Refinements of the Bell and Stirling numbers

Abstract

We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial, restricted Bell, and r-derangement numbers (and probably more!). By combining methods from analytic combinatorics, umbral calculus, and probability theory, we derive several recurrence relations and closed form expressions for these numbers. By specializing our results to the classical case, we recover explicit formulae for the Bell and Stirling numbers as sums over compositions.

Some applications of S-restricted set partitions

“…In this case the restriction is that every block contains at least m elements. Several combinatorial properties of these sequences can be found in [2,7,11,12,13,22,24,26,27,28,36]. Let E (resp.…”

“…Furthermore, generalizations were introduced that connect to other numbers, showing that Stirling numbers are special cases of a large family of interesting number sequences. Though some ideas were already mentioned in early works, as in the fundamental book of Comtet [13], recently the topic started to win again interests and more research groups are concerned with it ( [7], [12], [16], [21], [22], [26], [28], [36]). We adjoin this research investigating the case when the size of each block is contained in a given set S. We consider some applications of these generalization, studying the modifications of well-known formulas/identities of the classical Stirling numbers.…”