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# Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 68, Issue 4

# A generalized class of restricted Stirling and Lah numbers

Toufik Mansour
/ Mark Shattuck
• Corresponding author
• Institute for Computational Science & Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Published Online: 2018-08-06 | DOI: https://doi.org/10.1515/ms-2017-0140

## Abstract

In this paper, we consider a polynomial generalization, denoted by $\begin{array}{}{u}_{m}^{a,b}\end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{}{u}_{m}^{a,b}\end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{}{u}_{m}^{a,b}\end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{}{u}_{m}^{a,b}\end{array}$ (n, k) as well as for the associated Cauchy numbers.

MSC 2010: Primary 11B73; Secondary 05A19; 05A18

## References

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Accepted: 2017-04-11

Published Online: 2018-08-06

Published in Print: 2018-08-28

Communicated by Anatolij Dvurečenskij

Citation Information: Mathematica Slovaca, Volume 68, Issue 4, Pages 727–740, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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