scholarly journals Set partitions with successions and separations

2005 ◽  
Vol 2005 (3) ◽  
pp. 451-463 ◽  
Author(s):  
Augustine O. Munagi
Keyword(s):  
Stirling Numbers ◽  
Set Partitions ◽  
Consecutive Integers

Partitions of the set{1,2,…,n}are classified as having successions if a block contains consecutive integers, and separated otherwise. This paper constructs enumeration formulas for such set partitions and some variations using Stirling numbers of the second kind.

scholarly journals Enumerating set partitions by the number of positions between adjacent occurrences of a letter

2010 ◽  
Vol 4 (2) ◽  
pp. 284-308 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Asymptotic Estimates ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Minimal Elements

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

scholarly journals A Geometric Interpretation of the Intertwining Number

10.37236/7986 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Mahir Bilen Can ◽  
Yonah Cherniavsky ◽  
Martin Rubey
Keyword(s):  
Algebraic Geometry ◽  
Stirling Numbers ◽  
Geometric Interpretation ◽  
Bell Numbers ◽  
Set Partitions ◽  
Depth Index ◽  
Borel Orbits ◽  
Algebraic Monoid

We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining number to the algebraic geometry of Borel orbits. Furthermore, by studying the generating polynomials of our statistics, we determine the $q=-1$ specialization of a $q$-analogue of the Bell numbers. Finally, by using Renner's $H$-polynomial of an algebraic monoid, we introduce and study a $t$-analog of $q$-Stirling numbers.

scholarly journals On Certain Combinatorial Expansions of the Legendre-Stirling Numbers

10.37236/8060 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Shi-Mei Ma ◽  
Jun Ma ◽  
Yeong-Nan Yeh
Keyword(s):  
Spectral Theory ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Set Partitions ◽  
Combinatorial Code

The Legendre-Stirling numbers of the second kind were introduced by Everitt et al. in the spectral theory of powers of the Legendre differential expressions. As a continuation of the work of Andrews and Littlejohn (Proc. Amer. Math. Soc., 137 (2009), 2581-2590), we provide a combinatorial code for Legendre-Stirling set partitions. As an application, we obtain expansions of the Legendre-Stirling numbers of both kinds in terms of binomial coefficients.

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

10.37236/1837 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
J. B. Remmel ◽  
Michelle L. Wachs
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Set Partitions ◽  
Growth Functions ◽  
Rook Theory ◽  
Signed Permutations ◽  
Generalized Stirling Numbers ◽  
Alpha Beta ◽  
Natural Statistics ◽  
Appl Math

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.

scholarly journals Some identities for derangement and Ward number sequences and related bijections

2015 ◽  
Vol 25 (2) ◽  
pp. 132-143
Author(s):  
David Callan ◽  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Stirling Numbers ◽  
Set Partitions ◽  
Number Sequences ◽  
Free Set

Abstract We establish an alternating sum identity for three classes of singleton-free set partitions wherein the number of elements minus the number of blocks is fixed: (i) permutations, that is, partitions into cycles, (ii) unrestricted partitions, and (iii) contents-ordered partitions. Both algebraic and combinatorial proofs are given, the latter making use of a sign-changing involution in each ease. As a consequence, combinatorial proofs are found of specific cases of recent identities of Gould et al. involving both kinds of Stirling numbers.

scholarly journals Walks, Partitions, and Normal Ordering

10.37236/5181 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Askar Dzhumadil'daev ◽  
Damir Yeliussizov
Keyword(s):  
Weyl Algebra ◽  
Differential Operators ◽  
Stirling Numbers ◽  
Set Partitions ◽  
Normal Ordering ◽  
Graph Decompositions

We describe the relation between graph decompositions into walks and the normal ordering of differential operators in the $n$-th Weyl algebra. Under several specifications, we study new types of restricted set partitions, and a generalization of Stirling numbers, which we call the $\lambda$-Stirling numbers.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

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.

Sign in / Sign up

Export Citation Format

Share Document