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Augmented Rook Boards and General Product Formulas
"... There are a number of socalled 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 kth rook number of B ..."
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Cited by 3 (3 self)
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There are a number of socalled 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 kth 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 kth 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 qanalogues and (p, q)analogues of our general product formula. k=0
EULERMAHONIAN STATISTICS ON ORDERED SET PARTITIONS (II)
, 2007
"... We study statistics on ordered set partitions whose generating functions are related to p, qStirling 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 partitio ..."
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Cited by 3 (2 self)
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We study statistics on ordered set partitions whose generating functions are related to p, qStirling 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
On psiumbral extension of Stirling numbers and Dobinskilike formulas
 Advan. Stud. Contemp. Math
"... ψumbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the two well known qextensions.The further consecutive ψumbral extensions of CarlitzGould qStirling numbers are ..."
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ψumbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinskilike formulas are discovered. These extensions naturally encompass the two well known qextensions.The further consecutive ψumbral extensions of CarlitzGould qStirling numbers are therefore realized here in a twofold way. The fact that the umbral qextended Dobinski formula may also be interpreted as the average of powers of random variable Xq with the qPoisson distribution singles out the qextensions which appear to be a kind of ”bifurcation point ” in the domain of ψumbral extensions as expressed by Observations 2.1 and 2.2 as tackled with the paper closing down question.
and
, 2006
"... Twoparameter quantum algebras, twinbasic numbers, and associated generalized hypergeometric series † ..."
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Twoparameter quantum algebras, twinbasic numbers, and associated generalized hypergeometric series †
Statistics on Ordered . . .
, 2006
"... 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 ..."
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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 qanalogue 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
Statistics on ordered partitions of sets . . .
, 2006
"... 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 ..."
Abstract
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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 qanalogue 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