Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Arithmetic Properties of Odd Ranks and $k$-Marked Odd Durfee Symbols   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.

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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th