1 and Q 2 have respectively n 1 and n 2 blocks, one can reconstruct from (I 1 , I 2 ) exactly (n1+n2)! n1!n2! different labelled k-augmented m-Tamari intervals I = [P, Q], having n 1 + n 2 blocks. By
The exponential generating function of labelled m-Tamari intervals I 2 , counted by the size and the number of contacts, is

F (x, 1). Let F • k-1 (t; x, y) ≡ F • k-1 (x, y) be the exponential generating function of labelled proper pointed (k -1)-augmented m-Tamari intervals I 1 = [P ℓ 1 P r 1 , Q 1 ]
, counted by the size, the number of non-initial contacts in P r 1 and the number of blocks in the first ascent of Q 1 . Note that the generating function of labelled non-proper pointed 0-augmented m-Tamari intervals is simply F 0 (x, y). The above construction then implies that

F k (x, y) = F (x, 1) F • k-1 (x, y) + F 0 (x, y)½ k=1 . (12)
We claim that

F • k-1 (x, y) = ∆F k-1 (x, y). (13)
It follows from (12-13) that

F k (x, y) = F (x, 1) • ∆ xF 0 (x, y) if k = 1, F (x, 1) • ∆F k-1 (x, y) otherwise.
Now a simple induction on k > 0, combined with the case k = 0 of (10) and, proves the case k > 0 of [START_REF] Haglund | The q,t-Catalan numbers and the space of diagonal harmonics[END_REF]. So [START_REF] Haglund | The q,t-Catalan numbers and the space of diagonal harmonics[END_REF] will be proved if we establish [START_REF] Haiman | Vanishing theorems and character formulas for the Hilbert scheme of points in the plane[END_REF]. Write

F k-1 (x, y) = i≥0 F k-1,i (y)x i ,
so that F k-1,i (y) counts labelled (k -1)-augmented m-Tamari intervals having i non-initial contacts. By pointing a non-initial contact, such an interval gives rise to i labelled proper pointed (k -1)-augmented m-Tamari intervals [P ℓ 1 P r 1 , Q 1 ], having respectively 0, 1, . . . , i -1 non-initial contacts in P r 1 . Hence

F • k-1 (x, y) = i≥0 F k-1,i (y) 1 + x + • • • + x i-1 , = i≥0 F k-1,i (y) x i -1 x -1 = F k-1 (x, y) -F k-1 (1, y) x -1 = ∆F k-1 (x, y).
This coincides with [START_REF] Haiman | Vanishing theorems and character formulas for the Hilbert scheme of points in the plane[END_REF].

We finally want to prove [START_REF] Haglund | A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants[END_REF], and this will complete the proof of Proposition 6. A labelled m-augmented m-Tamari interval I = [P, Q] having n -1 blocks gets in F m (x, y) a weight

t n-1 (n -1)! x c(P )-1 y r(Q)-1 = n tx t n n! x c(P ) y r(Q)-1 ,
where r(Q) is the initial rise of Q, divided by m. Let us interpret the factor n as the choice of a label i ∈ 1, n assigned to the first m steps of P , while the labels of the blocks, which were 1, . . . , n -1, are redistributed so as to avoid i. The above identity shows that txF m (x, y) counts (by the number of blocks, the number of contacts, and the initial rise minus one), m-Tamari intervals [P, Q] in which the blocks are labelled in such a way the labels increase along sequences of consecutive blocks, except that the first block of the first ascent may have a larger label than the second block of the first ascent. Such intervals are obtained from usual labelled m-Tamari intervals by choosing a block in the first ascent and exchanging its label with the label of the very first block. In terms of power series, choosing a block of the first ascent boils down to differentiating with respect to y (this also decreases by 1 the exponent of y), and we thus obtain [START_REF] Haglund | A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants[END_REF].

3. Principle of the proof, and the case m = 1

Principle of the proof

Let us consider the functional equation ( 9), together with the initial condition F (t; x, 0) = x. Perform the change of variables (4), and denote G(z; u, y) ≡ G(u, y) = F (t; x, y). Then G(u, y) is a series in z with coefficients in Q[u, y], satisfying

∂G ∂y (u, y) = z(1 + u)e -mzu-m(m+1)z uG(u, 1) (1 + u)e -mzu -1 ∆ u (m) G(u, y), (14)
with

∆ u H(u) = H(u)-H(0) u
, and the initial condition

G(u, 0) = (1 + u)e -mzu . (15)
Observe that this pair of equations defines G(z; u, y) ≡ G(u, y) uniquely as a formal power series in z. Indeed, the coefficient of z n in G can be computed inductively from these equations (one first determines the coefficient of z n in ∂G ∂y , which can be expressed, thanks to [START_REF] Haiman | Conjectures on the quotient ring by diagonal invariants[END_REF], in terms of the coefficients of z i in G for i < n. Then the coefficient of z n in G is obtained by integration with respect to y, using the initial condition [START_REF] Huang | Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law[END_REF]). Hence, if we exhibit a series G(z; u, y) that satisfies both equations, then G(z; u, y) = G(z; u, y). We are going to construct such a series.

Let

G 1 (z; u) ≡ G 1 (u) = (1 + u)e (m+1)z-(m-1)zu 1 + 1 -e mzu u . (16)
Then G 1 (u) is a series in z with polynomial coefficients in u, which, as we will see, coincides with G(u, 1). Consider now the following equation, obtained from ( 14) by replacing G(u, 1) by its conjectured value G 1 (u):

∂ G ∂y (z; u, y) = z(1 + u)e -mzu-m(m+1)z uG 1 (u) (1 + u)e -mzu -1 ∆ u (m) G(z; u, y), (17)
with the initial condition

G(z; u, 0) = (1 + u)e -mzu . (18)
Eq. ( 17) can be rewritten as

∂ G ∂y (z; u, y) = z(1 + u)e -mzu Λ (m) G(z; u, y) ( 19
)
where Λ is the operator defined by

Λ(H)(u) = H(u) -H(0) A(u) (20)
with

A(u) = u 1 + u e -zu , (21)
and Λ (m) denotes the m-th iterate of Λ. Again, it is not hard to see that [START_REF] Stanley | A polytope related to empirical distributions, plane trees, parking functions, and the associahedron[END_REF] and the initial condition (18) define a unique series in z, denoted G(z; u, y) ≡ G(u, y). The coefficients of this series lie in Q[u, y]. The principle of our proof can be described as follows.

If we prove that G(u, 1) = G 1 (u), then the equation [START_REF] Loehr | Multivariate analogues of Catalan numbers, parking functions, and their extensions[END_REF] satisfied by G coincides with the equation ( 14) that defines G, and thus G(u, y) = G(u, y). In particular, G 1 (z; u) = G(z; u, 1) = G(z; u, 1) = F (t; x, 1), and Theorem 2 is proved.

The case m = 1

Take m = 1. In this subsection, we describe the three steps that, starting from [START_REF] Stanley | A polytope related to empirical distributions, plane trees, parking functions, and the associahedron[END_REF], prove that G(u, 1) = G 1 (u). In passing, we establish the expression (7) of F (t; x, 1) (equivalently, of G(z; u, 1)) given in Theorem 3.

3.2.1.

A homogeneous differential equation and its solution. When m = 1, the equation [START_REF] Stanley | A polytope related to empirical distributions, plane trees, parking functions, and the associahedron[END_REF] defining G(z; u, y) ≡ G(u, y) reads

∂ G ∂y (u, y) = z(1 + u)(1 + ū)( G(u, y) -G(0, y)), (22)
where ū = 1/u, with the initial condition

G(u, 0) = (1 + u)e -zu . (23)
These equations imply that G(-1, y) = 0. The coefficient of G(u, y) in the right-hand side of ( 22) is symmetric in u and ū. We are going to exploit this symmetry to eliminate the term G(0, y).

Replacing u by ū in (22) gives

∂ G ∂y (ū, y) = z(1 + u)(1 + ū)( G(ū, y) -G(0, y)), so that ∂ ∂y G(u, y) -G(ū, y) = z(1 + u)(1 + ū) G(u, y) -G(ū, y) .
This is a homogeneous linear differential equation satisfied by G(u, y) -G(ū, y). It is readily solved, and the initial condition (23) yields

G(u, y) -G(ū, y) = (1 + u)e yz(1+u)(1+ū) e -zu -ūe -z ū . (24)
3.2.2. Reconstruction of G(u, y). Recall that G(u, y) ≡ G(z; u, y) is a series in z with polynomial coefficients in u and y. Hence, by extracting from the above equation the positive part in u (as defined at the end of Section 1), we obtain

G(u, y) -G(0, y) = [u > ] (1 + u)e yz(1+u)(1+ū) e -zu -ūe -z ū .
For any Laurent polynomial P , we have

[u > ](1 + u)P (u) = (1 + u)[u > ]P (u) + u[u 0 ]P (u). (25)
Hence

G(u, y) -G(0, y) = (1 + u)[u > ] e yz(1+u)(1+ū) e -zu -ūe -z ū + u[u 0 ] e yz(1+u)(1+ū) e -zu -ūe -z ū .
Setting u = -1 in this equation gives, since G(-1, y) = 0, so that finally,

G(u, y) = (1 + u)[u > ] e yz(1+u)(1+ū) e -zu -ūe -z ū +(1 + u)[u 0 ] e yz(1+u)(1+ū) e -zu -ūe -z ū = (1 + u)[u ≥ ] e yz(1+u)(1+ū) e -zu -ūe -z ū . ( 26
)
As explained in Section 3.1, G(u, y) = G(u, y) will be proved if we establish that G(u, 1) = G 1 (u). This is the final step of our proof.

3.2.3.

The case y = 1. Equation ( 26) completely describes the solution of ( 22). It remains to check that G(u, 1) = G 1 (u), that is

G(u, 1) = (1 + u)e 2z 1 + 1 -e zu u . (27)
Let us set y = 1 in (26). We find

G(u, 1) = (1 + u)[u ≥ ] e z(2+ū) -ūe z(2+u) = (1 + u)e 2z 1 - e zu -1 u ,
which coincides with (27). Hence G(z; u, y) = G(z; u, y) = F (t; x, y) (with the change of variables ( 4)), and Theorem 3 is proved.

3.2.4. The trivariate series. We have now proved that G(u, y) = G(u, y), so that F (x, y) = G(u, y) after the change of variables [START_REF] Dvoretzky | A problem of arrangements[END_REF]. The expression (7) of F (x, y) given in Theorem 3 now follows from (26).

Solution of the functional equation: the general case

We now adapt to the general case the solution described for m = 1 in Section 3.2.

A homogeneous differential equation and its solution

Let us return to the equation ( 19) satisfied by G(u, y). The coefficient of G(u, y) in the right-hand side of this equation is zv(u), where

v(u) = (1 + u)e -mzu A(u) -m = (1 + u) m+1 ūm .
In the case m = 1, this (Laurent) polynomial was (1 + u)(1 + ū), and took the same value for u and ū. We are again interested in the series u i such that v(u i ) = v(u). Lemma 7. Denote v ≡ v(u) = (1 + u) m+1 u -m , and consider the following polynomial equation in U :

(1 + U ) m+1 = U m v. ( 28
)
This equation has no double root. We denote its m + 1 roots by u 0 = u, u 1 , . . . , u m .

Proof. A double root of (28) would also satisfy

(m + 1)(1 + u) m = mu m-1 v,
and this is easily shown to be impossible.

Remark. One can of course express the u i 's as Puiseux series in u (see [START_REF] Stanley | Enumerative combinatorics[END_REF]Ch. 6]), but this will not be needed here, and we will think of them as abstract elements of an algebraic extension of Q(u). In fact, all the series in z that involve the u i 's in this paper have coefficients that are symmetric rational functions of the u i 's, and hence, rational functions of v. At some point, we will have to prove that a symmetric polynomial in the u i 's (and thus a polynomial in v) vanishes at v = 0, that is, at u = -1, and we will consider series expansions of the u i 's around u = -1.

Proposition 8. Denote v = (1 + u) m+1 u -m , and let the series u i be defined as above. Denote A i = A(u i ), where A(u) is given by (21). Then m i=0 G(u i , y)

j =i (A i -A j ) = ve zvy . ( 29
)
By j =i (A i -A j ) we mean 0≤j≤m,j =i (A i -A j ) but we prefer the shorter notation when the bounds on j are clear. Observe that the A i 's are distinct since the u i 's are distinct (the coefficient of z 0 in A(u) is 1/(1 + ū)). Note also that when m = 1, then u 0 = u, u 1 = ū, and (29) coincides with (24). In order to prove the proposition, we need the following two lemmas.

Lemma 9. Let x 0 , x 1 , . . . , x m be m + 1 variables. Then

m i=0 x i m j =i (x i -x j ) = 1 ( 30
)
and

m i=0 1/x i j =i (x i -x j ) = (-1) m m i=0 1 x i . ( 31
)
Moreover, for any polynomial Q of degree less than m,

m i=0 Q(x i ) j =i (x i -x j ) = 0. ( 32
)
Proof. By the Lagrange interpolation, any polynomial R of degree at most m satisfies:

R(X) = m i=0 R(x i ) j =i X -x j x i -x j .
Equations ( 31) and (32) follow by evaluating this equation at X = 0, respectively with R(X) = 1 and R(X) = XQ(X). Equation ( 30) is obtained by taking R(X) = X m and extracting the leading coefficient.

Lemma 10. Let H(u) ≡ H(z; u) ∈ K(u)[[z]
] be a formal power series in z whose coefficients are rational functions in u over some field K of characteristic 0. Assume that these coefficients have no pole at u = 0. Then there exists a sequence g 0 (z), g 1 (z), . . . of formal power series in z such that for every k ≥ 0 one has:

Λ (k) H(u) = 1 A(u) k   H(u) - k-1 j=0 g j (z)A(u) j   . ( 33
)
where Λ is the operator defined by [START_REF] Yan | Generalized parking functions, tree inversions, and multicolored graphs[END_REF].

Proof. We denote by L the subring of K(u) [[z]] formed by formal power series whose coefficients have no pole at u = 0. By assumption, H(u) ∈ L. We use the notation

O(u k ) to denote an element of K(u)[[z]] of the form u k J(z; u) with J(z; u) ∈ L. First, note that A(u) = ue -zu /(1 + u) belongs to K(u)[[z]]. Moreover, A(u) = u + O(u 2 ). ( 34
)
We will first prove that there exists a sequence of formal power series (g j ) j≥0 ∈ K[[z]] N such that for all ℓ ≥ 0,

H(u) = ℓ-1 j=0 g j (z)A(u) j + O(u ℓ ). ( 35
)
We will then prove that these series g j satisfy (33). In order to prove (35), we proceed by induction on ℓ ≥ 0. The identity holds for ℓ = 0 since H(u) ∈ L. Assume it holds for some ℓ ≥ 0, i.e., that there exists series g 0 , . . . , g ℓ-1 in K[[z]] and J(u) ∈ L such that

H(u) = ℓ-1 j=0 g j (z)A(u) j + u ℓ J(u).
By (34) and by induction on r, we have u r = A(u) r + O(u r+1 ) for all r ≥ 0. Using this identity with r = ℓ, and rewriting J(u) = J(0) + O(u), we obtain u ℓ J(u) = J(0)A(u) ℓ + O(u ℓ+1 ), so that:

H(u) = ℓ j=0 g j (z)A(u) j + O(u ℓ+1 ),
with g ℓ (z) := J(0). Thus (35) holds for ℓ + 1.

Let us now prove (33). Again, we proceed by induction on k ≥ 0. The identity clearly holds for k = 0. Assume it holds for some k ≥ 0. In (33), replace H(u) by its expression (35) obtained with ℓ = k + 1, and let u tend to 0: this shows that g k (z) is in fact Λ (k) H(0). From the definition of Λ one then obtains

Λ (k+1) H(u) = Λ (k) H(u) -g k (z) A(u) = 1 A(u) k+1   H(u) - k j=0 g j (z)A(u) j   .
Thus (33) holds for k + 1.

Proof of Proposition 8. Thanks to Lemma 10 (applied with K = Q(y)), we can rewrite [START_REF] Stanley | A polytope related to empirical distributions, plane trees, parking functions, and the associahedron[END_REF] as

∂ G ∂y (u, y) = zv G(u, y) -zv m-1 j=0 g j (y)A(u) j , (36)
with v = (1 + u) m+1 ūm , and for all j ≥ 0 the series g j (y) ≡ g j (z; y) belongs to K(y)[[z]] (one can actually show that g j (y) ∈ K[y][[z]] but we will not need this). As was done in Section 3.2.1, we are going to use the fact that v(u i ) = v for all i ∈ 0, m to eliminate the m unknown series g j (y). For 0 ≤ i ≤ m, the substitution u → u i in (36) gives:

∂ G ∂y (u i , y) = zv G(u i , y) -zv m-1 j=0 g j (y)A j i = zv G(u i , y) -zvQ(A i ) (37)
where Q(X) = m-1 j=0 g j (y)X j is a polynomial in X of degree less than m. Consider the linear combination

L(u, y) := m i=0 G(u i , y) j =i (A i -A j ) . ( 38
)
Then by (37),

∂L ∂y (u, y) = zvL(u, y) -zv m i=0 Q(A i ) j =i (A i -A j ) = zvL(u, y) by (32).
This homogeneous linear differential equation is readily solved:

L(u, y) = L(u, 0)e zvy .
Recall the expression (38) of L in terms of G. The initial condition ( 18) can be rewritten G(u, 0) = vA(u) m , which yields

L(u, 0) = v m i=0 A i m j =i (A i -A j ) = v
by (30). Hence L(u, y) = ve zyv , and the proposition is proved.

Reconstruction of G(u, y)

We are now going to prove that (29), together with the condition G(-1, y) = 0 derived from [START_REF] Stanley | A polytope related to empirical distributions, plane trees, parking functions, and the associahedron[END_REF], characterizes the series G(u, y). We will actually obtain a (complicated) expression for this series, generalizing (26).

We first introduce some notation. Consider a formal power series in z, denoted H(z; u) ≡ H(u), having coefficients in K[u] for some field K of characteristic 0 (for instance Q(y)). We define a series H k in z whose coefficients are symmetric functions of k + 1 variables x 0 , . . . , x k :

H k (x 0 , . . . , x k ) = k i=0 H(x i ) 0≤j≤k,j =i (A(x i ) -A(x j ))
, where, as above, A is defined by (21).

Lemma 11. The series

H k (x 0 , . . . , x k ) has coefficients in K[x 0 , . . . , x k ]. If, moreover, H(-1) = 0, then the coefficients of H k are multiples of (1 + x 0 ) • • • (1 + x k ). Proof. Observe that 1 A(x i ) -A(x j ) = 1 x i -x j B(x i , x j ), (39)
where B(x i , x j ) is a series in z with polynomial coefficients in x i and x j . Hence

H k (x 0 , . . . , x k ) 0≤i<j≤k (x i -x j )
has polynomial coefficients in the x i 's. But these polynomials are anti-symmetric in the x i 's (since H k is symmetric), hence they must be multiples of the Vandermonde i<j (x i -x j ). Hence H k (x 0 , . . . , x k ) has polynomial coefficients.

A stronger property than (39) actually holds, namely:

1 A(x i ) -A(x j ) = (1 + x i )(1 + x j ) x i -x j C(x i , x j ),
where C(x i , x j ) is a series in z with polynomial coefficients in x i and x j . Hence, if

H(x) = (1 + x)K(x), H k (x 0 , . . . , x k ) = k i=0 K(x i )(1 + x i ) k+1 j =i (1 + x j )C(x i , x j ) x i -x j . Setting x 0 = -1 shows that H k (-1, x 1 , . . . , x k ) = 0, so that H k (x 0 , . . . , x k ) is a multiple of (1 + x 0 )
. By symmetry, it is also a multiple of all (1 + x i ), for 1 ≤ i ≤ k.

Our treatment of (29) actually applies to equations with an arbitrary right-hand side. We consider a formal power series series H(z; u) ≡ H(u) with coefficients in K[u], satisfying H(-1) = 0 and

m i=0 H(u i ) j =i (A i -A j ) = Φ m (v),
for some series Φ m (v) ≡ Φ m (z; v) with coefficients in vK [v], where v = (1 + u) m+1 ūm . Using the above notation, this equation can be rewritten as

H m (u 0 , . . . , u m ) = Φ m (v).
We will give an explicit expression of H(u) involving two standard families of symmetric functions [START_REF] Stanley | Enumerative combinatorics[END_REF]Chap. 7], namely the homogeneous functions h λ and the monomial functions m λ . We denote by ℓ(λ) the number of parts in a partition λ, and by S m+1 the symmetric group on {0, 1, . . . , m}.

Proposition 12. Let H(z; u) ≡ H(u) be a power series in z with coefficients in

K[u], satisfying H(-1) = 0 and H m (u 0 , . . . , u m ) = Φ m (v), (40)
where

Φ m (v) ≡ Φ m (z; v) is a series in z with coefficients in vK[v].
There exists a unique sequence Φ 0 , . . . , Φ m of series in z with coefficients in vK[v] such that for 0 ≤ k ≤ m, and for all permutation σ ∈ S m+1 ,

H k (u σ(0) , . . . , u σ(k) ) = m j=k Φ j (v)h j-k (A σ(0) , . . . , A σ(k) ). (41)
In particular, H(u) ≡ H 0 (u) is completely determined:

H(u) = m j=0 Φ j (v)A(u) j . (42)
The series Φ k (v) ≡ Φ k (z; v) can be computed by a descending induction on k as follows. Let us denote by

Φ > k-1 (u) the positive part in u of Φ k-1 (v), that is Φ > k-1 (u) := [u > ]Φ k-1 (ū m (1 + u) m+1
). Then for 1 ≤ k ≤ m, this series can be expressed in terms of Φ k , . . . , Φ m :

Φ > k-1 (u) = - 1 m k [u > ]   m j=k Φ j (v) λ⊢j-k+1 m -ℓ(λ) k -ℓ(λ) m λ (A 1 , . . . , A m )   , (43)
and Φ k-1 (v) can be expressed in terms of Φ > k-1 :

Φ k-1 (v) = m i=0 Φ > k-1 (u i ) -Φ > k-1 (-1) . (44)
We first establish three lemmas dealing with the symmetric functions of the series u i defined in Lemma 7.

Lemma 13. The elementary symmetric functions of u 0 = u, u 1 , . . . , u m are e j (u 0 , u 1 , . . . , u m ) = (-1)

j m + 1 j + v½ j=1 with v = u -m (1 + u) m .
The elementary symmetric functions of u 1 , . . . , u m are e m-j (u 1 , . . . , u m ) = (-1) m-j-1 j p=0 m + 1 p u p-j-1 .

In particular, they are polynomials in 1/u, and so is any symmetric polynomial in u 1 , . . . , u m . Finally,

m i=0 (1 + u i ) = v.
This is easily proved by collecting the coefficient of H(x i ), for all i ∈ 0, k , in both sides of the equation. We also have, for any indeterminates a 0 , . . . , a m , (a k-1 -a k )h j-k (a 0 , . . . , a k ) = h j-k+1 (a 0 , . . . , a k-2 , a k-1 ) -h j-k+1 (a 0 , . . . , a k-2 , a k ).

Hence, multiplying (41) by (A σ(k-1) -A σ(k) ) gives

H k-1 (u σ(0) , . . . , u σ(k-2) , u σ(k-1) ) - m j=k Φ j (v)h j-k+1 (A σ(0) , . . . , A σ(k-2) , A σ(k-1) ) = H k-1 (u σ(0) , . . . , u σ(k-2) , u σ(k) ) - m j=k Φ j (v)h j-k+1 (A σ(0) , . . . , A σ(k-2) , A σ(k) ),
for all σ ∈ S m+1 . This implies that the series

H k-1 (x 0 , . . . , x k-1 ) - m j=k Φ j (v)h j-k+1 (A(x 0 ), . . . , A(x k-1 ))
takes the same value at all points (u σ(0) , . . . , u σ(k-1) ), with σ ∈ S m+1 . Hence, by Lemma 15, there exists a series in z with rational coefficients in v, denoted Φ k-1 (v), such that for all σ ∈ S m+1 ,

H k-1 (u σ(0) , . . . , u σ(k-1) ) - m j=k Φ j (v)h j-k+1 (A σ(0) , . . . , A σ(k-1) ) = Φ k-1 (v). (46)
This is exactly (41) with k replaced by k -1.

The next point we will prove is that the coefficients of Φ k-1 belong to vK [v]. In order to do so, we symmetrize (46) over u 0 , . . . , u m . For any subset V = {i 1 , . . . , i k } of {0, . . . , m}, of cardinality k, we denote u V = (u i1 , . . . , u i k ) and similarly A V = (A i1 , . . . , A i k ). By (46),

m + 1 k Φ k-1 (v) = V ⊂{0,...,m},|V |=k H k-1 (u V ) - m j=k   Φ j (v) V ⊂{0,...,m},|V |=k h j-k+1 (A V )   .
(47) We will prove that both sums in the right-hand side of this equation are series in z with coefficients in vK [v].

Denote x V = (x i1 , . . . , x i k ). Observe that V ⊂{0,...,m},|V |=k

H k-1 (x V )
is a series in z with polynomial coefficients in x 0 , . . . , x m , which is symmetric in these variables ( Lemma 11). By Lemma 13, the first sum in (47) is thus a series in z with polynomial coefficients in v. We still need to prove that this series even vanishes at v = 0, that is, at u = -1. But this follows from Lemma 11, since u i = -1 for all i when u = -1.

Let us now consider the second sum in (47), and more specifically the term

Φ j (v) V ⊂{0,...,m},|V |=k h j-k+1 (A V ). (48)
Recall that

A i = u i 1 + u i e -zui .
But by Lemma 13,

1 1 + u i = 1 v 0≤j =i≤m (1 + u j ).
Hence (48) can be written as a series in z with coefficients in K[1/v, u 0 , . . . , u m ], symmetric in u 0 , . . . , u m . By the first part of Lemma 13, these coefficients belong to K[v, 1/v]. We want to prove that they actually belong to vK[v], that is, that they are not singular at v = 0 (equivalently, at u = -1) and even vanish at this point. >From the equation (1 + u i ) m+1 = vu m i , it follows that we can label u 1 , . . . , u m in such a way

1 + u i = ξ i (1 + u) + o(1 + u),
where ξ is a primitive (m + 1) st root of unity. Since Φ j (v) is a multiple of v = ūm (1 + u) m+1 , and the symmetric function h j-k+1 has degree j -k + 1 ≤ m, it follows that the series (48) is not singular at u = -1, and even vanishes at this point. Hence its coefficients belong to vK [v].

We finally want to obtain an explicit expression of Φ k-1 (v). Lemma 14, together with Φ k-1 (0) = 0, establishes (44). To express Φ > k-1 (u), we now symmetrize (46) over u 1 , . . . , u m . With the above notation,

m k Φ k-1 (v) = V ⊂{1,...,m},|V |=k H k-1 (u V ) - m j=k   Φ j (v) V ⊂{1,...,m},|V |=k h j-k+1 (A V )   . ( 49
)
As above,

V ⊂{1,...,m},|V |=k H k-1 (x V )
is a series in z with polynomial coefficients in x 1 , . . . , x m , which is symmetric in these variables. By the second part of Lemma 13, the first sum in (49) is thus a series in z with polynomial coefficients in 1/u. Since Φ k-1 (v) has coefficients in K[v], and hence in K[u, 1/u], the second sum in (49) is also a zeries in z with coefficients in K[u, 1/u]. We can now extract from (49) the positive part in u, and this gives

m k Φ > k-1 (u) = -[u > ]   m j=k   Φ j (v) V ⊂{1,...,m},|V |=k h j-k+1 (A V )     .
One easily checks that, for indeterminates a 1 , . . . , a m , V ⊂{1,...,m},|V |=k

h j-k+1 (a V ) = λ⊢j-k+1 m -ℓ(λ) k -ℓ(λ) m λ (a 1 , . . . , a m ),
so that the above expression of Φ > k-1 (u) coincides with (43).

The case y = 1

As explained in Section 3.1, Theorem 2 will be proved if we establish G(u, 1) = G 1 (u), where

G 1 (u) = (1 + u)e (m+1)z-(m-1)zu 1 + 1 -e mzu u .
A natural attempt would be to set y = 1 in the expression of G(u, y) that can be derived from Proposition 12, as we did when m = 1 in Section 3.2.3. However, we have not been able to do so, and will proceed differently. We have proved in Proposition 8 that the series G(u, y) satisfies (40) with Φ m (v) = ve zvy . In particular, G(u, 1) satisfies (40) with Φ m (v) = ve zv . By Proposition 12, this equation, together with the initial condition G(-1, 1) = 0, characterizes G(u, 1). It is clear that G 1 (-1) = 0. Hence it suffices to prove is the following proposition.

Proposition 16. The series G 1 (u) satisfies (40) with Φ m (v) = ve zv . Proof. Note that G 1 (u) = e (m+1)z vA(u) m-1 -1 A(u) . Using Lemma 9 with x i = A i , it follows that m i=0 G 1 (u i ) j =i (A i -A j ) = 0 + (-1) m+1 e (m+1)z m i=0 1 A i
(by ( 31) and (32))

= (-1) m+1 e (m+1)z+z i ui m i=0

(1 + u i ) u i = ve zv by Lemma 13.

The trivariate series

We have now proved that G(u, y) = G(u, y), so that F (x, y) = G(u, y) after the change of variables (4). As shown in Proposition 8, the series G(u, y) satisfies (40) with Φ m (v) = ve zvy . Hence Proposition 12 gives an explicit, although complicated, expression of the trivariate series F (t, x, y).

Theorem 17. Let F (m) (t; x, y) ≡ F (t; x, y) be the exponential generating function of labelled m-Tamari intervals, defined by (3). Let z and u be two indeterminates, and write t = ze -m(m+1)z and x = (1 + u)e -mzu . Then F (t; x, y) becomes a series in z with polynomial coefficients in u and y, and this series can be computed by an iterative extraction of positive parts. More precisely,

F (t; x, y) = m k=0 Φ k (v)A(u) k ,
where v = u -m (1 + u) m+1 , A(u) is defined by (21), and Φ k (v) ≡ Φ k (z; v) is a series in z with polynomial coefficients in v. This series can be computed by a descending induction on k as follows. First, Φ m (v) = ve zyv . Then for k ≤ m,

Φ k-1 (v) = m i=0 Φ > k-1 (u i ) -Φ > k-1 (-1)
where

Φ > k-1 (u) = [u > ]Φ k-1 (v) = - 1 m k [u > ]   m j=k Φ j (v)

Remark and examples

The case k = 1 of the above identity gives

[u > ]Φ 0 (v) = - 1 m [u > ]   m j=1 Φ j (v) m i=1 A(u i ) j   . (50)
Recall that F (t; x, y) = G(z; u, y) has polynomial coefficients in u and y. Hence • When m = 1, this is the expression (7) of F (1) (t; x, y) (recall that Φ m = ve zyv ).

• When m = 2, the generating function of labelled 2-Tamari intervals satisfies F (2) (t; x, y)

1 + u = [u ≥ ] Φ 1 (v) 1 + u A(u) - A(u 1 ) 2 - A(u 2 ) 2 + (1 + ū) 2 e zyv A(u) 2 - A(u 1 ) 2 2 - A(u 2 ) 2 2 ,
where

A(u) = u 1 + u e -zu , u 1,2 = 1 + 3u ± (1 + u) √ 1 + 4u 2u 2 ,
and

Φ 1 (v) = Φ > 1 (u) + Φ > 1 (u 1 ) + Φ > 1 (u 2 ) -3Φ > 1 (-1), with Φ > 1 (u) = -[u > ]
(1 + u) 3 ū2 e zy(1+u) 3 ū2 (A(u 1 ) + A(u 2 )) . This expression has been checked with Maple, after computing the first coefficients of F (t; x, y) from the functional equation (9).

Final comments

A constructive proof ? Our proof would not have been possible without a preliminary task consisting in guessing the expression of F (t; x, 1). This turned out to be difficult, in particular because the standard tools like the Maple package Gfun can only guess D-finite generating functions, while the generating function of the numbers (2) is not D-finite. More precisely, the expression of F (t; x, 1) becomes D-finite after the change of variables [START_REF] Bousquet-Mélou | The number of intervals in the m-Tamari lattices[END_REF], but what is hard to guess is this change of variables. A constructive proof of our result would be most welcome.

A q-analogue of the functional equation. As described in the introduction, the numbers (2) are conjectured to give the dimension of certain polynomial rings generalizing DR 3,n . These rings are tri-graded (with respect to the sets of variables {x i }, {y i } and {z i }), and it is conjectured [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF] that the dimension of the homogeneous component in the x i 's of degree k is the number of labelled intervals [P, Q] in T (m) n such that the longest chain from P to Q, in the Tamari order, has length k. One can recycle the recursive description of intervals described in Section 2 to generalize the functional equation of Proposition 6, taking into account (with a new variable q) this distance. Eq. ( 9) remains valid, upon defining the operator ∆ by ∆S(x) = S(qx) -S(1) qx -1 .

The coefficient of t n in the series F (t, q; x, y) does not seem to factor, even when x = y = 1. The coefficients of the bivariate series F (t, q; 1, 1) have large prime factors.

Further developments. There is a natural action of the symmetric group S n on labelled m-Tamari intervals of size n: it consists in permuting the labels according to the permutation one considers, and then to rearrange the labels in each sequence of consecutive steps so that they increase. The dimension of this representation of S n is the number (2) of labelled m-Tamari intervals of size n. Bergeron and Préville Ratelle have a refined conjecture that gives the character of this representation [START_REF] Bergeron | Higher trivariate diagonal harmonics via generalized Tamari posets[END_REF]. We have very recently proved this conjecture [START_REF] Bousquet-Mélou | The representation of the symmetric group on m-Tamari intervals[END_REF].

⊳ ¡ ⋄ £ ⊲ LFPR: LACIM, UQAM, C.P. 8888 Succ. Centre-Ville, Montréal H3C 3P8, Canada E-mail address: preville-ratelle.louis-francois@courrier.uqam.ca

Figure 1 .

 1 Figure 1. The covering relation between m-ballot paths (m = 2).

Figure 2 .

 2 Figure 2. The m-Tamari lattice T (m) n for m = 1 and n = 4 (left) and for m = 2 and n = 3 (right). The three walks surrounded by a line in T (1) 4

Figure 3 .

 3 Figure 3. A labelled 2-ballot path.

Figure 4 .

 4 Figure 4. A labelled 2-Tamari interval I = [P, Q] of size |I| = 3. It has c(P ) = 3 contacts, and its initial rise is r(Q) = 2.

  of T mn . Proposition 4 ([4, Prop. 4]). The set of m-Dyck paths with n blocks is the sublattice of T nm consisting of the paths that are larger than or equal to u m d m . . . u m d m . It is order isomorphic to T (m) n

m

  -ℓ(λ) k -ℓ(λ) m λ (A(u 1 ), . . . , A(u m ))   ,and u 0 = u, u 1 , . . . , u m are the m + 1 roots of the equation (1 + u i ) m+1 = u m i v.

F

  (t; x, y) = F (t; 1, y) + [u > ] m k=0 Φ k (v)A(u) k (since u = 0 when x = 1) = F (t; 1, y) + [u > ] i ) kby (25), and given that F (t; x, y) = 0 when u = -1.

Acknowledgements. We are grateful to François Bergeron for advertising in his lectures the conjectural interpretation of the numbers (2) in terms of labelled Tamari intervals. We also thank Éric Fusy and Gilles Schaeffer for interesting discussions on this topic, and thank Éric once more for allowing us to reproduce some figures of [4].

GC and LFPR were partially supported by the European project ExploreMaps -ERC StG 208471. LFPR is also partially supported by an Alexander Graham Bell Canada Graduate Scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC).

Proof. The symmetric functions of the roots of a polynomial can be read from the coefficients of this polynomial. Hence the first result follows directly from the equation satisfied by the u i 's, for 0 ≤ i ≤ m, namely

(1 + u i ) m+1 = vu m i . For the second one, we need to find the equation satisfied by u 1 , . . . , u m , which is

The second result follows.

The third one is obtained by evaluating at X = -1 the identity

Lemma 14. Denote v = ūm (1 + u) m+1 . Let P be a polynomial. Then P (v) is a Laurent polynomial in u. Let P > (u) denote its positive part:

Then

Proof. The right-hand side of ( 45) is a symmetric polynomial of u 0 , . . . , u m , and thus, by the first part of Lemma 13, a polynomial in v. Denote it by P (v). The second part of Lemma 13 implies that the positive part of P (v) in u is P > (u 0 ) = P > (u). That is, P (v) and P (v) have the same positive part in u. In other words, the polynomial

Laurent polynomial in u of non-positive degree. But since v = (1 + u) m+1 ūm , the degree in u of Q(v) coincides with the degree of Q, and so Q must be a constant. Finally, by setting u = -1 in P (v), we see that P (0) = P (0) (because u i = -1 for all i when u = -1, as follows for instance from Lemma 13). Hence Q = 0 and the lemma is proved.

Lemma 15. Let 0 ≤ k ≤ m, and let R(x 0 , . . . , x k ) be a rational function in k + 1 variables x 0 , . . . , x k , such that for any permutation σ ∈ S m+1 , R(u 0 , . . . , u k ) = R(u σ(0) , . . . , u σ(k) ).

Then there exists a rational fraction in v equal to R(u 0 , . . . , u k ).

Proof. Let R be the following rational function in x 0 , . . . , x m :

Then R is a symmetric function of x 0 , . . . , x m , and hence a rational function in the elementary symmetric functions e j (x 0 , . . . , x m ), say S(e 1 (x 0 , . . . , x m ), . . . , e m+1 (x 0 , . . . , x m )). By assumption, R(u 0 , . . . , u m ) = S(e 1 (u 0 , . . . , u m ), . . . , e m+1 (u 0 , . . . , u m )) = R(u 0 , . . . , u k ).

Since S is a rational function, it follows from the first part of Lemma 13 that R(u 0 , . . . , u k ) can be written as a rational function in v.

Proof of Proposition 12. We prove (41) by descending induction on k. For k = m, (41) holds by assumption when σ is the identity, and actually for any σ as H m (x 0 , . . . , x m ) is a symmetric function of the x i 's. Let us assume (41) holds for some k > 0, and prove it for k -1.

Observe that (A(x k-1 ) -A(x k ))H k (x 0 , . . . , x k ) = H k-1 (x 0 , . . . , x k-2 , x k-1 ) -H k-1 (x 0 , . . . , x k-2 , x k ).


Kafing Keita

Robert Frey

Philippe Delaye

Daniel Dolfi

Jean- Pierre Huignard

Gérald Roosen

Stimulated Raman scattering for variable gain amplification of small optically carried microwave signals

Stimulated Raman amplification of optically carried microwave signals in optical fibers is experimentally investigated. In the backward configuration which is shown to be less noisy than the forward one, net amplification factors larger than 20 dB are demonstrated in the low signal regime with better relative intensity noise (RIN) values than using erbiumdoped fiber amplifiers. Due to the RIN flatness with respect to the gain of the stimulated Raman amplifier high quality variable gain amplification of optically carried microwave signals could be implemented using optimized Raman amplifiers

Introduction:

Distribution and processing of optically carried microwave signals are among the basic functions for future active radar antennas [START_REF] Zmuda | Photonic Aspects of modern radars[END_REF]. However, due to the high linearity performances required for applications, the radio-frequency (RF) signal is encoded by means of integrated modulators on an optical carrier at low optical power with very small modulation depths [START_REF] Heismann | Optical fiber telecommunications[END_REF]. Moreover, carrier depletion (for instance using stimulated Brillouin scattering [START_REF] Williams | [END_REF]4]) further reduces the optical power, making optical amplification necessary. This amplification, generally operated using Erbium-doped-fiber-amplification (EDFA) [5,6], is accompanied by amplified spontaneous emission (ASE) noise. This contribution can be particularly large for the strong amplification of the small signals encountered in the field of optically carried microwave signals since, in that case, the amplifier is not saturated by the signal intensity and the noise level will be large.

Here we show that stimulated Raman amplification (SRA) in silica fiber can be used for low noise amplification of such small optically carried microwave signals. SRA in silica fibers has been widely studied [START_REF]Nonlinear fiber optics[END_REF] and successfully applied [START_REF] Zhu | [END_REF] in optical telecommunications essentially due its extremely large bandwidth (2.4 THz) which makes this amplification process very appropriate to dense wavelength-division-multiplexing (DWDM). However, even if several results obtained in these previous studies may be applied to microwave fiber optic links, the problem is fundamentally different essentially because of the nature of the optical signals to be amplified in RF photonics. Indeed, DWDM communications deal with digital short duration signals carried by discrete wavelengths in which the main aim is to discriminate two discrete intensity level representing "0" and "1" bits, while microwave optoelectronic links use optically carried analogical high frequency RF signals for which we want a minimum degradation of the frequency purity of the signal, with a large rejection of noise around the chosen RF. This means that the large bandwidth of SRA in silica fibers is not useful in this last case. However, due to its high gain SRA in fibers can be very attractive for the purpose of low noise small signal amplification of optically carried RF signals and to our knowledge little attention has been paid on this capability for microwave photonics applications. In particular, a significant advantage is that in SRA the small signal amplification gain and the total spontaneous noise power are both proportional to the pump power [START_REF] Shen | The principles of nonlinear optics[END_REF]. This peculiarity of SRA can ensure a constant output relative intensity noise (RIN) value with respect to gain variations, a property which cannot be achieved using EDFA [START_REF]Fiber optic communication systems[END_REF] thus leading to the development of high performance variable gain amplifiers. In the following we experimentally and theoretically demonstrate this new advantage of SRA over Erbium doped fibers amplifiers for microwave fiber optic links.

Experiment:

The Raman amplifiers studied in our analysis consisted of conventional singlemode fibers of different lengths pumped at a wavelength of 1480 nm by a CW fiber Raman laser (2 W power and 1 nm linewidth). In the CW regime two configurations with co-and counterpropagating signal and pump beams are possible (Fig. 1). Nevertheless, as well know, the copropagating configuration presents several constraints related to the presence of Modulation Instabilities [START_REF] Tai | [END_REF] and to the easier transfer of the pump noise towards the signal beam [12], that would give rise to troubles for the realization of a low noise amplification link. This additional noise was effectively observed (Fig. 2), that's why our experiments focus on the counter-propagating geometry (Fig. 1). that also injects the counter-propagating pump beam in the fiber. The signal beam was then sent in an optical spectrum analyzer. For comparison, measurement with a commercial EDFA (Keopsys, KPS-BT2-C27-PB-FA) were performed. Due to its architecture the EDFA replaces the whole system between wavelenth multiplexers (pump included too), and the measurement was made without any additional fiber.

Results:

In the different experiments we measure the net amplification factor (defined as the ratio of input and output signal powers), as well as the noise level. Figure 3 shows spectra of the injected and amplified signal beams modulated at a frequency of 20 GHz (dotted and continuous lines respectively) and the ASE obtained with no signal beam (dashed line) for a 22.5 km long fiber and a pump power of 1.6 W. The net amplification factor was 24 dB and no degradation of the signal beam was observed. Although a more quantitative analysis has to prove the spectral uniformity of the amplification factor which could be affected by four wave mixing processes, this first result is a strong indication of the usefulness of SRA to amplify optically carried microwave expected for an unsaturated amplifier, the noise level was not influenced by the presence of the signal beam (compare continuous and dashed lines out of the laser line regions in fig. 3).

- In order to compare the respective influence of ASE in EDFA and SRA experiments were also performed by replacing the SRA by an EDFA. This comparison is illustrated in figure 4 showing the mean relative intensity noise RIN=SNR -2 /ΔF plotted as a function of the net amplifier gain G=P Sout /P Sin (SNR=P Sout /P ASE is the optical signal to noise ratio, with P Sin = -20 dBm and P Sout the input and output signal powers and P ASE the noise power resulting from the amplification of photons emitted through spontaneous Raman generation, and ΔF=8.56 GHz is the bandwidth of the optical analyzer). For SRA, due to fiber losses, shorter was the fiber lower was the RIN value with minimum values around -168 dB/Hz for a fiber length of 6.5 km. In this case, however, the maximum gain achievable in our experiment was only 12 dB. Note, however, that with the same fiber length, higher Raman gains could be possible using the peak gain of the Raman transition, fibers with higher Raman nonlinearities such as Ge-doped fibers or higher pump powers. The measured values are also in very good agreement with those calculated using equations given in Ref. 13, as it can be seen in fig. 4 showing fits obtained for L=6.5 and 21 km (bolded lines). Let us also note the flatness of the RIN with respect to the net gain of the amplifier whatever the fiber length is. This RIN flatness which is obviously connected to the independence of the Raman-gain-tospontaneous-emission ratio from the pump power [START_REF] Shen | The principles of nonlinear optics[END_REF] is very interesting for high quality variable gain amplifiers of optically carried microwave signals. For comparison with existing amplification systems used in microwave fiber optic links [5,6], we make a measurement with an EDFA system. The RIN of the EDFA exhibits much higher values (see closed diamonds in fig. 4) and we clearly see that the RIN improvement found when using SRA was between 30 and 10 dB/Hz depending on the gain value. This is, in fact, directly connected to the decrease of the EDFA RIN value with increasing net gains (see fig. 4) which appears due to the decrease of the amplifier population inversion factor [START_REF]Fiber optic communication systems[END_REF].

The present study just considers SRA and EDFA independently of the system in which they will be inserted. The final choice may depend on the exact architecture of the microwave fiber optic link and essentially the length of the transmission fiber. In short links, the transmission fiber brings only negligible losses but can not be used to perform SRA, that will then use its own specific fiber for amplification (this case corresponds to the one described in Fig. 4). With long transmission fibers, the fiber will be directly used to implement the SRA, without additional fiber whereas the losses induced by the transmission fiber will bring additional parameters in the evaluation of the EDFA performances. In this last case the influence of the transmission line will depend on the position of the EDFA in the system and will need a careful evaluation. If positioned after the transmission fiber, the input signal power will be small and the EDFA will be used far from saturation with a higher noise figure.

On the other hand if used as a booster before the launching of the signal in the transmission fiber, the EDFA may be operated closer to the saturated regime (the reduced net gain due to losses of the fiber would then correspond to a shift of the EDFA curve to the Fig. 4) but the high power signal may then induce nonlinear effects in the transmission fiber (such as Stimulated Brillouin Scattering) which will decrease the transmission performances.

The low RIN value measured for SRA and the flatness of this RIN with the gain value demonstrate the superiority of Raman amplification for small signal amplification of optically carried microwave signals with variable gain. Evidently, this optical measurement which is averaged on a large frequency bandwidth does not provide as much information as electrical measurements can and these last should be done for definite conclusions. However, as SRA does not suffer from relaxation oscillations our measurements provide a good first indication.

Conclusion:

Amplification of optically carried microwave signals have been demonstrated using stimulated Raman amplification in conventional telecom optical fibers. For backward SRA which is known to be less noisy than forward SRA, net amplification factors larger than 20dB have been obtained in the small signal regime together with an improvement of the RIN value around 10 to 30 dB/Hz when compared to EDFA. Even better results are expected using an optimized SRA. Preliminary results presented in this paper tend to prove that SRA can provide an attractive mechanism of low noise and variable gain amplification for the transmission and the processing of optically carried RF signals.

Figure 1 :

 1 Figure 1 : Experimental setup for counter-propagating SRA. MZM : Mach-Zehnder modulator. WM :wavelength multiplexer. OSA : optical spectrum analyser. In the co-propagating scheme, we only had to exchange the position of the pump laser and trap.

Figure 2 :

 2 Figure 2 : Signal power spectrum for co-and counter-propagating SRA. The fiber length is 22.5km, and the pump power is 1.7W in both cases, leading to a net gain of 25dB.

Figure 3 :

 3 Figure 3 : Input and output signal and ASE spectra.

Figure 4 :

 4 Figure 4 : Compared RIN for SRA and EDFA as a function of amplifier net gain for a -20 dBm input signal. Light solid lines are guides for the eyes while bolded solid lines correspond to fits obtained using equations giving the amplitude of the amplified spontaneous emission responsible for the noise given in Ref. 13


T Shimura

F Grappin

P Delaye

S Iwamoto

Y Arakawa

K Kuroda

G Roosen

Simultaneous determination of the index and absorption gratings in multiple quantum well photorefractive devices designed for laser ultrasonic sensor

A new method to determine complex amplitudes of index and absorption gratings in photorefractive multiple quantum wells is presented. In usual method to evaluate these parameters, spectral measurement of absorption change due to the applied field over a wide wavelength range is required and the index change is calculated through the Kramers-Krönig's relationship. In our method, only a set of measured data for photorefractive two-wave mixing set-up with small sinusoidal phase modulation for fixed wavelength is needed. This is thus well adapted to insitu characterization of systems that operate at a given wavelength. Real and imaginary part of the gains owing to the index and the absorption gratings are calculated from the output signals of the system. We apply this method to our photorefractive multiple quantum devices operating at 1064 nm, and obtain consistent results.

R e v i e w C o p y

T. Shimura et al.

page2/13

Laser ultrasonic sensor with photorefractive two-wave mixing is quite useful because of its high sensitivity and adaptive nature. [1][2][3][4][5][6] Slow noise component compared to the response time of the photorefractive material is compensated, and wave-front matching is automatically maintained with real time recording of the interference fringe. High sensitivity is also one of the advantages of this method. Sub-nanometer vibration of the scattering surface is easily detected by this system with small power CW-lasers.

Within various photorefractive materials, bulk semiconductors such as CdTe:V and InP:Fe have shown quite good performance for these systems, and the sensitivity of the systems with these materials is close to interferometric limit [4,5]. Photorefractive multiple quantum wells (PR-MQWs) have also been proved that they have good properties for this purpose, especially a higher cut off frequency for compensating the turbulences due to their faster response speed [6,7]. Recently, InGaAs/GaAs PR-MQWs sensitive at 1064 nm have been developed [8] and applied to the vibration detection system [9]. The advantage of this material is, of course, that compact, high power, and high coherence Nd:YAG laser is usable as a laser source.

It is rather easy to consider the optimum operating condition in these systems when the only index grating exists. In this case, photorefractive phase shift p , which is the phase shift between the fringe pattern and the space charge field grating, should be 0 or , and simply larger amplitude of the index grating gives the larger output signal [5]. On the other hand, in PR-MQWs, the situation is more complicated because both index and absorption gratings exist at the same time [7]. To find the optimum operating condition, we need to measure the electro-absorption spectrum over a wide range of wavelength , and calculate index change through the Kramers-Krönig's relationship, then the excitonic spectral phase is obtained from the ratio of the index to absorption variation. The knowledge of this spectral phase allows to choose the operating wavelength that maximizes the devices performances, i.e. the detection sensitivity. In the case of system operating with Nd:YAG lasers the tunability of the laser is not sufficient to see sizeable variation of the spectral phase, and this parameter even if useful for characterization of the structure, brings too much information when regarding device performances.

In this paper, we present a complementary method to determine the amplitude and the phase of both index and absorption gratings, that brings all the required information in a simple manner using the known symmetries of the grating written in the PR-MQW structure. We make a simple two-wave mixing and electro-absorption measurements for positive and negative electric field to determine the amplitudes of the index and absorption gratings simultaneously and photorefractive phase shift without any wavelength scan.

In PR-MQWs, both absorption and index gratings, will contribute to the phase demodulation in the photorefractive laser ultrasonic sensor. Using an usual two -wave mixing architecture , we can calculate the optical field for the probe beam at the output of the crystal at time t, in case of the high pump to signal intensity ratio :

( ) ( ) ( ) ( ) ( ) [ ] 1 0 2 + = + L t i / L s s n e e e E t E , (1)
where n and are the photorefractive gain due to the index and absorption gratings respectively, is the absorption of the material, (t) is the phase modulation carried by the probe beam, and L is the length of the crystal. The photorefractive gains n, , is related to the amplitude of the index and the absorption gratings n and as :

= n i n 2 and 2 = . (2)
In the index grating definition, the -i term deals with the -/2 phase shift acquired by the diffracted beam due to the diffraction on a phase grating. It adds to the phase shift between the space charge field grating and illumination pattern p, which equals to /2 when only the diffusion contributes to the carrier migration. We can then define a new phase shift 0= p-/2. For the absorption grating, this additional phase shift does not exist and the phase shift is only due to the nature of the absorption grating. In the case of PR-MQW the absorption grating is an electro-absorption grating that is in phase with space charge field grating and the phase shift is equal to p (= 0+ /2). This point differs from the case of bulk photorefractive materials in which the absorption grating is due to a population grating in the deep traps and that is /2 phase shifted with the space charge field grating. The electro-absorption nature of the gratings (both index and absorption) has another consequence linked to the quadratic nature of the electro-absorption phenomenon. The index and the absorption grating are proportional to the applied field and the space charge field : E•Esc. Thus the photorefractive gains n, , which are complex quantities, write as :

( )

0 ' = + = i n n n n e E E " i , (3) and
( ) ( ) 2 0 ' / i e E E " i + = + = . ( 4
)
Knowing the symmetry of the phase shift 0, we can deduce the symmetries of the gain regarding the sign of the applied field. The applied field induces a displacement of the space charge field from its /2 phase shifted position at zero field, in a direction that depends on the sign of the electric field. That leads to the symmetry relation : 0(-E)=-0(E), from which we deduce :

n ' E ( ) = n ' E ( ) = E E ( ) n cos 0 n " E ( )= n " E ( )= E E ( ) n sin 0 , (5)
and, ' E

( ) = ' E ( ) = E E ( ) sin 0 " E ( )= " E ( )= E E ( ) cos 0 . ( 6
)
These symmetry relations will be the main point that will allow the independent extraction of the real and imaginary parts of the gains from the experimental data.

From Eq. ( 1), we calculate the intensity at the output of the crystal with the assumption of the small phase modulation ( (t)<< /2) [10] :

( ) ( ) ( ) ( ) [ ] ( ) { } t L " sin e e e I t I n L L x s s n n + = + + " 2 ' ' ' ' 2 0 ,. (7)
This is a classical relation in which we simply add the absorption grating term. The gains n, depends on the applied electric field, as well as the absorption of the structure, that has to be independently determined.

Experimentally, we applied on the probe beam a known sinusoidal phase modulation t ( ) = 2 RMS sin t ( ) , and measure its transformation into an intensity modulation through the written grating. The detected signal is thus given by (using (7)) :

( ) ( ) ( ) ( ) t sin E I E I t I RMS M s + = 2 , (8)
where we also consider the energy transfer from the pump beam to the probe beam, that causes a change in the mean intensity I M (E) of the probe beam. The mean probe intensity also changes due to the electroabsorption, that is determined with the same experimental setup by a simple measurement of the mean probe intensity with cross polarized beams (i.e. without grating):

I (E) = I s0 e E ( )L . ( 9
) Comparing the Eq. ( 7) and ( 8), we obtain, using relation (9) :

I M E ( ) = I (E)e 2 ' E ( )+ ' E ( ) ( ) L , (10)
I RMS (E) = 2 RMS I (E)I M (E) sin n "(E) + "(E) ( )L [ ] .
(11) Now using the symmetry relation of the gains, we can easily show that a measurement of these three quantities ( I (E) , I M (E) , I RMS (E)) with equal positive and negative values of the applied electric field allows to determine :

n ' E ( ) + ' E ( ) [ ] L = 1 2 Log I M E
( ) ( )

I (E) n ' E ( )+ ' E ( ) [ ] L = 1 2 Log I M E ( ) I ( E) (12)
2 RMS I (E)I M (E) n "(E) "(E) [ ] L = arcsin I RMS E ( ) 2 RMS I ( E)I M ( E) (13)
from which the real and imaginary part of the two gains are easily calculated. We evaluated the real and imaginary components of index and absorption gratings with the ultrasonic detection measurement and the method above. The samples were two InGaAs/GaAs PR-MQWs devices. These two samples named sample 1 and 2 have the same structure and made through same procedure as described in ref [8]. They have 100 periods of In 0.25 Ga 0.75 As wells and GaAs barriers whose thicknesses are 10 nm and 5 nm respectively. After the sample growth by metal-organic chemical-vapor deposition, 3 MeV protons are irradiated to make samples semi-insulating. Au-Ge alloy electrodes, whose separation is 1 mm, are deposited upon the MQWs. The samples operating in the Franz-Keldysh geometry [START_REF] Nolte | Photorefractive Effects and Materials[END_REF], in which the external field is parallel to the MQW layers.

Experimental setup was based on two-wave mixing configuration. Probe beam was s-polarized. Pump beam was s-polarized for the measurement of I M (E) and I RMS (E) in Eq. ( 8), and was p-polarized to measure I (E) defined in Eq. ( 9). Pump beam covered the entire region of gap of the two electrodes, which is 1 mm, and the intensity was 20 mW/cm 2 . Probe beam was issued from a scattering target imaged just in front of the sample giving a spot radius on the sample of 0.5 mm. The pump probe intensity ratio was 100 approximately. Grating spacing was 3 µm, which is slightly larger than the cut off grating period [8]. We applied the external field for 50 ms every 1 second.

The measured values for I M (E), I (E) normalized to the probe intensity I s0 when no external field was applied are shown in Fig. 1. The normalized output intensity without grating, I (E) has the expected quadratic shape due to the change of the electro-absorption proportional to the square of the electric field. Electro-absorption was decreased when the field is applied and the amount of the change was almost the same for both samples. The mean intensity with grating I M (E) contains the effect of electro-absorption and energy transfer due to the two-wave mixing (and to real part of the index and the absorption gains). The behaviour of the two tested sample is very different. The sign of the energy transfer changed with the change of the direction of the field in sample 1, whereas the sign does not change in sample 2. This result implies that the index grating (real part is an odd function of the field) is dominant for sample 1 and that the absorption grating (even function) is dominant for sample 2.

Root mean square (RMS) values of the AC part of the output probe beams I RMS for the samples 1 and 2 are shown in Fig. 2. The amplitudes of the phase modulation in RMS, RMS were 67.9 and 282 mrad for the measurement of sample 1 and 2 respectively. Sample 2 requires larger phase modulation in order to have enough signal for the measurement due to its relatively low sensitivity. Signs of the output signal were opposite for the reversed field for sample 2, on the contrary to the sample 1. Output signal was very small when the applied field was near 10 kV for sample 1 and from zero to -5 kV for the sample 2. It corresponds to the field where

0 = + ) ( " ) ( " E E n
. The output signal contains small 2 component in these points, especially in the case of the measurement of sample 2 because of large phase modulation. Only component was extracted by Fourier analysis of the signal for the data shown in Fig. 2. Also, the noise component of each data point is compensated, under the assumption that the RMS value of the output for E=0 comes from the noise component only.

Real and imaginary parts of the index and absorption gratings are calculated from these data by Eqs. (12) and (13). Both index and absorption gratings existed in sample 1 and almost only absorption grating were seen in sample 2, especially the contribution of the index grating to the phase detection is quite small because n "L is almost zero. From the results of I (E) shown is Fig. 1, the sign and the magnitude of absorption change is the same for both samples, and these results are consistent with the results shown in Fig. 3, which show similar values of the absorption gain in both samples. Imaginary parts of the index and absorption gratings have same amplitude and opposite sign for sample 1 as shown in Fig 3(a), these two gratings works cooperatively under the negative field and destructively for the positive field. These results well agree with the results shown in Fig. 2 becomes small for the positive field.

From the experimental data we can also estimate the phase shift 0, using equations ( 5) and (6). For an applied electric field around 10kV.cm -1 , we deduce for sample 1 a phase shift 0=-0.8 ±0.1 for both index and absorption gratings. In the same condition we calculate for sample 2, 0=-0.8 ±0.1 for the absorption grating and 0=0 for the index grating. The difference in the value of the phase shift for the index grating in the two samples corresponds to a change of the sign of the index grating variation with the applied electric field. The origin of this difference is mainly due to a shift of the excitonic absorption peak compared to the fixed used wavelength. We operate close to a minimum value of the index change and the slight change of the exciton wavelength is sufficient to cross the zero level of the index change with wavelength. This corresponds to a value of the excitonic spectral phase, as defined by Nolte et al. [7], close to zero and that changes its sign between the two samples.

In the two samples, the phase shift of the space charge field grating p, is the same and is close to /2. This parameter linked to the carrier migration mechanism in PR-MQW, is expected to be identical for every sample, if we suppose that they have similar value of the trap density.

In conclusion, we have introduced a new method to evaluate the photorefractive properties of PR-MQWs at fixed wavelength which is used in the laser ultrasonic detection system. The phase and amplitude of the index and absorption gratings is determined from a pair of experimental data for positive and negative applied field without the usage of any spectroscopic measurement. Experimental results were all consistent each other and showed good agreement with the photorefractive properties in PR-MQWs.

  (a), in which the sensitivity

Fig. 1 .

 1 Figure captions

Fig. 2 .Fig. 3 .

 23 Fig. 2. Experimental results of the detected signals I RMS (E), i.e. rms value of the ac component of the output probe intensity, for (a) sample 1, and (b) sample 2. Values are normalized to the output probe intensity with no applied field.

of 13 Friday , June 13, 2003 Elsevier

of 13 Friday , June 13, 2003 Elsevier


est destinée au dépôt

Ferricretes in New Caledonia: synthesis of age constraints by paleomagnetic and radiometric techniques. Implication on the morphogenesis of 'Grande Terre'

Caroline Ricordel-Prognon, Brice Sevin, Florence Quesnel, Johan Yans, Pierre

Maurizot

To cite this version: Although the description of the emplacement and the weathering of the New Caledonia peridotites are well documented, the reconstructionof the planation surfaces formed upon the ultrabasic massifs are poorly documented. Ambiguities remain on the timing and modality of weathering and related supergene nickel ore. The weathering being still active on the peridotites in the current differentiated landscape and tropical climate, the beginning of the ferruginisationis estimated to have occurred between 34 Ma (end of the obduction) or 25 Ma (last granitic intrusions) and present. The data and interpretations presented here are based on paleomagnetic analysis of lateritic ferricretesofGoro and Tiébaghiand radiometric datingsobtain by K/Ar on Mnoxides fromthe Saint-Louis granite weathering profile. In Tiébaghi an age of 25 Ma has been obtained along a section crossing the plateau. Well constrained ages of 0-5 Ma to 25 Ma have been obtained in Goro, highlighting possible stepped paleosurfaces. A consistentage of 18.25 +/-0.6Mahasbeen obtained at Saint-Louis. The Tiébaghi Massif displays a single episode of ferricrete development during latest Oligocene times, whereas the Goro site reveals several episodes of ferricretedevelopment from latest Oligocene to Pliocene-Quaternary times, the weathering of the Saint-Louis granite being recorded in between. These data allowto constrain the post-obduction morphogenesis of New Caledonia since uplift probably prevented regolith development until latest Oligocene times. Moreover, the later uplift and deep erosion of the West Coast klippes during the Early Miocene probably prevented further regolith preservation.

00682038 Ferricretes in New Caledonia: synthesis of age constraints by paleomagnetic and radiometric techniques. Implication on the morphogenesis of 'Grande Terre'

	Caroline RICORDEL-PROGNON MAURIZOT 2 ,	1 , Brice SEVIN	2 , Florence QUESNEL	1 , Johan YANS	3 , Pierre
	1 BRGM (French Geological Survey), Orléans, France, c.prognon@brgm.fr; New Caledonia, Nouméa, New Caledonia; 3 FUNDP -Institution -rue de Bruxelles 61, B-5000 Namur, 2 Geological Survey of Belgium; 4 Centre National de RechercheTechnologique « Nickel et son Environnement Nouméa, New
	Caledonia

Caroline Ricordel-Prognon, Brice Sevin, Florence Quesnel, Johan Yans, Pierre Maurizot. Ferricretes in New Caledonia: synthesis of age constraints by paleomagnetic and radiometric techniques. Implication on the morphogenesis of 'Grande Terre'. 34th International Geological Congress, Aug 2012, Brisbane, Australia. hal-