The \((r_1,\dots ,r_p)\)-Stirling numbers of the second kind. (English) Zbl 1310.11031

Summary: Let \(R_1, \ldots, R_p\) be subsets of the finite set \([n]=\{1,\ldots,n\}\) with \(| R_i| =r_i\) and \(R_i\cap R_j=\emptyset\) for all \(i,j=1,\ldots,p\), \(i\neq j\). The \((r_1,\ldots, r_p)\)-Stirling number of the second kind, \(p\geq 1\) introduced in this paper and denoted by \({n\brace k}_{r_1,\ldots,r_p}\), counts the number of partitions of the set \([n]\) into \(k\) classes (or blocks) such that the elements in each \(R_i\), \(i=1,\ldots,p\), are in different classes (or blocks). Combinatorial and algebraic properties of these numbers are explored.


11B73 Bell and Stirling numbers
05A15 Exact enumeration problems, generating functions
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