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Some generating functions and inequalities for the andrews–stanley partition functions

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  27 December 2021

Na Chen ,
Shane Chern ,
Yan Fan  and
Ernest X. W. Xia
Na Chen
Affiliation:
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, Jiangsu215009, P. R. China (chenna1360@163.com, ernestxwxia@163.com)
Shane Chern
Affiliation:
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova ScotiaB3H 4R2, Canada (chenxiaohang92@gmail.com)
Yan Fan
Affiliation:
Department of Mathematics, Jiangsu University, Zhenjiang, Jiangsu212013, P. R. China (free1002@126.com)
Ernest X. W. Xia
Affiliation:
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, Jiangsu215009, P. R. China (chenna1360@163.com, ernestxwxia@163.com)
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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.

Keywords

Andrews–Stanley partition functionrankcrankpartition inequalityasymptotic formula

MSC classification

Primary: 11P81: Elementary theory of partitions
Secondary: 05A17: Partitions of integers
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 1 , February 2022 , pp. 120 - 135
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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