Almost partition identities

George E. Andrews, Cristina Ballantine

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Abstract

An almost partition identity is an identity for partition numbers that is true asymptotically 100% of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.

Original languageEnglish (US)
Pages (from-to)5428-5436
Number of pages9
JournalProceedings of the National Academy of Sciences of the United States of America
Volume116
Issue number12
DOIs
StatePublished - 2019

All Science Journal Classification (ASJC) codes

  • General

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