On The Combinatorics of Multi-Restricted Numbers

Abstract

The so-called multi-restricted numbers generalize and extend the role of Stirling numbers and Bessel numbers in various problems of combinatorial enumeration. Multi-restricted numbers of the second kind count set partitions with a given number of parts, none of whose cardinalities may exceed a fixed threshold or “restriction”. The numbers are shown to satisfy a three-term recurrence relation. Both analytic and combinatorial proofs for this relation are presented. Multi-restricted numbers of both the first and second kinds provide connections between the orbit decompositions of subsets of powers of a finite group permutation representation, in which the number of occurrences of elements is restricted. An exponential generating function for the number of orbits on such restricted powers is given in terms of powers of partial sums of the exponential function.