Combinatorial perspectives on identities for partitions with distinct even parts

TL;DR

This paper offers new combinatorial perspectives on partitions with distinct even parts, connecting them with signed and bicolored partitions, and provides bijective proofs to establish several new identities.

Contribution

It introduces novel combinatorial approaches and bijections linking partitions with distinct even parts to signed and bicolored partitions, addressing open problems.

Findings

Derived new partition identities involving distinct even parts.

Constructed bijections connecting different classes of partitions.

Partially answered open combinatorial problems posed by other researchers.

Abstract

Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored partitions, thereby obtaining several partition identities. We construct bijective proofs for each of our results. Furthermore, these bijections will partially answer the combinatorial problems posed by Andrews-El Bachraoui and K$\imath$l$\imath$\c{c}-Kur\c{s}ung\"oz. respectively.