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A generalized class of restricted Stirling and Lah numbers

Toufik Mansour and Mark Shattuck
From the journal Mathematica Slovaca

Abstract

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.

  1. Communicated by Anatolij Dvurečenskij

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Received: 2016-09-04
Accepted: 2017-04-11
Published Online: 2018-08-06
Published in Print: 2018-08-28

© 2018 Mathematical Institute Slovak Academy of Sciences