The spt-function of Andrews.
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- Record sourced from PubMed, PMID 19091951.
- Also identified by DOI 10.1073/pnas.0809431105 and PMC identifier 2629266.
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Abstract
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 These congruences confirm unpublished conjectures of Garvan and Sellers.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.