Given positive integers , , and , the -th -restrained Stirling number of the first kind is the number of permutations of an -set with disjoint cycles of length . By inverting the matrix consisting of the -th -restrained Stirling number of the first kind as the -th entry, the -th -restrained Stirling number of the second kind is defined. In this paper, we study the multi-restrained Stirling numbers of the first and second kinds to derive their explicit formulae, recurrence relations, and generating functions. Additionally, we introduce a unique expansion of multi-restrained Stirling numbers for all integers and , and a new generating function for the Stirling numbers of the first kind.
