Partitions weighted by the parity of the crank

Dohoon Choi; Soon Yi Kang; Jeremy Lovejoy

doi:10.1016/j.jcta.2009.02.002

Partitions weighted by the parity of the crank

Dohoon Choi, Soon Yi Kang, Jeremy Lovejoy

Research output: Contribution to journal › Article › peer-review

12 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 M_e (n) minus the number of partitions of n with odd crank M_o (n). We also discuss the combinatorial implications of q-series identities involving M_e (n) - M_o (n). Finally, we determine the exact values of M_e (n) - M_o (n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.

Original language	English
Pages (from-to)	1034-1046
Number of pages	13
Journal	Journal of Combinatorial Theory. Series A
Volume	116
Issue number	5
DOIs	https://doi.org/10.1016/j.jcta.2009.02.002
Publication status	Published - 2009 Jul
Externally published	Yes

Keywords

Congruences
Crank
Partitions

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jcta.2009.02.002

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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",

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).",

year = "2009",

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doi = "10.1016/j.jcta.2009.02.002",

language = "English",

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T1 - Partitions weighted by the parity of the crank

AU - Choi, Dohoon

AU - Kang, Soon Yi

AU - Lovejoy, Jeremy

N1 - 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).

PY - 2009/7

Y1 - 2009/7

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

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U2 - 10.1016/j.jcta.2009.02.002

DO - 10.1016/j.jcta.2009.02.002

M3 - Article

AN - SCOPUS:64649102257

SN - 0097-3165

VL - 116

SP - 1034

EP - 1046

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

IS - 5

ER -

Partitions weighted by the parity of the crank

Abstract

Keywords

ASJC Scopus subject areas

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