On Product Partitions of Integers

Abstract

Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

Enumerating finite groups and their defining relations II

“…t k (n/t)). Given n, the number of such factorizations is at most n 2 [3]. Given the factorization, the number of possibilities for each T i is at most two, so the number of possibilities for T is at most 2 k n. Given T , we can consider G as a subgroup of Aut(T ) containing T , and Aut(T ) is a subgroup of (Aut(T 1 ) × · · · × Aut(T k ))S, where S is the subgroup of S k , the symmetric group on k letters, consisting of the permutations that Aut(T ) induces on the factors T 1 , .…”

Profinite groups with nonabelian crowns of bounded rank and their probabilistic zeta function

“…The number of possibilities for K 1 is at most m c 1 ; given K 1 , the number of possibilities for K 2 is at most m c 2 , etc., and so the number of possibilities for chains as above is at most |G : K| c . The number of possible ordered factorizations |G : K| = m 1 · · · m t is at most |G : K| 2 [5], so we get the bound |μ(K, G)| ≤ |G : K| c+2 .…”

On subgroups with non-zero Möbius numbers in the alternating and symmetric groups

“…We know that the number of ordered factorizations of m is at most m 2 (see [3]). Fix a factorization m = β 1 β 2 · · · β r−1 .…”