The multiset partitions and the generalized Stirling numbers

2020 ◽  
Vol 31 (5-6) ◽  
pp. 813-831
Author(s):  
Miloud Mihoubi ◽  
Asmaa Rahim ◽  
Said Taharbouchet
Keyword(s):  
Stirling Numbers ◽  
Generalized Stirling Numbers

scholarly journals Unimodality properties of generalized Stirling numbers

2015 ◽  
Vol 45 (9) ◽  
pp. 1583-1586
Author(s):  
Yi WANG ◽  
BaoXuan ZHU ◽  
Lily Li LIU
Keyword(s):  
Stirling Numbers ◽  
Generalized Stirling Numbers

scholarly journals On a Class of Combinatorial Sums Involving Generalized Factorials

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
W.-S. Chou ◽  
L. C. Hsu ◽  
P. J.-S. Shiue
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Sums ◽  
Generalized Stirling Numbers

The object of this paper is to show that generalized Stirling numbers can be effectively used to evaluate a class of combinatorial sums involving generalized factorials.

scholarly journals Modified approach to generalized Stirling numbers via differential operators

2010 ◽  
Vol 23 (1) ◽  
pp. 115-120 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad P. Cakić ◽  
Toufik Mansour
Keyword(s):  
Differential Operators ◽  
Stirling Numbers ◽  
Generalized Stirling Numbers

scholarly journals A Newton Interpolation Approach to Generalized Stirling Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
Explicit Expression ◽  
Recurrence Relation ◽  
Generating Function ◽  
Stirling Numbers ◽  
Divided Differences ◽  
Newton Interpolation ◽  
Basic Properties ◽  
Generalized Stirling Numbers

We employ the generalized factorials to define a Stirling-type pair{s(n,k;α,β,r),S(n,k;α,β,r)}which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated.

scholarly journals Lagrange and Wilson theorems for the generalized Stirling numbers

1938 ◽  
Vol 5 (4) ◽  
pp. 171-173 ◽  
Author(s):  
E. T. Bell
Keyword(s):  
Stirling Numbers ◽  
Generalized Stirling Numbers ◽  
Image Position

If m, n are integers, m > 0, n > 1, the generalized Stirling numbers are defined by the identity in x,

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 The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

Sign in / Sign up

Export Citation Format

Share Document