Arithmetic Properties of Odd Ranks and $k$-Marked Odd Durfee Symbols

Abstract

Let $N^0(m,n)$ be the number of odd Durfee symbols of $n$ with odd rank $m$, and $N^0(a,M;n)$ be the number of odd Durfee symbols of $n$ with odd rank congruent to $a$ modulo $M$. We give explicit formulas for the generating functions of $N^0(a,M;n)$ and their $\ell$-dissections where $0\le a\le M-1$ and $M,\ell \in \{2,4,8\}$. From these formulas, we obtain some interesting arithmetic properties of $N^0(a,M;n)$. Furthermore, let ${\mathcal{D}}_k^0(n)$ denote the number of $k$-marked odd Durfee symbols of $n$. Andrews (2007) conjectured that ${\mathcal{D}}_2^0(n)$ is even if $n\equiv 4$ or 6 (mod 8) and ${\mathcal{D}}_3^0(n)$ is even if $n\equiv 1,9,11$ or 13 (mod 16). Using our results on odd ranks, we prove Andrews' conjectures.

23 pages

References in corpus (1)

• On ranks and cranks of partitions modulo $4$ and $8$