Congruences for Partition Pairs with Conditions

jennings-shaffer, chris;

We prove congruences for the number of partition pairs (π1, π2) such that π1 is non-empty, s(π1) ≤ s(π2) and l(π2) < 2s(π1), where s(π) is the smallest part and l(π) 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.