Almost partition identities

An almost partition identity is an identity for partition numbers that is true asymptotically [Formula: see text] 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.

Medienart:

E-Artikel

Erscheinungsjahr:

2019

Erschienen:

2019

Enthalten in:

Zur Gesamtaufnahme - volume:116

Enthalten in:

Proceedings of the National Academy of Sciences of the United States of America - 116(2019), 12 vom: 19. März, Seite 5428-5436

Sprache:

Englisch

Beteiligte Personen:

Andrews, George E [VerfasserIn]
Ballantine, Cristina [VerfasserIn]

Links:

Volltext

Themen:

Asymptotics
Identities
Journal Article
Partitions

Anmerkungen:

Date Revised 20.11.2019

published: Print-Electronic

Citation Status PubMed-not-MEDLINE

doi:

10.1073/pnas.1820945116

funding:

Förderinstitution / Projekttitel:

PPN (Katalog-ID):

NLM294575561