On odd Durfee symbols and their odd ranks

Speaker:   夏先伟 教授 (江苏大学数学科学学院)

Inviter:  陈绍示 （中国科学院数学机械化重点实验室）

Title:  On odd Durfee symbols and their odd ranks

Time & Venue:  202091 15:00-16:00

Abstract:

In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let $N^{0}(m,n)$ denote the number of odd Durfee symbols of $n$ with odd rank $m$,  and $N^{0}(r,m;n)$ be the number of odd Durfee symbols of $n$ with odd rank congruent to $r$ modulo $m$. Recently, Wang established explicit formulas for the generating functions of $N^{0} (r,8;n)$ and their $8$-dissections. In this talk, we give the generating functions for $N^{0}(r,12;n)$ by utilizing some identities involving Appell-Lerch sums $m(x,q,z)$ and a universal mock theta function $g(x,q)$. Based on these formulas, we determine the signs of $N^{0} (r,12;4n+t)-N^{0}(s,12;4n+t)$ for all $0\leq r, s\leq 6$ and $0\leq t \leq 3$. Moreover, let $D_k^0(n)$ denote the number of the number of $k$-marked odd Durfee symbols of $n$ which was introduced by Andrew. He also conjectured that $D_2^{0}(8n+r)$ ($r=4,6$) and $D_3^0(16n+s)$ ($s=1,9,11,13$) are even. These two conjectures were confirmed by Wang. Motivated by Wang's work, we prove new congruences on $D_k^0(n)$ which are stronger than Andrews' congruences.