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

TL;DR

This paper derives explicit formulas for generating functions related to odd Durfee symbols and their ranks, revealing arithmetic properties and proving conjectures about the parity of certain marked Durfee symbols.

Contribution

It provides explicit formulas for generating functions of odd Durfee symbols' ranks and proves conjectures on their parity properties.

Findings

Explicit formulas for generating functions of $N^{0}(a,M;n)$.

Arithmetic properties of odd Durfee symbols derived from these formulas.

Proof of Andrews' conjectures on the parity of $k$-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.