Mixed restricted Stirling numbers

Abstract

In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.2000 Mathematics Subject Classification. 05A18, 11B73. Key words and phrases. Multiplicative partition function; Stirling numbers of the second kind; mixed partitions of a set.

Mixed coloured permutations

“…. ),respectively B * n,k,t (c S ), counts the number of mixed partitions defined and studied in [1,12], respectively the mixed partitions with block sizes contained in a given set of positive integers S.…”

Vector weighted Stirling numbers and an application in graph theory

“…Choi and Smith [5] considered r-restricted Stirling numbers of the second kind and those partitions that each block contains at most r members. To see other researches on the generalization of Stirling numbers, we refer the reader to [13,14,9,7,2,1,15,11].…”