On Partition Classes Arising from Parity, Differences, and Repeated Smallest Parts

TL;DR

This paper explores various classes of partition functions related to parity, differences, and repeated smallest parts, establishing identities that extend classical theorems like Euler's and Legendre's, with combinatorial and $q$-series proofs.

Contribution

It introduces new identities connecting different partition classes and extends classical partition theorems with novel combinatorial and $q$-series proofs.

Findings

Established identities linking partition classes based on parity, differences, and smallest parts.

Extended Euler's partition theorem using new identities.

Derived an analogue of Legendre's theorem with combinatorial and $q$-series proofs.

Abstract

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities connecting these various classes of partitions. Moreover, our identities help us to extend the Euler's partition theorem. An analogue of Legendre's theorem of the partition-theoretic interpretation of Euler's pentagonal number theorem is also derived. Both combinatorial and $q$-series proofs are given for our results.