Congruences for Partition Pairs with Conditions
Chris Jennings-Shaffer

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 such that is non-empty, , and where is the smallest part and 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.
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