Relations for partitions with distinct even parts except the largest part which is even

TL;DR

This paper addresses how partition statistics counting even parts interact with $4$-regular partitions by introducing generalized functions $DE_{ ege n}(n)$ and $DE_k(n)$, and three new families $DE_e(n)$, $DE_{ek}(n)$, and $DE_{e\ge k}(n)$. Employing $q$-series methods, the authors derive generating-function identities and extract connections to the $4$-regular partition function $reg_4(n)$, including explicit recurrences such as $DE_e(n)=\sum_{i=0}^{\left\lfloor (n-1)/3\right\rfloor} (-1)^i reg_4(n-1-3i)$. They establish congruences modulo $2$, $4$, and $8$ for certain new functions and extend prior work of Andrews and Bachraoui on DE-type relations. The results provide a unified framework linking partitions with distinct even parts to $reg_4(n)$, offering pathways for further generalizations and modular applications in partition theory.

Abstract

In this paper, we prove some new \(q\)-series identities connecting \(4\)-regular partitions and partitions with distinct even parts with largest part being odd. We also define three new partition functions with distinct even parts except the largest part which is even, and prove identities connecting the three partitions with \(4\)-regular partitions. Moreover, we also offer some congruence for the three newly defined partitions.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5