Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extension of Stanley's Theorem for Partitions   In this paper we present an extension of Stanley's theorem related to
partitions of positive integers. Stanley's theorem states a relation between
"the sum of the numbers of distinct members in the partitions of a positive
integer $n$" and "the total number of 1's that occur in the partitions of $n$".
Our generalization states a similar relation between "the sum of the numbers of
distinct members in the partitions of $n$" and the total number of 2's or 3's
or any general $k$ that occur in the partitions of $n$ and the subsequent
integers. We also apply this result to obtain an array of interesting
corollaries, including alternate proofs and analogues of some of the very
well-known results in the theory of partitions. We extend Ramanujan's results
on congruence behavior of the 'number of partition' function $p(n)$ to get
analogous results for the 'number of occurrences of an element $k$ in
partitions of $n$'. Moreover, we present an alternate proof of Ramanujan's
results in this paper.

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The categories of this paper are: acc-phys
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: math-ph
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The categories of this paper are: eess
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an