There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White [6] showed that for any Ferrers board B = F (b1, b2,..., bn), n∏ (x + bi − (i − 1)) = i=1 n∑ rk(B)(x) ↓n−k where rk(B) is the k-th rook number of B and (x) ↓k = x(x − 1) · · · (x − (k − 1)) is the usual falling factorial polynomial. Similar formulas where rk(B) is replaced by some appropriate generalization of the k-th rook number and (x) ↓k is replaced by polynomials like (x) ↑k,j = x(x + j) · · · (x + j(k − 1)) or (x) ↓k,j = x(x − j) · · · (x − j(k − 1)) can be found in the work of Goldman and Haglund [5], Remmel and Wachs [9], Haglund and Remmel [7], and Briggs and Remmel [3]. 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. k=0

A so called ψ-umbral extensions of the Stirling numbers of the second kind are considered and the resulting Dobinski-like various formulas- including new ones- are presented. These extensions naturally encompass the two well known q-extensions. The further consecutive ψ- umbral extensions of Carlitz-Gould-Milne q-Stirling numbers are therefore realized here in a two-fold way. The fact that the umbral q-extended Dobinski formula may also be interpreted as the average of powers of random variable Xq with the q-Poisson distribution singles out the q-extensions which appear to be a kind of ”singular point ” in the domain of ψ-umbral extensions as expressed by Observations 2.1 and 2.2. Other relevant possibilities are tackled with the paper‘s closing down questions and suggestions with respect to other already existing extensions while a brief limited survey of these other type extensions is being delivered. There the Newton interpolation formula and divided differences appear helpful and inevitable along with umbra symbolic language in describing properties of general exponential polynomials of Touchard and their possible generalizations. Exponential structures or algebraically equivalent prefabs with their exponential formula appear to be also naturally relevant.

Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series †

Abstract. We study statistics on ordered set partitions whose generating functions are related to p, q-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of Steingrímsson(Arxiv:math.CO/0605670). Our basic idea is to encode ordered partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. We also give a partition version of MacMahon’s theorem on the equidistribution of the statistics inversion number and

An ordered partition of [n]: = {1, 2,..., n} is a sequence of its disjoint subsets whose union is [n]. The number of ordered partitions of [n] with k blocks is k!S(n, k), where S(n, k) is the Stirling number of second kind. In this paper we prove some refinements of this formula by showing that the generating function of some statistics on the set of ordered partitions of [n] with k blocks is a natural q-analogue of k!S(n, k). In particular, we prove several conjectures of Steingrímsson. To this end, we construct a mapping from ordered partitions to walks in some digraphs and then, thanks to transfermatrix method, we determine the corresponding generating functions by determinantal