Two partition functions with congruences modulo 3, 5, 7, and 13

TL;DR

This paper introduces two new partition functions with specific size restrictions and proves they satisfy Ramanujan-type congruences modulo 3, 5, 7, and 13 using advanced q-series techniques.

Contribution

The paper presents two novel partition functions and establishes their Ramanujan-type congruences using generalized Lambert series identities.

Findings

Both functions satisfy congruences modulo 3, 5, 7, and 13.

Use of generalized Lambert series identities to prove congruences.

Advancement in understanding partition functions with size restrictions.

Abstract

We introduce two new integer partition functions, both of which are the number of partition quadruples of $n$ with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$ by use of generalized Lambert series identities and $q$-series techniques.