The Andrews spt-function and higher-order generalizations

Abstract

Andrews' spt-function can be written as the difference between the second symmetrized crank and rank moment functions. The spt-function is a partition function that satisfies some very unexpected arithmetic properties. For example, it satisfies explicit Ramanujan-type congruences modulo all primes TeX Embedding failed!. We explain some of these conruences and how they fit in with the theory of harmonic Maass forms. We show how the spt-function may generalized to higher order moments. The construction relies on the hypergeometric machinery of Bailey pairs. This leads to an inequality between crank and rank moments that was only know previously for sufficiently large n and fixed order.

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