$m$-submultisets and $m$-permutations of multisets elements

Abstract

The article is devoted to two classical combinatorial problems on multisets, which in the existing literature are given unjustifiably little space. Namely, the calculation of the number of all submultisets of power $m$ for an arbitrary multiset and the number of $m$-permutations of such multisets. The first problem is closely related to the width of a partially ordered set of all submultisets of a multiset with the inclusion $\subseteq$. The article contains some important classes of multisets. Combinatorial proofs of problems on the number of $m$-submultisets and $m$-permutations of multiset elements are considered. In the article, on the basis of the generatrix method, economical algorithms for calculating $m$-submultisets and $m$-permutations of multiset elements are constructed. The paper also provides a brief overview of the results that are related to this area of research.