Statistics on Lattice Walks and q-Lassalle Numbers

Lenny Tevlin

doi:10.46298/dmtcs.2528

Abstract

This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

Highlights

Introduction toLassalle’s Sequences and their q-analogs.Michel Lassalle [Las12] has discussed two related sequences of numbers Ak and αk. {Ak} is generated by the following recurrence: n−1An = (−1)n−1Cn + (−1)n−j−1 2n − 1 2j − 1 Aj Cn−j, j=1 A1 = 1The second sequence is αn =1365–8050 c 2015 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, FranceHe proved that both An and αn are positive integers and each sequence is increasing. (It turns out that the second sequence is related to power sums of zeros of Bessel function J0(z))
To understand the positivity of (4) combinatorially, i.e. as a generating function of certain weighted lattice walks, I first interpret the q-version of the generating function of all NSEW walks as a generating function of a certain inversion statistics on lattice walks
I continue with the lattice walk part of the story

Summary

Introduction to q-enumeration of Lattice walks

To understand the positivity of (4) combinatorially, i.e. as a generating function of certain weighted lattice walks, I first interpret the q-version of the generating function of all NSEW walks as a generating function of a certain (new) inversion statistics on lattice walks. The total number of lattice walks from (0, 0) to (c, d) of length n is given by [DR84]. Denote the set of all lattice walks from (a, b) to (c, d) in n letters (steps) by Pn((a, b) | (c, d)). The q-analog of the walk enumeration formula is the following generating function. The purpose of this section is to derive a bilinear recursion relations for Ak(q) and αk(q), from which the positivity and integrality follow. As may be seen from (6), the positivity and integrality of cn,k(q) follows from its combinatorial interpretation as the generating function of winv statistics of upper half-plane lattice walks, namely the walks that start at (0, 0) and end up at (2k − n − 1, 1) in (n − 1) steps. I continue with the lattice walk part of the story

Walk Inversion Generating Function

Upper Half-Plane walks

Major Walk Index