Proofs of some conjectures of Keith and Zanello on $t$-regular partition

TL;DR

This paper proves Keith and Zanello's conjectures on the self-similarity and congruences modulo 2 for the number of t-regular partitions for specific values of t, advancing understanding of partition congruences.

Contribution

It confirms conjectures on self-similarity modulo 2 for b_t(n) at t=3, 25, and establishes a new self-similarity result at t=21.

Findings

Proved conjectures for b_3(n) and b_25(n)

Established self-similarity for b_21(n) modulo 2

Extended the understanding of partition congruences

Abstract

For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$ for $b_{t}(n)$ for certain values of $t$. Further, they proposed some conjectures on self-similarities of $b_t(n)$ modulo $2$ for certain values of $t$. In this paper, we prove their conjectures on $b_3(n)$ and $b_{25}(n)$. We also prove a self-similarity result for $b_{21}(n)$ modulo $2$.