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Mathematica Slovaca

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Volume 68, Issue 4


A generalized class of restricted Stirling and Lah numbers

Toufik Mansour / Mark Shattuck
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  • 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


In this paper, we consider a polynomial generalization, denoted by uma,b (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 uma,b (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 uma,b (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 uma,b (n, k) as well as for the associated Cauchy numbers.

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

Keywords: restricted Stirling numbers; Cauchy numbers; polynomial generalization; combinatorial identities


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About the article

Received: 2016-09-04

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, DOI: https://doi.org/10.1515/ms-2017-0140.

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