Restricted integer partition functions

Research output: Chapter in Book/Report/Conference proceedingChapter


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∈Amaa where ma ∈ M∪{0} for all a, and all numbers ma 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 = {kk}k≥1, 0 < p(n, A, M) ≤ nc for all n > n0. This answers a question of Ljujić and Nathanson.

Original languageEnglish (US)
Title of host publicationIntegers
Subtitle of host publicationAnnual Volume 2013
PublisherWalter de Gruyter GmbH
Number of pages9
ISBN (Electronic)9783110298161
ISBN (Print)9783110298116
StatePublished - Jan 1 2014
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics


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