Congruences for Partition Pairs with Conditions

We prove congruences for the number of partition pairs $(π_1,π_2)$ such that $π_1$ is non-empty, $s(π_1)\le s(π_2)$, and $\ell(π_2)< 2s(π_1)$ where $s(π)$ is the smallest part and $\ell(π)$ 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.

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