Identities involving partitions with distinct even parts and $4$-regular partitions

TL;DR

This paper establishes new identities linking partitions with distinct even parts to restricted 4-regular partitions, expanding understanding of their combinatorial relationships.

Contribution

It introduces novel identities connecting restricted partitions with distinct even parts to specific classes of 4-regular partitions.

Findings

Proved identities relating restricted partitions with distinct even parts to 4-regular partitions.

Extended classical results on partition enumeration.

Provided combinatorial proofs for new partition identities.

Abstract

It is well known that the number of partitions into distinct even parts equals the number of $4$-regular partitions. In this paper we prove identities relating certain restricted partitions into distinct even parts with restricted $4$-regular partitions.