Restricted integer partition functions

2012/07/13 by Alon, Noga
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1207.3317

Abstract

For two sets A and M of positive integers and for a positive integer n, let p(n,A,M) denote the number of partitions of n with parts in A and multiplicities in M, that is, the number of representations of n in the form n=∑_{a ∈ A} m_a a where m_a ∈ M ∪ 0 for all a, and all numbers m_a but finitely many are 0. It is shown that there are infinite sets A and M so that p(n,A,M)=1 for every positive integer n. This settles (in a strong form) a problem of Canfield and Wilf. It is also shown that there is an infinite set M and constants c and n₀ so that for A=k!_{k ≥ 1} or for A=\k^k\_{k ≥ 1}, 0n₀. This answers a question of Ljujić and Nathanson.

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