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A Generalized Class of Restricted Stirling and Lah Numbers

A Generalized Class of Restricted Stirling and Lah Numbers

In this paper, we consider a polynomial generalization, denoted by u^{a,b}_{m} (*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 u^{a,b}_{m} (*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 u^{a,b}_{m} (*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 u^{a,b}_{m} (*n*, *k*) as well as for the associated Cauchy numbers.