$6$-regular partitions: new combinatorial properties, congruences, and linear inequalities

TL;DR

This paper explores new properties, congruences, and inequalities related to 6-regular partitions, connecting them with other partition functions and providing combinatorial interpretations and open problems.

Contribution

It introduces new infinite families of congruences for $b_6(n)$, links between $b_6(n)$ and $Q_2(n)$, and discovers novel linear inequalities involving partition functions.

Findings

Infinite congruences modulo 3 for $b_6(n)$

Connections between $b_6(n)$ and $Q_2(n)$ with combinatorial interpretations

New linear inequalities involving Euler's partition function $p(n)$

Abstract

We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts not congruent to $\pm 2$ modulo $6$, $Q_2(n)$, and investigate connections between $b_6(n)$ and $Q_2(n)$ providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler's partition function $p(n)$. Infinite families of linear inequalities involving the $6$-regular partition function $b_6(n)$ and the distinct partition function $Q_2(n)$ are proposed as open problems.