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Some inequalities for Garvan's bicrank function of 2-colored partitions

Volume 190 / 2019

Shane Chern, Dazhao Tang, Liuquan Wang Acta Arithmetica 190 (2019), 171-191 MSC: Primary 05A17; Secondary 11P55, 11P82, 11P83. DOI: 10.4064/aa180507-7-11 Published online: 28 June 2019


In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities between the bicrank counts $M^{*}(r,m,n)$ for $m=2$, $3$ and $4$ via their asymptotic formulas and some $q$-series techniques. These inequalities are parallel to Andrews and Lewis’ results on the rank and crank counts for ordinary partitions.


  • Shane ChernDepartment of Mathematics
    The Pennsylvania State University
    University Park, PA 16802, U.S.A.
  • Dazhao TangCollege of Mathematics and Statistics
    Chongqing University
    401331, Chongqing, P.R. China
  • Liuquan WangSchool of Mathematics and Statistics
    Wuhan University
    430072, Wuhan, P.R. China

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