Algorithmic approach to counting of certain types m-ary partitions

Abstract

Partitions of integers of the type mⁿ as a sum of powers of m (the so-called m-ary partitions) and their counting is considered in this paper. Two algorithms for counting of m-ary partitions of sums, where each addend is mⁿ, are developed. On the base of these algorithms some arithmetical and combinatorial properties, and also polynomial form representations of the number of such partitions are derived. An algorithm with a polynomial running time, which produces the coefficients of this polynomial and next computes the number of considered partitions, is proposed. Two applications, concerning counting problems of special types of m-ary trees and partitions of the Boolean cube, are given.