Partitions weighted by the parity of the crank

Dohoon Choi, Soon Yi Kang, Jeremy Lovejoy

Research output: Contribution to journalArticlepeer-review

14 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
Externally publishedYes

Bibliographical note

Funding Information:
✩ The first author was supported by the Korea Research Foundation Grant funded by the Korean government (KRF-2008-331-C00005), the second author was supported by the SRC program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean government (MEST) (R11-2007-035-01002-0), and the third author was supported by an ACI “Jeunes Chercheurs et Jeunes Chercheuses”. E-mail addresses: choija@kau.ac.kr (D. Choi), s2kang@kaist.ac.kr (S.-Y. Kang), lovejoy@liafa.jussieu.fr (J. Lovejoy).

Keywords

  • Congruences
  • Crank
  • Partitions

ASJC Scopus subject areas

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

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