Some generating functions and inequalities for the andrews–stanley partition functions. Issue 1 (27th February 2022)

• Record Type: Journal Article
• Authors: Chen, Na; Chern, Shane; Fan, Yan; Xia, Ernest X. W.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 65:Issue 1(2022)
• Page Start: 120
• Page End: 135
• Publication Date: 2022-02-27
• Subjects: Andrews–Stanley partition function -- rank -- crank -- partition inequality -- asymptotic formula; 11P81 -- 05A17; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091521000833
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: Let $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$ . In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$, where $\pi '$ is the conjugate of $\pi$ . Let $p(r, \, m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$ . Generating function identities, congruences and inequalities for $p(0, \, 4;n)$ and $p(2, \, 4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r, \, m;n)$ with $m=16$ and $24$ . These results are refinements of some inequalities due to Swisher.