Separable integer partition (SIP) classes

Abstract: Three of the most classical and well-known identities in the theory of partitions concern: (1) the generating function for $p(n)$ (Euler); (2) the generating function for partitions into distinct parts (Euler), and (3) the generating function for partitions in which parts differ by at least 2 (Rogers-Ramanujan). The lovely, simple argument used to produce the relevant generating functions is mostly never seen again. Actually, there is a very general theorem here which we shall present. We then apply it to prove two familiar theorems; (1) G\" ollnitz-Gordon, and (2) Schur 1926. We also consider an example where the series representation for the partitions in question is new. We close with an application to "partitions with n copies of n."

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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