scholarly journals Combinatorial Interpretations of the Jacobi-Stirling Numbers

10.37236/342 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Yoann Gelineau ◽  
Jiang Zeng
Keyword(s):  
Spectral Theory ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Combinatorial Interpretations ◽  
Central Factorial Numbers

The Jacobi-Stirling numbers of the first and second kinds were introduced in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.

scholarly journals Combinatorial Interpretations of Particular Evaluations of Complete and Elementary Symmetric Functions

10.37236/2131 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Pietro Mongelli
Keyword(s):  
Spectral Theory ◽  
Symmetric Functions ◽  
Stirling Numbers ◽  
Elementary Symmetric Functions ◽  
Combinatorial Interpretations

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial  interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.

scholarly journals A new approach to the r-Whitney numbers by using combinatorial differential calculus

2019 ◽  
Vol 11 (2) ◽  
pp. 387-418
Author(s):  
Miguel A. Méndez ◽  
José L. Ramírez
Keyword(s):  
Differential Calculus ◽  
Differential Operators ◽  
Family Study ◽  
Stirling Numbers ◽  
Umbral Calculus ◽  
Combinatorial Interpretation ◽  
New Approach ◽  
Whitney Numbers ◽  
Combinatorial Interpretations ◽  
Binomial Type

Abstract In the present article we introduce two new combinatorial interpretations of the r-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar G := {y → yxm, x → x}. By specializing m = 1 we obtain also a new combinatorial interpretation of the r-Stirling numbers of the second kind. Again, by specializing to the case r = 0 we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard’s polynomials. Moreover, we recover several known identities involving the r-Dowling polynomials and the r-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The r-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the r-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce [m]-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

scholarly journals On Pattern-Avoiding Partitions

10.37236/763 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Stirling Numbers ◽  
Equivalence Classes ◽  
Catalan Numbers ◽  
Set Partition ◽  
Combinatorial Interpretations

A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$. We choose the ordering of the blocks so that they satisfy $\min B_1 < \min B_2 < \cdots < \min B_d$. We represent such a set partition by a canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, with $\pi_i=j$ if $i\in B_j$. We say that a partition $\pi$ contains a partition $\sigma$ if the canonical sequence of $\pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $\sigma$. Two partitions $\sigma$ and $\sigma'$ are equivalent, if there is a size-preserving bijection between $\sigma$-avoiding and $\sigma'$-avoiding partitions. We determine all the equivalence classes of partitions of size at most $7$. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most $3$. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

scholarly journals On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

10.37236/6699 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Yu-Chang Liang ◽  
Tsai-Lien Wong
Keyword(s):  
Weyl Algebra ◽  
Stirling Numbers ◽  
Stirling Number ◽  
Combinatorial Structures ◽  
Normal Ordering ◽  
Rook Placements ◽  
Threshold Graphs ◽  
Combinatorial Interpretations ◽  
Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

scholarly journals On r-Central Incomplete and Complete Bell Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 724 ◽  
Author(s):  
Dae San Kim ◽  
Han Young Kim ◽  
Dojin Kim ◽  
Taekyun Kim
Keyword(s):  
Stirling Numbers ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Central Factorial Numbers

Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions of the r- Stirling numbers of the second kind and the extended r-Bell polynomials. In this paper, we study several properties, some identities and various explicit formulas about these polynomials and their connections as well.

scholarly journals A simple combinatorial interpretation of certain generalized Bell and Stirling numbers

2014 ◽  
Vol 318 ◽  
pp. 53-57 ◽  
Author(s):  
Pietro Codara ◽  
Ottavio M. D’Antona ◽  
Pavol Hell
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Interpretation

scholarly journals The Hankel Transform ofq-Noncentral Bell Numbers

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Cristina B. Corcino ◽  
Roberto B. Corcino ◽  
Jay M. Ontolan ◽  
Charrymae M. Perez-Fernandez ◽  
Ednelyn R. Cantallopez
Keyword(s):  
Stirling Numbers ◽  
Hankel Transform ◽  
Convolution Type ◽  
Combinatorial Interpretation ◽  
Bell Numbers

We define two forms ofq-analogue of noncentral Stirling numbers of the second kind and obtain some properties parallel to those of noncentral Stirling numbers. Certain combinatorial interpretation is given for the second form of theq-analogue in the context of 0-1 tableaux which, consequently, yields certain additive identity and some convolution-type formulas. Finally, aq-analogue of noncentral Bell numbers is defined and its Hankel transform is established.

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