On Finite Analogs of Schmidt's Problem and Its Variants

TL;DR

This paper refines Schmidt's problem and related partition identities using advanced combinatorial tools, introducing new variants involving sums of parts, hook lengths, and weighted counts, with implications for Rogers-Ramanujan partitions.

Contribution

It presents novel Schmidt-type theorems incorporating even and odd parts, hook lengths, and weighted partition counts, expanding the scope of classical partition problems.

Findings

New Schmidt-type theorems with refined conditions

Connections between weighted partitions and Rogers-Ramanujan identities

Formulas involving Boulet-Stanley weights and Rogers-Szeg ext{"o} polynomials

Abstract

We refine Schmidt's problem and a partition identity related to 2-color partitions which we will refer to as Uncu-Andrews-Paule theorem. We will approach the problem using Boulet-Stanley weights and a formula on Rogers-Szeg\H{o} polynomials by Berkovich-Warnaar, and present various Schmidt's problem alike theorems and their refinements. Our new Schmidt type results include the use of even-indexed parts' sums, alternating sum of parts, and hook lengths as well as the odd-indexed parts' sum which appears in the original Schmidt's problem. We also translate some of our Schmidt's problem alike relations to weighted partition counts with multiplicative weights in relation to Rogers-Ramanujan partitions.