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.

Source : oai:HAL:hal-01337769v1

Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)

Section: Proceedings

Published on: January 1, 2015

Submitted on: November 21, 2016

Keywords: lattice walks,statistics on words,$q$-integers,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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