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A Generalized Class of Restricted Stirling and Lah Numbers

Authors: 

Toufik Mansour, Mark Shattuck

Source title: 
Mathematica Slovaca, 68(4): 727-740, 2018 (ISI)
Academic year of acceptance: 
2018-2019
Abstract: 

In this paper, we consider a polynomial generalization, denoted by ua,bm (nk), 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 ua,bm (nk) 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 ua,bm (nk) 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 ua,bm (nk) as well as for the associated Cauchy numbers.