Monotonicity properties for ranks of overpartitions

Abstract

The rank of partitions play 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(m, n) and N2(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 (m, n) and N2(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(m, n) ≤ N2(m, n + 1); and for (m, n) = (0, 4) with n = |m| + 2, we have N(m, n) ≤ N (m, n + 1). Furthermore, when m increases, we prove that N (m, n) ≥ N(m + 2, n) and N2(m, n) ≥ N2(m + 2, n) for any m, n ≥ 0, which is an analogue of Chan and Mao's result for partitions.

Asymptotic formula for the $M_2$-ranks of overpartitions

“…Let N 2 (m, n) denote the number of overpartitions of n with the M 2 -rank m. Xiong and Zang [21] proved the following inequality on N 2 (m, n) conjectured by Chan and Mao [7]. We highlight this result to confirm the monotonicity of N 2 (a, c, n) in Theorem 2.1.…”

“…The paper is organized as follows. In Section 2, we introduce the Ingham Tauberian Theorem [4,17] and the monotonicity of rank of overpartition obtained by Xiong and Zang [21]. In Section 3, we obtain the asymptotic behavior of the Appell function associated with the generating function of N 2 (a, c, n).…”