Partitions weighted by the parity of the crank

Dohoon Choi, Soon Yi Kang, Jeremy Lovejoy

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

The 'crank' is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank Me (n) minus the number of partitions of n with odd crank Mo (n). We also discuss the combinatorial implications of q-series identities involving Me (n) - Mo (n). Finally, we determine the exact values of Me (n) - Mo (n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.

Original languageEnglish
Pages (from-to)1034-1046
Number of pages13
JournalJournal of Combinatorial Theory. Series A
Volume116
Issue number5
DOIs
Publication statusPublished - 2009 Jul 1
Externally publishedYes

Fingerprint

Parity
Partition
Statistics
Ramanujan
Congruence
Q-series
Asymptotic Formula
Statistic
Odd
Distinct
Zero

Keywords

  • Congruences
  • Crank
  • Partitions

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Cite this

Partitions weighted by the parity of the crank. / Choi, Dohoon; Kang, Soon Yi; Lovejoy, Jeremy.

In: Journal of Combinatorial Theory. Series A, Vol. 116, No. 5, 01.07.2009, p. 1034-1046.

Research output: Contribution to journalArticle

Choi, Dohoon ; Kang, Soon Yi ; Lovejoy, Jeremy. / Partitions weighted by the parity of the crank. In: Journal of Combinatorial Theory. Series A. 2009 ; Vol. 116, No. 5. pp. 1034-1046.
@article{33a7e4f87386484088a970dc3aeef1b9,
title = "Partitions weighted by the parity of the crank",
abstract = "The 'crank' is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank Me (n) minus the number of partitions of n with odd crank Mo (n). We also discuss the combinatorial implications of q-series identities involving Me (n) - Mo (n). Finally, we determine the exact values of Me (n) - Mo (n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.",
keywords = "Congruences, Crank, Partitions",
author = "Dohoon Choi and Kang, {Soon Yi} and Jeremy Lovejoy",
year = "2009",
month = "7",
day = "1",
doi = "10.1016/j.jcta.2009.02.002",
language = "English",
volume = "116",
pages = "1034--1046",
journal = "Journal of Combinatorial Theory - Series A",
issn = "0097-3165",
publisher = "Academic Press Inc.",
number = "5",

}

TY - JOUR

T1 - Partitions weighted by the parity of the crank

AU - Choi, Dohoon

AU - Kang, Soon Yi

AU - Lovejoy, Jeremy

PY - 2009/7/1

Y1 - 2009/7/1

N2 - The 'crank' is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank Me (n) minus the number of partitions of n with odd crank Mo (n). We also discuss the combinatorial implications of q-series identities involving Me (n) - Mo (n). Finally, we determine the exact values of Me (n) - Mo (n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.

AB - The 'crank' is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank Me (n) minus the number of partitions of n with odd crank Mo (n). We also discuss the combinatorial implications of q-series identities involving Me (n) - Mo (n). Finally, we determine the exact values of Me (n) - Mo (n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.

KW - Congruences

KW - Crank

KW - Partitions

UR - http://www.scopus.com/inward/record.url?scp=64649102257&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=64649102257&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2009.02.002

DO - 10.1016/j.jcta.2009.02.002

M3 - Article

AN - SCOPUS:64649102257

VL - 116

SP - 1034

EP - 1046

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 5

ER -