The odd moments of ranks and cranks

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:120

Article

Andrews, George E.¹  Chan, Song Heng²  Kim, Byungchan^3,4 

[1] Penn State Univ, Dept Math, University Pk, PA 16802 USA

[2] Nanyang Technol Univ, Sch Phys & Math Sci, Div Math Sci, Singapore 637371, Singapore

[3] Seoul Natl Univ Sci & Technol, Sch Liberal Arts, Seoul 139743, South Korea

[4] Seoul Natl Univ Sci & Technol, Inst Convergence Fundamental Studies, Seoul 139743, South Korea

关键词: Partitions;    Rank;    Crank;    Rank moments;    Crank moments;    Smallest part function;    Strings;   

DOI  :  10.1016/j.jcta.2012.07.001

来源: Elsevier

【 摘 要 】

In this paper, we modify the standard definition of moments of ranks and cranks such that odd moments no longer trivially vanish. Denoting the new k-th rank (resp. crank) moments by (N) over bar (k)(n) (resp. (M) over bar (k)(n)), we prove the following inequality between the first rank and crank moments: (M) over bar (1)(n) > (N) over bar (1)(n). This inequality motivates us to study a new counting function, ospt(n), which is equal to (M) over bar (1)(n) - (N) over bar (1)(n). We also discuss higher order moments of ranks and cranks. Surprisingly, for every higher order moments of ranks and cranks, the following inequality holds: (M) over bar (k)(n) > (N) over bar (k)(n). This extends F.G. Garvan's result on the ordinary moments of ranks and cranks. (C) 2012 Elsevier Inc. All rights reserved.

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