Monotonicity properties for ranks of overpartitions

JOURNAL OF NUMBER THEORY

卷:209

Article

Xiong, Huan¹  Zang, Wenston J. T.² 

[1] Univ Vienna, Fac Math, A-1090 Vienna, Austria

[2] Harbin Inst Technol, Inst Adv Study Math, Harbin 150001, Heilongjiang, Peoples R China

关键词: Overpartition;    Partition;    Rank;    Monotonicity;   

DOI  :  10.1016/j.jnt.2019.08.025

来源: Elsevier

【 摘 要 】

The rank of partitions plays an important role in the combinatorial interpretations of several Ramanujan's famous congruence formulas. In 2005 and 2008, the D-rank and M-2-rank of an overpartition were introduced by Lovejoy, respectively. Let (N) over bar (m, n) and (N2) over bar (m, n) denote the number of overpartitions of n with D-rank m and M-2-rank m, respectively. In 2014, Chan and Mao proposed a conjecture on monotonicity properties of (N) over bar (m, n) and (N2) over bar (m, n). In this paper, we prove the Chan-Mao monotonicity conjecture. To be specific, we show that for any integer m and nonnegative integer n, (N2) over bar (m, n) <= (N2) over bar (m, n + 1); and for (m, n) not equal (0,4) with n not equal vertical bar m vertical bar + 2, we have N(m,n) <= N(m,n + 1). Furthermore, when m increases, we prove that (N) over bar (m,n) >= (N) over bar (m + 2,n) and (N2) over bar (m, n) >= ( N2) over bar (m + 2, n) for any rn, n >= 0, which is an analogue of Chan and Mao's result for partitions. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free