Shifted distinct-part partition identities in arithmetic progressions

TL;DR

This paper investigates shifted partition identities involving distinct-part partitions with congruence conditions, using modular functions to establish conditions for their existence and extending known identities to new arithmetic progressions.

Contribution

It provides necessary and sufficient conditions for shifted partition identities and extends existing theorems to additional arithmetic progressions using modular function theory.

Findings

Identifies conditions for the existence of shifted partition identities.

Extends Alladi's theorem to new arithmetic progressions.

Uses modular functions to analyze partition identities.

Abstract

The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence conditions. A shifted partition identity is an identity of the form $p_{S_1}(n-H) = p_{S_2}(n)$ for all $n$ in some arithmetic progression. Several identities of this type have been discovered, including two infinite families found by Alladi. In this paper, we use the theory of modular functions to determine the necessary and sufficient conditions for such an identity to exist. In addition, for two specific cases, we extend Alladi's theorem to other arithmetic progressions.