Bijective Approaches for Schmidt-Type Theorems

TL;DR

This paper introduces new bijective methods to prove Schmidt-type theorems, leading to refined results and new generating function identities in partition theory.

Contribution

It develops novel bijections, including a generalization of existing identities and connections between partitions with specific index constraints and colored partitions.

Findings

Refined Schmidt-type results using bijections.

New generating function identities for partitions.

Generalization of Bridges and Uncu's identity.

Abstract

We provide new Schmidt-type results through an investigation of two bijections, which are results involving partitions with parts counted only at given indices. Mork's bijection, the first of these, was originally given as a proof of Schmidt's theorem. We show that a version of Sylvester's bijection is equivalent to Mork's bijection applied to 2-modular diagrams, which implies refinements of existing results and new generating function identities. We then develop a bijection based on an idea appearing in a recent paper of Andrews and Keith, that places partitions counted at the indices $r$, $t+r$, $2t+r, \dots$ in correspondence with $t$-colored partitions. This leads to a substantial generalization of an identity of Bridges and Uncu, and complements a similar investigation of Li and Yee.