Schmidt-type theorems for partitions with uncounted parts

TL;DR

This paper generalizes Schmidt's theorem to include partitions with selectively counted or uncounted parts, providing a new framework that links such partitions to colored partitions and aids in deriving sum-product $q$-series identities.

Contribution

It introduces a broad generalization of Schmidt's theorem for partitions with arbitrary counted parts, connecting them to colored partitions for new identity derivations.

Findings

Generalized Schmidt's theorem for arbitrary counted parts

Established connections to colored partitions for sum-product identities

Provided examples of new $q$-series identities

Abstract

Schmidt's theorem is significantly generalized, to partitions in which periodic but otherwise arbitrary subsets of parts are counted or uncounted. The identification of such sets of partitions with colored partitions satisfying certain specifications may be a generally useful tool for establishing sum-product $q$-series identities, examples of which are given.