A q-rious positivity

Abstract

The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of $\qbinom{n}{m}$. In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of $q$-factorials that happen to be polynomials.

6 pages

References in corpus (4)

• The generalized Borwein conjecture. II. Refined q-trinomial coefficients
• Factorial ratios, hypergeometric series, and a family of step functions
• Factors of Alternating Sums of Products of Binomial and q-Binomial Coefficients
• Critère pour l'intégralité des coefficients de Taylor des applications miroir

Cited by in corpus (13)

• Congruences for $q$-binomial coefficients
• Some divisibility properties of binomial and q-binomial coefficients
• Similarity of matrices over local rings of length two
• Complexity problems in enumerative combinatorics
• Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series
• Factors of sums and alternating sums involving binomial coefficients and powers of integers
• On super Catalan polynomials
• Factors of sums involving $q$-binomial coefficients and powers of $q$-integers
• Proof of Sun's conjecture on the divisibility of certain binomial sums
• Factors of sums and alternating sums of products of $q$-binomial coefficients and powers of $q$-integers
• $q$-rious and $q$-riouser
• $q$-Super Catalan Numbers: Combinatorial identities, Generating Functions, and Narayana Refinements
• Some observations about super Catalan numbers, corresponding orthogonal polynomials, and their q-analogues