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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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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
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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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
The generalized Stirling and Bell numbers revisited
 J. Integer Seq
"... The generalized Stirling numbers Ss;h(n,k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n,k;α,β,r) considered by Hsu and Shiue. From this relation, several properties of Ss;h(n,k) and the associated Bell numbers Bs; ..."
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The generalized Stirling numbers Ss;h(n,k) introduced recently by the authors are shown to be a special case of the three parameter family of generalized Stirling numbers S(n,k;α,β,r) considered by Hsu and Shiue. From this relation, several properties of Ss;h(n,k) and the associated Bell numbers Bs;h(n) and Bell polynomials B s;hn(x) are derived. The particular case s = 2 and h = −1 corresponding to the meromorphic Weyl algebra is treated explicitly and its connection to Bessel numbers and Bessel 1 polynomials is shown. The dual case s = −1 and h = 1 is connected to Hermite polynomials. For the general case, a close connection to the Touchard polynomials of higher order recently introduced by Dattoli et al. is established, and Touchard polynomials of negative order are introduced and studied. Finally, a qanalogue Ss;h(n,kq) is introduced and first properties are established, e.g., the recursion relation and an explicit expression. It is shown that the qdeformed numbers Ss;h(n,kq) are special cases of the typeII p,qanalogue of generalized Stirling numbers introduced by Remmel and Wachs, providing the analogue to the undeformed case (q = 1). Furthermore, several special cases are discussed explicitly, in particular, the case s = 2 and h = −1 corresponding to the qmeromorphic Weyl algebra considered by Diaz and Pariguan. 1
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.
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
EULERMAHONIAN STATISTICS ON ORDERED PARTITIONS
"... Abstract. An ordered partition with k blocks of [n]: = {1, 2,..., n} is a sequence of k disjoint and nonempty subsets, called blocks, whose union is [n]. Clearly the number of such ordered partitions is k!S(n, k), where S(n, k) is the Stirling number of the second kind. A statistic on ordered partit ..."
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Abstract. An ordered partition with k blocks of [n]: = {1, 2,..., n} is a sequence of k disjoint and nonempty subsets, called blocks, whose union is [n]. Clearly the number of such ordered partitions is k!S(n, k), where S(n, k) is the Stirling number of the second kind. A statistic on ordered partitions of [n] with k blocks is called EulerMahonian statistics if its generating polynomial is [k]q!Sq(n, k), which is a natural qanalogue of k!S(n, k). Motivated by Steingŕımsson’s conjectures, we consider two different methods to produce EulerMahonian statistics on ordered partitions: (a) we give a bijection between ordered partitions and weighted Motzkin paths by using a variant of FrançonViennot’s bijection to derive many EulerMahonian statistics by expanding the generating function of [k]q!Sq(n, k) as an explicit continued fraction; (b) we encode ordered partitions by walks in some digraphs and then derive new EulerMahonian statistics by computing their generating functions using the transfermatrix method. In particular, we prove several conjectures of Steingŕımsson. Contents
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 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 ..."
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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
Note on WardHoradam H(x) binomials ’ recurrences and related interpretations, II
"... This note is a continuation of [48, 2010]. Firstly, we propose H(x)binomials’ recurrence formula appointed by WardHoradam H(x) = 〈Hn(x)〉n≥0 functions’ sequence i.e. any functions ’ sequence solution of the second order recurrence with functions ’ coefficients. As a method this comprises H ≡ H(x = ..."
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This note is a continuation of [48, 2010]. Firstly, we propose H(x)binomials’ recurrence formula appointed by WardHoradam H(x) = 〈Hn(x)〉n≥0 functions’ sequence i.e. any functions ’ sequence solution of the second order recurrence with functions ’ coefficients. As a method this comprises H ≡ H(x = 1) number sequences Vbinomials ’ recurrence formula determined by the primordial Lucas sequence of the second kind V = 〈Vn〉n≥0 as well as its well elaborated companion fundamental Lucas sequence of the first kind U = 〈Un〉n≥0 which gives rise in its turn to the known Ubinomials ’ recurrence as in [1, 1878] , [6,