Restricted k-color partitions

TL;DR

This paper introduces a new class of overpartition generalizations called (k,j)-colored partitions, explores their properties using advanced mathematical tools, and uncovers various congruences, symmetries, and connections to other mathematical concepts.

Contribution

It extends overpartition theory to (k,j)-colored partitions and develops new results using diverse methods like modular forms and combinatorial bijections.

Findings

Discovered numerous congruences and symmetries in (k,j)-colored partitions.

Established connections to divisor sums and hook length formulas.

Proposed approaches to finitization and further research directions.

Abstract

We generalize overpartitions to (k,j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems: generating function dissections, modular forms, bijections, and other combinatorial maps. We find connections to divisor sums, the Han/Nekrasov-Okounkov hook length formula and a possible approach to a finitization, and other topics, suggesting that a rich mine of results is available.