Asymptotics of Higher Order Ospt-Functions for Overpartitions

Abstract

In this paper we obtain asymptotic formulas for positive crank and rank moments for overpartitions. Moreover, we show that crank and rank moments are asymptotically equal while the difference is asymptotically positive. This indicates that there exist analogous higher ospt-functions for overpartitions, which we define.

The asymptotic profile of $\chi_y$-genera of Hilbert schemes of points on K3 surfaces

“…To obtain our results we use and develop techniques from analytic number theory. See for example the closely related papers [4,5,6,7,9,14,18,19,27]. We expect that the techniques in this paper might in turn motivate and be relevant for questions in analytic number theory.…”

“…We note that m∈Z a m,k (n) = p k (n), with p k (n) the number of partitions of n in k colors. Our first result is obtained using the approach of Wright [26] also used in [4,6,7,19,27].…”

Higher order SPT functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and without repeated odd parts

“…In Corollary 2.10 we use this to prove the inequality M 2k (n) > N 2k (n) for all k ≥ 1 and all n ≥ 1. Previously this inequality was known to hold for each fixed k for sufficiently large n, due to the work of Zapata Rolon [26] in determining the asymptotics for M Unlike in [18], here k = 1, 2 as a subscript in spt (n) does not specify the smallest part being odd or even.…”

Higher order smallest parts functions and rank-crank moment inequalities from Bailey pairs

“…The advantage to these positive moments is that while N 2k (n) = 2N + 2k (n) and M 2k (n) = 2M + 2k (n), it is no longer the case that the odd moments are zero. It is true that M + 2k+1 (n) > N + 2k+1 (n), and there have been several studies of the positive moments corresponding to the rank and crank of ordinary partitions as well as overpartitions, see [1,8,18,28,35]. As such we should expect that analogous results and inequalities hold for the moments of this article, however our methods do not directly apply and it is not clear if one can handle positive moments in the generality we have managed for the original moments.…”