Crossings, Motzkin paths and moments

Abstract

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain q-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulas for the moments of the q-Laguerre and the q-Charlier polynomials, in the style of the Touchard-Riordan formula (which gives the moments of some q-Hermite polynomials, and also the distribution of crossings in matchings).Our method mainly consists in the enumeration of weighted Motzkin paths, which are naturally associated with the moments. Some steps are bijective, in particular we describe a decomposition of paths which generalises a previous construction of Penaud for the case of the Touchard-Riordan formula. There are also some non-bijective steps using basic hypergeometric series, and continued fractions or, alternatively, functional equations.

Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

“…There are known formulas for the 2nth moments of H n (x; 1, q) and H n (x; q, q 2 ). For the 2nth moment of H n (x; 1, q), we have the Touchard-Riordan formula which has various proofs, see [2,6,7,[13][14][15]18]:…”

“…There is a well-known bijection ζ : M(2n) → H(2n), see [20] or [7]. For π ∈ M(2n), the corresponding Hermite history ζ(π ) = (μ, H) is defined as follows.…”

A q-analog of Schläfli and Gould identities on Stirling numbers

“…Our goal is to show that both (1) and (2) remains true when S 1 (n, k) and S 2 (n, k) are replaced with natural q-analogs, denoted S 1 [n, k] and S 2 [n, k] (see Theorems 6.1 and 7.1). The q-analog S 2 [n, k] was studied in [10], and the q-analog S 1 [n, k] is new and gives a natural companion to S 2 [n, k]. Indeed, they share similar properties: they can be defined combinatorially in terms of inversions (see Definitions 4.3 and 5.3), and their ordinary generating functions have continued fraction expansions (see Theorems 4.4 and 5.4), from which we can obtain closed formulas in terms of binomial and q-binomial coefficients (see Theorems 4.5 and 5.5).…”

Crossings of signed permutations and $q$-Eulerian numbers of type $B$

“…The theorem follows from the two lemmas below (and a third lemma is needed to prove the second lemma). The first lemma was essentially present in [19]. Lemma 8.2.…”

“…Note that the jth step is of type 3 or 7 if and only if |π i−1 | < π i . This leads us to define an ascent statistic hasc(π) as (19) hasc(π) = 2 × #{ i : 0 ≤ i ≤ n − 1 and |π i | < π i+1 } + neg(π).…”