Matthieu Josuat-Verg Ès
email: matthieu.josuat-verges@univ-mlv.fr

CUMULANTS OF THE q-SEMICIRCULAR LAW, TUTTE POLYNOMIALS, AND HEAPS

The q-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of its moments in terms of matchings where q follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case q = 0 of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier-Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

where [i] q = 1-q i 1-q . For example, m 0 (q) = m 2 (q) = 1, m 4 (q) = 2 + q, and the odd values are 0. The generating function being a Stieltjes continued fraction, m n (q) is the nth moment of a symmetric probability measure on R (at least when 0 ≤ q ≤ 1). An explicit formula for the density w(x) such that m n (q) = x n w(x)dx is given by Szegő [START_REF] Szegő | Ein Beitrag zur Theorie der Thetafunktionen[END_REF]:

w(x) =    1 π √ 1 -q sin θ ∞ n=1 (1 -q n )|1 -q n e 2iθ | 2 if -2 ≤ x √ 1 -q ≤ 2, 0 otherwise,
where θ ∈ [0, π] is such that 2 cos θ = x √ 1q. At q = 0, it is the semicircular distribution with density (2π) -1 √ 4x 2 supported on [-2, 2], whereas at the limit q → 1 it becomes the Gaussian distribution with density (2π) -1/2 e -x 2 /2 . This law is therefore known either as the q-Gaussian or the q-semicircular law. It can be conveniently characterized by its orthogonal polynomials, defined by the relation xH n (x|q) = H n+1 (x|q)+[n] q H n-1 (x|q) together with H 1 (x|q) = x and H 0 (x|q) = 1, and called the continuous q-Hermite polynomials (but we do not insist on this point of view since the notion of cumulant is not particularly relevant for orthogonal polynomials).

The semicircular law is the analogue in free probability of the Gaussian law [START_REF] Hiai | The Semicircle Law, Free Random Variables and Entropy[END_REF][START_REF] Nica | Lectures on the combinatorics of free probability[END_REF]. More generally, the q-semicircular measure plays an important role in noncommutative probability theories [START_REF] Anshelevich | Free infinite divisibility for q-Gaussians[END_REF][START_REF] Blitvić | On the norm of q-circular operators[END_REF][START_REF] Bożejko | q-Gaussian processes: non-commutative and classical aspects[END_REF][START_REF] Bożejko | An example of a generalized brownian motion[END_REF][START_REF] Van Leeuwen | A q-deformation of the Gauss distribution[END_REF][START_REF] Van Leeuwen | An obstruction for q-deformation of the convolution product[END_REF]. This was initiated by Bożejko and Speicher [START_REF] Bożejko | q-Gaussian processes: non-commutative and classical aspects[END_REF][START_REF] Bożejko | An example of a generalized brownian motion[END_REF] who used creation and annihilation operators on a twisted Fock space to build generalized Brownian motions.

The goal of this article is to examine the combinatorial meaning of the classical cumulants k n (q) of the q-semicircular law (we recall the definition in the next section). The first values lead to the observation that k2n (q) = k 2n (q) (q -1) n-1 is a polynomial in q with nonnegative coefficients. For example: k2 (q) = k4 (q) = 1, k6 (q) = q + 5, k8 (q) = q 3 + 7q 2 + 28q + 56.

We actually show in Theorem 3.5 that this k2n (q) can be given a meaning as a generating function of connected matchings, i.e. the same objects that give a combinatorial meaning to the free cumulants of the q-semicircular law. However, the weight function that we use here on connected matching is not as simple as in the case of free cumulants, it is given by the value at (1, q) of the Tutte polynomial of a graph attached to each connected matching, called the crossing graph.

There are various points where the evaluation of a Tutte polynomials has combinatorial meaning, in particular (1, 0), [START_REF] Aigner | A course in enumeration[END_REF][START_REF] Aigner | A course in enumeration[END_REF] and [START_REF] Aigner | A course in enumeration[END_REF][START_REF] Amdeberhan | A new proof of a conjecture by D. Zeilberger about Catalan numbers[END_REF]. In the first and third case (q = 0 and q = 2), they can be used to give an alternative proof of Theorem 3.5. These will be provided respectively in Section 5 and Section 6. The integers k2n (0) were recently considered by Lassalle [START_REF] Lassalle | Two integer sequences related to Catalan numbers[END_REF] who defines them as a sequence simply related with Catalan numbers, and further studied in [START_REF] Amdeberhan | A new proof of a conjecture by D. Zeilberger about Catalan numbers[END_REF]. Being the (classical) cumulants of the semicircular law, it might seem unnatural to consider this quantity since this law belongs to the world of free probability, but on the other side the free cumulants of the Gaussian have numerous properties (see [START_REF] Belinschi | The normal distribution is ⊞-infinitely divisible[END_REF]). The interesting feature is that this particular case q = 0 can be proved via the theory of heaps [START_REF] Cartier | Problèmes combinatoires de commutation et réarrangements[END_REF][START_REF] Viennot | Heaps of pieces I. Basic definitions and combinatorial lemmas[END_REF]. As for the case q = 2, even though the q-semicircular is only defined when |q| < 1 its moments and cumulants and the link between still exist because (1) can be seen as an identity between formal power series in z. The particular proof for q = 2 is an application of the exponential formula.

Preliminaries

Let us first precise some terms used in the introduction. Besides the moments {m n (q)} n≥0 , the q-semicircular law can be characterized by its cumulants {k n (q)} n≥1 formally defined by (1)

n≥1 k n (q) z n n! = log n≥0 m n (q) z n n! ,
or by its free cumulants {c n (q)} n≥1 [START_REF] Nica | Lectures on the combinatorics of free probability[END_REF] formally defined by

C(zM (z)) = M (z) where M (z) = n≥0 m n (q)z n , C(z) = 1 + n≥1 c n (q)z n .
These relations can be reformulated using set partitions.

For any finite set V , let P(V ) denote the lattice of set partitions of V , and let P(n) = P({1, . . . , n}). We will denote by 1 the maximal element and by µ the Möbius function of these lattices, without mentioning V explicitly. Although we will not use it, let us mention that µ(π, 1) = (-1) #π-1 (#π -1)! where #π is the number of blocks in π. See [START_REF] Stanley | Enumerative Combinatorics[END_REF]Chapter 3] for details. When we have some sequence (u n ) n≥1 , for any π ∈ P(V ) we will use the notation:

u π = b∈π u #b .
Then the relations between moments and cumulants read:

(2)

m n (q) = π∈P(n) k π (q), k n (q) = π∈P(n)
m π (q)µ(π, 1).

These are equivalent via the Möbius inversion formula and both can be obtained from (1) using Faà di Bruno's formula. When V ⊂ N, let N C(V ) ⊂ P(V ) denote the subset of noncrossing partitions, which form a sublattice with Möbius function µ N C . Then we have [START_REF] Hiai | The Semicircle Law, Free Random Variables and Entropy[END_REF][START_REF] Nica | Lectures on the combinatorics of free probability[END_REF]:

(3) m n (q) = π∈N C(n) c π (q), c n (q) = π∈N C(n) m π (q)µ N C (π, 1).
Equations ( 2) and ( 3) can be used to compute the first non-zero values:

k 2 (q) = 1, k 4 (q) = q -1, k 6 (q) = q 3 + 3q 2 -9q + 5, c 2 (q) = 1, c 4 (q) = q, c 6 (q) = q 3 + 3q 2 .
Let M(V ) ⊂ P(V ) denote the set of matchings, i.e. set partitions into blocks of size 2. As is customary, a block of σ ∈ M(V ) will be called an arch. When V ⊂ N, a crossing [START_REF] Ismail | The combinatorics of q-Hermite polynomials and the Askey-Wilson integral[END_REF] of σ ∈ M(V ) is a pair of arches {i, j} and {k, ℓ} such that i < k < j < ℓ. Let cr(σ) denote the number of crossings of σ ∈ M(V ). Let N (V ) = M(V ) ∩ N C(V ) denote the set of noncrossing matchings, i.e. those such that cr(σ) = 0. Let also M(2n) = M({1, . . . , 2n}) and N (2n) = N ({1, . . . , 2n}). Let P c (n) ⊂ P(n) denote the set of connected set partitions, i.e. π such that no proper interval of {1, . . . , n} is a union of blocks of π, and let M c (2n) = M(2n) ∩ P c (2n) denote the set of connected matchings.

It is known [START_REF] Ismail | The combinatorics of q-Hermite polynomials and the Askey-Wilson integral[END_REF] that for any n ≥ 0, the moment m 2n (q) count matchings on 2n points according to the number of crossings:

(4) m 2n (q) = σ∈M(2n)
q cr(σ) .

It was showed by Lehner [18] that (3) and (4) gives a combinatorial meaning for the free cumulants:

c 2n (q) = σ∈M c (2n)
q cr(σ) .

See [START_REF] Belinschi | The normal distribution is ⊞-infinitely divisible[END_REF] for various properties of connected matchings in the context of free probability. Let us also mention that both quantities m 2n (q) and c 2n (q) are considered in an article by Touchard [START_REF] Touchard | Sur un problème de configurations et sur les fractions continues[END_REF].

3. A combinatorial formula for k n (q)

We will use the Möbius inversion formula in Equation ( 2), but we first need to consider the combinatorial meaning of the products m π (q). Lemma 3.1. For any σ ∈ M(2n) and π ∈ P(2n) such that σ ≤ π, let cr(σ, π) be the number of crossings ({i, j}, {k, ℓ}) of σ such that {i, j, k, ℓ} ⊂ b for some b ∈ π. Then we have:

(5) m π (q) = σ∈M(2n) σ≤π q cr(σ,π) .

Proof. Denoting σ| b = {x ∈ σ : x ⊂ b}, the map σ → (σ| b ) b∈π is a natural bijection between the set {σ ∈ M(2n) : σ ≤ π} and the product Π b∈π M(b), in such a way that cr(σ, π) = b∈π cr(σ| b ). This allows to factorize the right-hand side in [START_REF] Belinschi | The normal distribution is ⊞-infinitely divisible[END_REF] and obtain m π (q).

From Equation (2) and the previous lemma, we have:

k 2n (q) = π∈P(2n) m π (q)µ(π, 1) = π∈P(2n) σ∈M(2n) σ≤π q cr(σ,π) µ(π, 1) = σ∈M(2n) π∈P(2n) π≥σ q cr(σ,π) µ(π, 1) = σ∈M(2n) W (σ), (6)
where for each σ ∈ M(2n) we have introduced:

(7) W (σ) = π∈P(2n) π≥σ q cr(σ,π) µ(π, 1).
A key point is to note that W (σ) only depends on how the arches of σ cross with respect to each other, which can be encoded in a graph. This leads to the following: Definition 3.2. Let σ ∈ M(2n). The crossing graph G(σ) = (V, E) is as follows. The vertex set V contains the arches of σ (i.e. V = σ), and the edge set E contains the crossings of σ (i.e. there is an edge between the vertices {i, j} and {k, ℓ} if and only if i < k < j < ℓ). See Figure 1 for an example. Note that the graph G(σ) is connected if and only if σ is a connected matching in the sense of the previous section.

We also need the following definition for a general graph.

Definition 3.3. Let G = (V, E) be a graph, and π ∈ P(V ). Then we denote i(E, π) the number of elements in the edge set E such that both endpoints are in the same block of π.

Lemma 3.4. Let σ ∈ M(2n) and G(σ) = (V, E) be its crossing graph. Then we have:

(8) W (σ) = π∈P(V ) q i(E,π) µ(π, 1).
Proof. There is a natural bijection between the interval [σ, 1] in P(2n) and the set P(V ), in such a way that cr(σ, π) = i(E, π). Hence Equation ( 8) is just a rewriting of ( 7) in terms of the graph G(σ).

Now we can use Proposition 4.1 from the next section. It allows to recognize (q -1) -n+1 W (σ) as an evaluation of the Tutte polynomial T G(σ) , except that it is 0 when the graph is not connected.

Gathering Equations ( 6), [START_REF] Bożejko | An example of a generalized brownian motion[END_REF], and Proposition 4.1 from the next section, we have proved:

Theorem 3.5. For any n ≥ 1, k2n (q) = σ∈M c (2n) T G(σ) (1, q).
In particular k2n (q) is a polynomial in q with nonnegative coefficients.

The Tutte polynomial of a connected graph

For any graph G = (V, E), let T G (x, y) denote its Tutte polynomial, we give here a short definition and refer to [START_REF] Aigner | A course in enumeration[END_REF]Chapter 9] for details. This graph invariant can be computed recursively via edge deletion and edge contraction. Let e ∈ E, let G\e = (V, E\e) and G/e = (V /e, E\e) where V /e is the quotient set where both endpoints of the edge e are identified. Then the recursion is:

(9) T G (x, y) =      xT G/e (x, y) if e is a bridge, yT G\e (x, y) if e is a loop, T G/e (x, y) + T G\e (x, y) otherwise.
The initial case is that T G (x, y) = 1 if the graph G has no edge. Here, a bridge is an edge e such that G\e has one more connected component than G, and a loop is an edge with identical endpoints. Proposition 4.1. Let G = (V, E) be a graph (possibly with multiple edges and loops). Let n = #V . We have:

(10) 1 (q -1) n-1 π∈P(V ) q i(E,π) µ(π, 1) = T G (1, q) if G is connected, 0 otherwise.
Proof. Denote by U G the left-hand side in [START_REF] Cartier | Problèmes combinatoires de commutation et réarrangements[END_REF] and let e be an edge of

G. Suppose e ∈ E is a loop, it is then clear that i(E\e, π) = i(E, π) -1, so U G = qU G\e .
Then suppose e is not a loop, and let x and y be its endpoints. We have:

U G -U G\e = 1 (q -1) n-1 π∈P(V ) q i(E,π) -q i(E\e,π) µ(π, 1).
In this sum, all terms where x and y are in different blocks of π vanish. So we can keep only π such that x and y are in the same block, and these can be identified with elements of P(V /e) and satisfy i(E\e, π) = i(E, π) -1. We obtain:

U G -U G\e = 1 (q -1) n-2 π∈P(V /e) q i(E\e,π) µ(π, 1) = U G/e .
This is a recurrence relation which determines U G , and it remains to describe the initial case. So, suppose the graph G has n vertices and no edge, i.e. G = (V, ∅). We have i(∅, π) = 0. By the definition of the Möbius function, we have:

π∈P(V ) µ(π, 1) = δ n1 ,
hence U G = δ n1 as well in this case.

We have thus a recurrence relation for U G , and it remains to show that the righthand side of (10) satisfies the same relation. This is true because when x = 1, and when we consider a variant of the Tutte polynomial which is 0 for a non-connected graph, then the first case of (9) becomes a particular case of the third case.

Remark 4.2. The proposition of this section can also be derived from results of Burman and Shapiro [START_REF] Burman | Around matrix-tree theorem[END_REF], at least in the case where G is connected. More precisely, in the light of [START_REF] Burman | Around matrix-tree theorem[END_REF]Theorem 9] we can recognize the sum in the left-hand side of [START_REF] Cartier | Problèmes combinatoires de commutation et réarrangements[END_REF] as the external activity polynomial C G (w), where all edge variables are specialized to q -1. It is known to be related with T G (1, q), see for example [24, Section 2.5].

5.

The case q = 0, Lassalle's sequence and heaps

In the case q = 0, the substitution z → iz recasts Equation ( 1) as [START_REF] Fédou | Sur les fonctions de Bessel[END_REF] -

log n≥0 (-1) n C n z 2n (2n)! = n≥1 k2n (0) z 2n (2n)! , where C n = 1 n+1 2n
n is the nth Catalan number, known to be the cardinal of N (2n), see [START_REF] Stanley | Enumerative Combinatorics[END_REF]. The integer sequence { k2n (0)} n≥1 = (1, 1, 5, 56, . . . ) was previously defined by Lassalle [START_REF] Lassalle | Two integer sequences related to Catalan numbers[END_REF] via an equation equivalent to [START_REF] Fédou | Sur les fonctions de Bessel[END_REF], and Theorem 1 from [START_REF] Lassalle | Two integer sequences related to Catalan numbers[END_REF] states that the integers k2n (0) are positive and increasing (stronger results are also true, see [START_REF] Lassalle | Two integer sequences related to Catalan numbers[END_REF][START_REF] Amdeberhan | A new proof of a conjecture by D. Zeilberger about Catalan numbers[END_REF]).

The goal of this section is to give a meaning to [START_REF] Fédou | Sur les fonctions de Bessel[END_REF] in the context of the theory of heaps [START_REF] Viennot | Heaps of pieces I. Basic definitions and combinatorial lemmas[END_REF] [START_REF] Cartier | Problèmes combinatoires de commutation et réarrangements[END_REF]Appendix 3]. This will give an alternative proof of Theorem 3.5 for the case q = 0, based on a classical result on the evaluation T G (1, 0) of a Tutte polynomial in terms of some orientations of the graph G. Definition 5.1. A graph G = (V, E) is rooted when it has a distinguished vertex r ∈ V , called the root. An orientation of G is root-connected, if for any vertex v ∈ V there exists a directed path from the root to v. Proposition 5.2 (Greene & Zaslavsky [START_REF] Greene | On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs[END_REF]). If G is a rooted and connected graph, T G (1, 0) is the number of its root-connected acyclic orientations.

The notion of heap was introduced by Viennot [START_REF] Viennot | Heaps of pieces I. Basic definitions and combinatorial lemmas[END_REF] as a geometric interpretation of elements in the Cartier-Foata monoid [START_REF] Cartier | Problèmes combinatoires de commutation et réarrangements[END_REF], and has various applications in enumeration. We refer to [10, Appendix 3] for a modern presentation of this subject (including a comprehensive bibliography).

Let M be the monoid built on the generators (x ij ) 1≤i<j subject to the relations

x ij x kℓ = x kℓ x ij if i < j < k < ℓ or i < k < ℓ < j.
We call it the Cartier-Foata monoid (but in other contexts it could be called a partially commutative free monoid or a trace monoid as well). Following [START_REF] Viennot | Heaps of pieces I. Basic definitions and combinatorial lemmas[END_REF], we call an element of M a heap.

Any heap can be represented as a "pile" of segments, as in the left part of Figure 2 (this is remindful of [START_REF] Bousquet-Mélou | Empilements de segments et q-énumération de polyominos convexes dirigés[END_REF]). This pile is described inductively: the generator x ij correspond to a single segment whose extremities have abscissas i and j, and multiplication m 1 m 2 is obtained by placing the pile of segments corresponding to m 2 above the one corresponding to m 1 . In terms of segments, the relation x ij x kℓ = x kℓ x ij if i < j < k < ℓ has a geometric interpretation: segments are allowed to move vertically as long as they do not intersect (this is the case of x 34 and x 67 in Figure 2). Similarly, the other relation x ij x kℓ = x kℓ x ij if i < k < ℓ < j can be treated by thinking of each segment as the projection of an arch as in the central part of Figure 2. In this three-dimensional representation, all the commutation relations are translated in terms of arches that are allowed to move along the dotted lines as long as they do not intersect.

A heap can also be represented as a poset. Consider two segments s 1 and s 2 in a pile of segments, then the relation is defined by saying that s 1 < s 2 if s 1 is always below s 2 , after any movement of the arches (along the dotted lines and as long as they do not intersect, as above). This way, a heap can be identified with a poset where each element is labeled by a generator of M , and two elements whose labels do not commute are comparable. See the right part of Figure 2 for an example and [10, Appendice 3] for details. Note that M • contains the neutral element of M so that the sum in the left-hand side is invertible, being a formal power series with constant term equal to 1. Definition 5.4. An element m ∈ M is called a pyramid if the associated poset has a unique maximal element. Let P ⊂ M denote the subset of pyramids.

A fundamental result of the theory of heaps links the generating function of pyramids with the one of all heaps [START_REF] Cartier | Problèmes combinatoires de commutation et réarrangements[END_REF][START_REF] Viennot | Heaps of pieces I. Basic definitions and combinatorial lemmas[END_REF]. It essentially relies on the exponential formula for labeled combinatorial objects, and reads: [START_REF] Gioan | Enumerating degree sequences in digraphs and a cycle-cocycle reversing system[END_REF] log

m∈M m = comm p∈P 1 |p| p,
where the sign = comm means that the equality holds in any commutative quotient of Z[[M ]]. Combining ( 12) and ( 13), we obtain:

(14) -log m∈M • (-1) |m| m = comm p∈P 1 |p| p.
Now, let us examine how to apply this general equality to the present case.

The following lemma is a direct consequence of the definitions, and permits to identify trivial heaps with noncrossing matchings.

Lemma 5.5. The map [START_REF] Hiai | The Semicircle Law, Free Random Variables and Entropy[END_REF] Φ :

x i1j1 • • • x injn → {{i 1 , j 1 }, . . . , {i n , j n }}
defines a bijection between the set of trivial heaps M • and the disjoint union of N (V ) where V runs through the finite subsets (of even cardinal) of N >0 .

For a general heap m ∈ M , we can still define Φ(m) via (15) but it may not be a matching, for example Φ(x 1,2 x 2,3 ) = {{1, 2}, {2, 3}}. Let us first consider the case of m ∈ M such that Φ(m) is really a matching. Lemma 5.6. Let σ ∈ M(V ) for some V ⊂ N >0 . Then the heaps m ∈ M such that Φ(m) = σ are in bijection with acyclic orientations of G(σ). Thus, such a heap m ∈ M can be identified with a pair (σ, r) where r is an acyclic orientation of the graph G(σ).

Proof. An acyclic orientation r on G(σ) defines a partial order on σ by saying that two arches x and y satisfy x < y if there is a directed path from y to x. In this partial order, two crossing arches are always comparable since they are adjacent in G(σ). We recover the description of heaps in terms of posets, as described above, so each pair (σ, r) corresponds to a heap m ∈ M with Φ(m) = σ.

To treat the case of m ∈ M such that Φ(m) is not a matching, such as x 12 x 23 , we are led to introduce a set of commuting variables (a i ) i≥1 such that a 2 i = 0, and consider the specialization x ij → a i a j which defines a morphism of algebras

ω : Z[[M ]] → Z[[a 1 ,
]. This way, for any m ∈ M we have either ω(m) = 0, or Φ(m) ∈ M(V ) for some V ⊂ N >0 .

Let m ∈ M such that ω(m) = 0. As seen in Lemma 5.6, it can be identified with the pair (σ, r) where σ = Φ(m), and r is an acyclic orientation of G(σ). Then the condition defining pyramids is easily translated in terms of (σ, r), indeed we have m ∈ P if and only if the acyclic orientation r has a unique source (where a source is a vertex having no ingoing arrows).

Under the specialization ω, the generating function of trivial heaps is: ( 16)

ω m∈M • (-1) |m| m = n≥0 (-1) n C n e 2n ,
where e 2n is the 2nth elementary symmetric functions in the a i 's. Indeed, let V ⊂ N >0 with #V = 2n, then the coefficient of i∈V a i in the left-hand side of ( 16) is (-1) n #N (V ) = (-1) n C n , as can be seen using Lemma 5.5. In particular, it only depends on n so that this generating function can be expressed in terms of the e 2n . Moreover, since the variables a i have vanishing squares their elementary symmetric functions satisfy

e 2n = 1 (2n)! e 2n 1 ,
so that the right-hand side of ( 16) is actually the exponential generating of the Catalan numbers (evaluated at e 1 ). It remains to understand the meaning of taking the logarithm of the left-hand side of ( 16) using pyramids and Equation [START_REF] Greene | On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs[END_REF]. Note that the relation = comm becomes a true equality after the specialization x ij → a i a j . So taking the image of ( 14) under ω and using [START_REF] Ismail | The combinatorics of q-Hermite polynomials and the Askey-Wilson integral[END_REF], this gives

-log n≥0 (-1) n C n e 2n = p∈P 1 |p| ω(p).
The argument used to obtain [START_REF] Ismail | The combinatorics of q-Hermite polynomials and the Askey-Wilson integral[END_REF] shows as well that the right-hand side of the previous equation is

xn n e 2n where x n = #{p ∈ P :

ω(p) = a 1 • • • a 2n }. So we have -log n≥0 (-1) n C n e 2n = n≥0 x n n e 2n ,
and comparing this with [START_REF] Fédou | Sur les fonctions de Bessel[END_REF], we obtain k2n (0) = xn n . Clearly, a graph with an acyclic orientation always has a source, and it has a unique source only when it is root-connected (for an appropriate root, viz. the source). So a pyramid p such that ω(p) = 0 can be identified with a pair (σ, r) where r is a root-connected acyclic orientation of G(σ). Then using Proposition 5.2, it follows that

x n = n σ∈M c (2n) T G(σ) (1, 0).
Here, the factor n in the right-hand side accounts for the n possible choices of the source in each graph G(σ). Eventually, we obtain

(17) k2n (0) = σ∈M c (2n)
T G(σ) (1, 0), i.e. we have proved the particular case q = 0 of Theorem 3.5. Let us state again the result in an equivalent form. We can consider that if σ ∈ M(2n), the graph G(σ) has a canonical root which the arch containing 1. Then, Equation ( 17) gives a combinatorial model for the integers k2n (0): Theorem 5.7. The integer k2n (0) counts pairs (σ, r) where σ ∈ M c (2n), and r is an acyclic orientation of G(σ) whose unique source is the arch of σ containing 1.

From this, it is possible to give a combinatorial proof that the integers k2n (0) are increasing, as suggested by Lassalle [START_REF] Lassalle | Two integer sequences related to Catalan numbers[END_REF] who gave an algebraic proof. Indeed, we can check that pairs (σ, r) where {1, 3} is an arch of σ are in bijection with the same objects but of size one less, hence k2n (0) ≤ k2n+2 (0).

Before ending this section, note that the left-hand side of ( 11) islog( 1 z J 1 (2z)) where J 1 is the Bessel function of order 1. There are quite a few other cases where the combinatorics of Bessel functions is related with the theory of heaps, see the articles of Fédou [START_REF] Fédou | Sur les fonctions de Bessel[END_REF][START_REF] Fédou | Combinatorial objects enumerated by q-Bessel functions[END_REF], Bousquet-Mélou and Viennot [START_REF] Bousquet-Mélou | Empilements de segments et q-énumération de polyominos convexes dirigés[END_REF].

6. The case q = 2, the exponential formula

The specialization at (1, 2) of a Tutte polynomial has combinatorial significance in terms of connected spanning subgraphs (see [START_REF] Aigner | A course in enumeration[END_REF]Chapter 9]), so it is natural to consider the case q = 2 of Theorem 3.5. This case is particular because the factor (q -1) n-1 disappears, so that k2n (2) = k 2n (2). We can then interpret the logarithm in the sense of combinatorial species, by showing that k2n (2) counts some primitive objects and m 2n (2) counts assemblies of those, just like permutations that are formed by assembling cycles (this is the exponential formula for labeled combinatorial objects, see [START_REF] Aigner | A course in enumeration[END_REF]Chapter 3]). What we obtain is another more direct proof of Theorem 3.5, based on an interpretation of T G (1, 2) as follows. Proposition 6.1 (Gioan [START_REF] Gioan | Enumerating degree sequences in digraphs and a cycle-cocycle reversing system[END_REF]). If G is a rooted and connected graph, T G (1, 2) is the number of its root-connected orientations.

This differs from the more traditional interpretation of T G (1, 2) in terms of connected spanning subgraphs mentioned above, but it is what naturally appears in this context. Definition 6.2. Let M + (2n) be the set of pairs (σ, r) where σ ∈ M(2n) and r is an orientation of the graph G(σ). Such a pair is called an augmented matching, and is depicted with the convention that the arch {i, j} lies above the arch {k, ℓ} if there is an oriented edge {i, j} → {k, ℓ}, and behind it if there is an oriented edge {k, ℓ} → {i, j} . See Figure 3 for example. Clearly, #M + (2n) = m 2n (2). Indeed, each graph G(σ) = (V, E) has 2 #E orientations, and #E = cr(σ), so this follows from [START_REF] Bousquet-Mélou | Empilements de segments et q-énumération de polyominos convexes dirigés[END_REF].

Notice that if there is no directed cycle in the oriented graph (G(σ), r), the augmented matching (σ, r) can be identified with a heap m ∈ M as defined in the previous section. The one in Figure 3 would be x 3,5 x 4,11 x 10,12 x 1,6 x 7,9 x 2,8 . Actually, the application of the exponential formula in the present section is quite reminiscent of the link between heaps and pyramids as seen in the previous section. Definition 6.3. Recall that each graph G(σ) is rooted with the convention that the root is the arch containing 1. Let I(2n) ⊂ M + (2n) be the set of augmented matchings (σ, r) such that σ is connected and r is a root-connected orientation of T G(σ) [START_REF] Aigner | A course in enumeration[END_REF][START_REF] Amdeberhan | A new proof of a conjecture by D. Zeilberger about Catalan numbers[END_REF], so that the particular case q = 2 of Theorem 3.5 is the equality #I(2n) = k 2n (2).

To prove this from (1) and using the exponential formula, we have to see how an augmented matching can be decomposed into an assembly of primitive ones, as stated in Proposition 6.4 below. This decomposition thus proves the case q = 2 of Theorem 3.5. Note also that the bijection given below is equivalent to the first identity in (2).

Proposition 6.4. There is a bijection

M + (2n) -→ π∈P(n) V ∈π I(V ).
Proof. Let (σ, r) ∈ M + (2n), the bijection is defined as follows. Consider the vertices of G(σ) which are accessible from the root. This set of vertices defines a matching on a subset V 1 ⊂ {1, . . . , 2n}. For example, in the case in Figure 3, the root is {1, 6} and the only other accessible vertex is {2, 8}, so V 1 = {1, 2, 6, 8}. Together with the restriction of the orientation r on this subset of vertices, this defines an augmented matching (σ 1 , r 1 ) ∈ M + (V 1 ) which by construction is primitive. By repeating this operation on the set {1, . . . , 2n}\V 1 , we find V 2 ⊂ {1, . . . , 2n}\V 1 and (σ 2 , r 2 ) ∈ I(V 2 ), and so on. See Figure 4 for the result, in the case of the augmented matching in Figure 3. The inverse bijection is easily described.

If (σ i , r i ) ∈ I(V i ) for any 1 ≤ i ≤ k where π = {V 1 , . . . , V k }, let σ = σ 1 ∪ • • • ∪ σ k ,
and the orientation r of G(σ) is as follows. Let e be an edge of G(σ) and x 1 , x 2 be its endpoints, with x 1 ∈ σ j1 and x 2 ∈ σ j2 . If j 1 = j 2 , the edge e is oriented in accordance with the orientation r j1 = r j2 . Otherwise, say j 1 < j 2 , then the edge e is oriented in the direction x 1 ← x 2 .

Cumulants in terms of free cumulants

In this section, {µ n } n≥0 is any sequence of moments, and {k n } n≥1 and {c n } n≥1 are the corresponding cumulants and free cumulants defined by ( 2) and (3). It was shown by Lehner [18] that (2) and ( 3) imply ( 18)

c n = π∈P c (n) k π .
By a "triangularity" argument, this can clearly be inverted to have k n in terms of c 1 , c 2 , etc. But it cannot be done using a Möbius function as in ( 2) and ( 3), because in this case we have that

c π = ρ∈P c (n) ρ≤π k ρ
in general, contrary to:

m π = ρ∈P(n) ρ≤π k ρ if π ∈ P(n), m π = ρ∈N C(n) ρ≤π c ρ if π ∈ N C(n).
Still, we obtain an explicit inverse formula for [START_REF] Lehner | Free cumulants and enumeration of connected partitions[END_REF] in the theorem below.

For any set partition π ∈ P(V ) for some V ⊂ N, we can define a crossing graph G(π), whose vertices are the blocks of π, and there is an edge between b, c ∈ π if {b, c} is not a noncrossing partition. Note that π is connected if and only if the graph G(π) is connected. The two different proofs for the semicircular cumulants show as well the following: Theorem 7.1. For any n ≥ 1, we have:

k n = π∈P c (n) c π (-1) 1+#π T G(π) (1, 0).
Let us sketch the proofs. If π ∈ P(n), similar to Lemma 3.1 we have:

m π = b∈π ρ∈N C(b) c ρ = ρ∈P(n) ρ¢π c ρ
where the relation ρ ¢ π means that ρ ≤ π and ρ| b is a noncrossing partition for each b ∈ π. Indeed the map ρ → (ρ| b ) b∈π is a bijection between {ρ ∈ P(n) : ρ ¢ π} and b∈π N C(b). The same computation as in ( 6) and ( 7) gives ( 19)

k n = π∈P(n) m π µ(π, 1) = ρ,π∈P(n) ρ¢π c ρ µ(π, 1) = ρ∈P(n) c ρ W (ρ), where W (ρ) = π∈P(n) ρ¢π µ(π, 1).
Denoting G(ρ) = (V, E) the crossing graph of ρ, the previous equality is rewritten W (ρ) = µ(π, 1) where the sum is over π ∈ P(V ) such that for any b ∈ π, e ∈ E, the block b does not contain both endpoints of the edge e. Then the case q = 0 of Proposition 4.1 shows that

W (ρ) = (-1) 1+#ρ T G(ρ) (1, 0) if ρ ∈ P c (n), 0 otherwise.
Together with [START_REF] Van Leeuwen | A q-deformation of the Gauss distribution[END_REF], this completes the first proof of Theorem 7.1.

As for the second proof, we follow the outline of Section 5, but with another definition for M , M • , P and ω. Let M be the monoid with generators (x V ) where V runs through finite subsets of N >0 , and with relations x V x W = x W x V if {V, W } is a noncrossing partition. We also denote M • ⊂ M the corresponding set of trivial heaps, i.e. products of pairwise commuting generators, and an element We still understand that V ⊂ N >0 is finite, (a i ) i≥1 are commuting variables with vanishing squares, and e n is the nth elementary symmetric function in the a i 's. Equation ( 14) is still valid as such with the new definition of M • and P , and taking the image by ω gives:

x b1 • • • x bi ∈ M • is identified
log

Final remarks

It would be interesting to explain why the same combinatorial objects appear both for c 2n (q) and k 2n (q). This suggests that there exists some quantity that interpolates between the classical and free cumulants of the q-semicircular law, however, building a noncommutative probability theory that encompasses the classical and free ones appear to be elusive (see [START_REF] Van Leeuwen | An obstruction for q-deformation of the convolution product[END_REF] for a precise statement). It means that building such an interpolation would rely not only on the q-semicircular law and its moments, but on its realization as a noncommutative random variable. This might be feasible using q-Fock spaces [START_REF] Bożejko | q-Gaussian processes: non-commutative and classical aspects[END_REF][START_REF] Bożejko | An example of a generalized brownian motion[END_REF] but is beyond the scope of this article.

Figure 1 .

 1 Figure 1. A matching σ and its crossing graph G(σ).

Figure 2 .

 2 Figure 2. The heap m = x 46 x 67 x 34 x 16 x 47 as a pile of segments and the Hasse diagram of the associated poset.

Figure 3 .

 3 Figure 3. An augmented matching (σ, r) and the corresponding orientation of G(σ).

Figure 4 .

 4 Figure 4. Decomposition of an augmented matching into primitive ones.

  with {b 1 , . . . , b i } ∈ N C(∪ i j=1 b j ). The subset P ⊂ M is characterized by Definition 5.4. Now, we consider the morphism ω defined onZ[[M ]] by ω(x V ) = -c #V i∈V a i .

1 )

 1 By extracting the coefficient of a 1 • • • a n in both sides, we get[START_REF] Van Leeuwen | An obstruction for q-deformation of the convolution product[END_REF] -k n a 1 • • • a n = 1 |p| ω(p)where the sum is over p ∈ P that can be writtenp = x V1 • • • x V k and π = {V 1 , . . . , V k } ∈ P(n).We can gather the terms corresponding to a same π. Note thatω(p) = (-1) #π c π a 1 • • • a n . From (20), we obtain (21) k n a 1 • • • a n = #π c π a 1 • • • a nwhere x π is the number of p ∈ P that are a product of x b where b runs through the blocks of π. Following the ideas in Section 5, such a p is characterized by π together with an acyclic orientation of the graph G(π) having a unique source. So x π is the number of such orientations, which is equal to #π • T G(π) (1, 0) if π is connected and 0 otherwise. Knowing the value of x π , from Equation (21) we complete the second proof of Theorem 7.1.

Acknowledgment

This work was initiated during the trimester "Bialgebras in Free Probability" at the Erwin Schrödinger Institute in Vienna. In particular I thank Franz Lehner, Michael Anshelevich and Natasha Blitvić for their conversation.