Arithmetic properties of $5$-regular partitions into distinct parts

TL;DR

This paper investigates the arithmetic properties of 5-regular partitions into distinct parts, including parity, congruences, and lacunarity of related generating functions, revealing new modular and combinatorial insights.

Contribution

It provides a complete characterization of the parity of certain partition counts, establishes new congruences modulo 4, and proves lacunarity of a generating function modulo powers of 5.

Findings

Parity characterization of b'_5(2n+1)

Several congruences modulo 4 for b'_5(n)

Lacunarity of the generating function of b'_5(5n+1) modulo powers of 5

Abstract

A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of $b^\prime_5(n)$. We provide full characterization of the parity of $b^\prime_5(2n+1)$, present several congruences modulo 4, and prove that the generating function of the sequence $(b^\prime_5(5n+1))$ is lacunary modulo any arbitrary positive powers of 5.