Integer Partitions With Restricted Distinct Parts

TL;DR

This paper investigates the enumeration of integer partitions with specific restrictions on parts, establishing Ramanujan-type congruences using advanced q-series and theta function identities.

Contribution

It introduces new Ramanujan-type congruences for partition functions with restricted parts based on modular conditions, expanding the understanding of partition theory.

Findings

Proved Ramanujan-type congruences for specific restricted partition functions

Utilized q-series and theta function identities to derive results

Extended classical partition congruences to new restricted cases

Abstract

For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for $Q_{t}^{s}(n)$ for some particular values of $s$ and $t$ by employing $q$-series and theta function identities.