Multi-Restrained Stirling Numbers

Given positive integers n, k, and m, the (n, k)-th mrestrained Stirling number of the first kind is the number of permutations of an n-set with k disjoint cycles of length ≤ m. Inverting the matrix consisting of the (n, k)-th m-restrained Stirling number of the first kind as the (n+1, k +1)-th entry, the (n, k)-th m-restrained Stirling number of the second kind is defined. In this paper, the multi-restrained Stirling numbers of the first and the second kinds are studied to find their explicit formulae, recurrence relations, and generating functions. Also, a unique expansion of multi-restrained Stirling numbers for all integers n and k, and a new generating function for the Stirling numbers of the first kind are introduced.