Congruences for Partition Pairs with Conditions

الأصل · EN

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.