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Proc Natl Acad Sci U S A. 2008 Dec 23;105(51):20152-6. doi: 10.1073/pnas.0809431105. Epub 2008 Dec 17.

The spt-function of Andrews.

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  • 1Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.


Recently, Andrews introduced the function s(n) = spt(n) which counts the number of smallest parts among the integer partitions of n. We show that its generating function satisfies an identity analogous to Ramanujan's mock theta identities. As a consequence, we are able to completely determine the parity of s(n). Using another type of identity, one based on Hecke operators, we obtain a complete multiplicative theory for s(n) modulo 3. These congruences confirm unpublished conjectures of Garvan and Sellers. Our methods generalize to all integral moduli.

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