Some combinatorial identities involving noncommuting variables

International audience We derive combinatorial identities for variables satisfying specific sets of commutation relations. The identities thus obtained extend corresponding ones for $q$-commuting variables $x$ and $y$ satisfying $yx=qxy$. In particular, we obtain weight-dependent binomial theorems, functional equations for generalized exponential functions, we propose a derivative of noncommuting variables, and finally utilize one of the considered weight functions to extend rook theory. This leads us to an extension of the $q$-Stirling numbers of the second kind, and of the $q$-Lah numbers. Nous obtenons des identités combinatoires pour des variables satisfaisant des ensembles spécifiques de relations de commutation. Ces identités ainsi obtenues généralisent leurs analogues pour des variables $q$-commutantes $x$ et $y$ satisfaisant $yx=qxy$. En particulier, nous obtenons des théorèmes binomiaux dépendant du poids, des équations fonctionnelles pour les fonctions exponentielles généralisées, nous proposons une dérivée des variables non-commutatives, et finalement nous utilisons l’une des fonctions de poids considérées pour étendre la théorie des tours. Nous en déduisons une généralisation des $q$-nombres de Stirling de seconde espèce et des $q$-nombres de Lah.

Partitions, Rooks, and Symmetric Functions in Noncommuting Variables

Let $\Pi_n$ denote the set of all set partitions of $\{1,2,\ldots,n\}$. We consider two subsets of $\Pi_n$, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let ${\cal E}_n\subseteq\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, ${\cal T}_{n-1}$. Given $\pi\in\Pi_m$ and $\sigma\in\Pi_n$, define their slash product to be $\pi|\sigma=\pi\cup(\sigma+m)\in\Pi_{m+n}$ where $\sigma+m$ is the partition obtained by adding $m$ to every element of every block of $\sigma$. Call $\tau$ atomic if it can not be written as a nontrivial slash product and let ${\cal A}_n\subseteq\Pi_n$ denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of $NCSym$, the symmetric functions in noncommuting variables. We show that, despite their very different definitions, ${\cal E}_n={\cal A}_n$ for all $n\ge0$. Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to $NCSym$. We end with some remarks.

Augmented Rook Boards and General Product Formulas

There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White showed that for any Ferrers board $B = F(b_1, b_2, \ldots, b_n)$, $$\prod_{i=1}^n (x+b_i-(i-1)) = \sum_{k=0}^n r_k(B) (x)\downarrow_{n-k}$$ where $r_k(B)$ is the $k$-th rook number of $B$ and $(x)\downarrow_k = x(x-1) \cdots (x-(k-1))$ is the usual falling factorial polynomial. Similar formulas where $r_k(B)$ is replaced by some appropriate generalization of the $k$-th rook number and $(x)\downarrow_k$ is replaced by polynomials like $(x)\uparrow_{k,j} = x(x+j) \cdots (x+j(k-1))$ or $(x)\downarrow_{k,j} = x(x-j) \cdots (x-j(k-1))$ can be found in the work of Goldman and Haglund, Remmel and Wachs, Haglund and Remmel, and Briggs and Remmel. We shall refer to such formulas as product formulas. The main goal of this paper is to develop a new rook theory setting in which we can give a uniform combinatorial proof of a general product formula that includes, as special cases, essentially all the product formulas referred to above. We shall also prove $q$-analogues and $(p,q)$-analogues of our general product formula.

Rook Theory, Generalized Stirling Numbers and $(p,q)$-Analogues

In this paper, we define two natural $(p,q)$-analogues of the generalized Stirling numbers of the first and second kind $S^1(\alpha,\beta,r)$ and $S^2(\alpha,\beta,r)$ as introduced by Hsu and Shiue [Adv. in Appl. Math. 20 (1998), 366–384]. We show that in the case where $\beta =0$ and $\alpha$ and $r$ are nonnegative integers both of our $(p,q)$-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We also show how our $(p,q)$-analogues of the generalized Stirling numbers of the second kind can be interpreted in terms of colored set partitions and colored restricted growth functions. Finally we show that our $(p,q)$-analogues of the generalized Stirling numbers of the first kind can be interpreted in terms of colored permutations and how they can be related to generating functions of permutations and signed permutations according to certain natural statistics.