Congruences for Partition Pairs with Conditions

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 $\mathrm{\ell }({\pi }_2)<2s({\pi }_1)$ where $s(\pi )$ is the smallest part and $\mathrm{\ell }(\pi )$ 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.