Congruences for Partition Pairs with Conditions

TL;DR

This paper establishes new congruences for specific partition pairs with certain size constraints, using advanced analytical tools and combinatorial refinements to deepen understanding of partition theory.

Contribution

It introduces novel congruences for partition pairs under specific conditions and employs Bailey's Lemma and Lambert series identities for proofs.

Findings

Proved congruences for partition pairs with size constraints

Used Bailey's Lemma and Lambert series identities in proofs

Introduced a partition pair crank for combinatorial refinements

Abstract

We prove congruences for the number of partition pairs $(\pi_1,\pi_2)$ such that $\pi_1$ is non-empty, $s(\pi_1)\le s(\pi_2)$, and $\ell(\pi_2)< 2s(\pi_1)$ where $s(\pi)$ is the smallest part and $\ell(\pi)$ is the largest part of a partition. The proofs use Bailey's Lemma and a generalized Lambert series identity of Chan. We also discuss how a partition pair crank gives combinatorial refinements of these congruences.