Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Franklin Type Involution for Squares   We find an involution as a combinatorial proof of a Ramanujan's partial theta
identity. Based on this involution, we obtain a Franklin type involution for
squares in the sense that the classical Franklin involution provides a
combinatorial interpretation of Euler's pentagonal number theorem. This
Franklin type involution can be considered as a solution to a problem proposed
by Pak concerning the parity of the number of partitions of n into distinct
parts with the smallest part being odd. Using a weighted form of our
involution, we give a combinatorial proof of a weighted partition theorem
derived by Alladi from Ramanujan's partial theta identity. This answers a
question of Berndt, Kim and Yee. Furthermore, through a different weight
assignment, we find combinatorial interpretations for another partition theorem
derived by Alladi from a partial theta identity of Andrews. Moreover, we obtain
a partition theorem based on Andrews' identity and provide a combinatorial
proof by certain weight assignment for our involution. A specialization of our
partition theorem is relate to an identity of Andrews concerning partitions
into distinct nonnegative parts with the smallest part being even. Finally, we
give a more general form of our partition theorem which in return corresponds
to a generalization of Andrews' identity.

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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: stat
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ph