On new polynomials related to bell numbers and stirling numbers

2014 ◽  
Vol 8 ◽  
pp. 763-769 ◽  
Author(s):  
H. Y. Lee ◽  
C. S. Ryoo
Keyword(s):  
Stirling Numbers ◽  
Bell Numbers

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 Identities and derivative formulas for the combinatorial and Apostol-Euler type numbers by their generating functions

Filomat ◽  
2018 ◽  
Vol 32 (20) ◽  
pp. 6879-6891
Author(s):  
Irem Kucukoglu ◽  
Yilmaz Simsek
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Stirling Numbers ◽  
Bell Numbers ◽  
New Family ◽  
Derivative Formulas ◽  
Combinatorial Sums ◽  
Computation Algorithm ◽  
Fractional Derivative Operators ◽  
And Inversion

The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.

scholarly journals A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Miguel Méndez ◽  
Adolfo Rodríguez
Keyword(s):  
Stirling Numbers ◽  
Direct Application ◽  
Bell Numbers ◽  
Bijective Correspondence ◽  
Rook Placements ◽  
Combinatorial Model ◽  
International Audience ◽  
Generalized Stirling Numbers ◽  
Dual Graded Graphs ◽  
Combinatorial Methods

International audience We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.

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.

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.

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