The Stirling number of the second kind {k} counts the number of ways to partition a set of n labeled balls into k non-empty unlabeled cells. We extend this problem and give a new statement of the r-Stirling numbers of the second kind and r-Bell numbers. We also introduce the r-mixed Stirling number of the second kind and r-mixed Bell numbers. As an application of our results we obtain a formula...

Sequences of generalized Stirling numbers of both kinds are introduced. These sequences of triangles (i.e. infinite-dimensional lower triangular matrices) of numbers will be denoted by S2(k;n,m) and S1(k;n,m) with k ∈ Z. The original Stirling number triangles of the second and first kind arise when k = 1. S2(2;n,m) is identical with the unsigned S1(2;n,m) triangle, called S1p(2;n,m), which also...

We present computer-generated proofs of some summation identities for (q-)Stirling and (q-)Eulerian numbers that were obtained by combining a recent summation algorithm for Stirling number identities with a recurrence solver for difference fields.

We consider a class of sequences defined by triangular recurrence equations. This class contains Stirling numbers and Eulerian numbers of both kinds, and hypergeometric multiples of those. We give a sufficient criterion for sums over such sequences to obey a recurrence equation, and present algorithms for computing such recurrence equations efficiently. Our algorithms can be used for verifying ...

The q-classical polynomials are orthogonal polynomial sequences that are eigenfunctions of a second order q-differential operator of a certain type. We explicitly construct q-differential equations of arbitrary even order fulfilled by these polynomials, while giving explicit expressions for the integer composite powers of the aforementioned second order q-differential operator. The latter is ac...

For a simple finite graph G denote by { G k } the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If En is the graph on n vertices with no edges then { En k } coincides with { n k } , the ordinary Stirling number of the second kind, and so we refer to { G k } as a graph Stirling number. Harper showed that the ...