Congruences for 9-regular partitions modulo 3

TL;DR

This paper establishes new divisibility properties for 9-regular partitions, proving specific congruences modulo 3 and 6, and introduces an infinite family of related congruences and conjectures involving powers of 2 and 5.

Contribution

It presents new proven congruences for 9-regular partitions and proposes an infinite family of related congruences and conjectures for these partitions.

Findings

Number of 9-regular partitions divisible by 3 when n ≡ 3 mod 4.

Number of 9-regular partitions divisible by 6 when n ≡ 13 mod 16.

Existence of an infinite family of congruences mod 3 for other progressions.

Abstract

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4 and 5. A collection of conjectures includes two congruences modulo higher powers of 2 and a large family of "congruences with exceptions" for these and other regular partitions mod 3.