Negative qq-Stirling numbers

The notion of the negative $q$-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative $q$-binomial, we show the classical $q$ -Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $(1+q)$. We extend this enumerative result via a decomposition of the Stirling poset, as well as a homological version of Stembridge’s $q=-1$ phenomenon. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting $t=1+q$ we give a bijective combinatorial argument à la Viennot showing the $(q; t)$-Stirling numbers of the first and second kind are orthogonal

Negative qq-Stirling numbers

International audienceThe notion of the negative $q$-binomial was recently introduced by Fu, Reiner, Stanton and Thiem. Mirroring the negative $q$-binomial, we show the classical $q$ -Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $(1+q)$. We extend this enumerative result via a decomposition of the Stirling poset, as well as a homological version of Stembridge’s $q=-1$ phenomenon. A parallel enumerative, poset theoretic and homological study for the $q$-Stirling numbers of the first kind is done beginning with de Médicis and Leroux’s rook placement formulation. Letting $t=1+q$ we give a bijective combinatorial argument à la Viennot showing the $(q; t)$-Stirling numbers of the first and second kind are orthogonal.La notion de la $q$-binomial négative était introduite par Fu, Reiner, Stanton et Thiem. Réfléchissant la $q$-binomial négative, nous démontrons que les classiques $q$-nombres de Stirling de deuxième espèce peuvent être exprimés comme une paire de statistiques sur un sous-ensemble des mots de croissance restreinte. Les expressions résultantes sont les polynômes en $q$ et $1+q$. Nous étendons ce résultat énumératif via une décomposition du poset de Stirling, ainsi que d’une version homologique du $q=-1$ phénomène de Stembridge. Un parallèle énumératif, poset théorique et étude homologique des $q$-nombres de Stirling de première espèce se fait en commençant par la formulation du placement des tours par suite des auteurs de Médicis et Leroux. On laisse $t=1+q$ et on donne les arguments combinatoires et bijectifs à la Viennot qui démontrent que les $(q;t)$-nombres de Stirling de première et deuxième espèces sont orthogonaux

Computation of q-Binomial Coefficients with the P(n,m)P(n,m) Integer Partition Function

Using $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, which was the subject of an earlier paper, $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, can be computed in $O(n^2)$, and the q-binomial coefficient can be computed in $O(n^3)$. Using the definition of the q-binomial coefficient, some properties of the q-binomial coefficient and $P(n,m,p)$ are derived. The q-multinomial coefficient can be computed as a product of q-binomial coefficients. A formula for $Q(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ distinct parts with each part at most $p$, is given. Some formulas for the number of integer partitions with each part between a minimum and a maximum are derived. A computer algebra program is listed implementing these algorithms using the computer algebra program of the earlier paper

New Perspectives of Quantum Analogues

In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective argument showing the (q, t)-Stirling numbers of the first and second kind are orthogonal. In the second part we give combinatorial proofs of q-Stirling identities via restricted growth words. This includes new proofs of the generating function of q-Stirling numbers of the second kind, the q-Vandermonde convolution for Stirling numbers and the q-Frobenius identity. A poset theoretic proof of Carlitz’s identity is also included. In the last part we discuss a new expression for q-binomial coefficients based on the weighting of certain 01-permutations via a new bistatistic related to the major index. We also show that the bistatistics between the inversion number and major index are equidistributed. We generalize this idea to q-multinomial coefficients evaluated at negative q values. An instance of the cyclic sieving phenomenon related to flags of unitary spaces is also studied

Non-central generalized q-factorial coefficients and q-Stirling numbers

AbstractThe qs-differences of the non-central generalized q-factorials of t of order n, scale parameter s and non-centrality parameter r, at t=0, are thoroughly examined. These numbers for s→0 and s→∞ converge to the non-central q-Stirling numbers of the first and the second kind, respectively. Explicit expressions, recurrence relations, generating functions and other properties of these q-numbers are derived. Further, a sequence of Bernoulli trials is considered in which the conditional probability of success at the nth trial, given that k successes occur before that trial, varies geometrically with n and k. Then, the probability functions of the number of successes in n trials and the number of trials until the occurrence of the kth success are deduced in terms of the qs-differences of the non-central generalized q-factorials of t of order n, scale parameter s and non-centrality parameter r