Combinatorics of Exterior Peaks on Pattern-Avoiding Symmetric Transversals

Let \(\mathcal{S}\mathcal{T}_{\lambda }(\tau )\) denote the set of symmetric transversals of a self-conjugate Young diagram \(\lambda \) which avoid the permutation pattern \(\tau \). Given two permutations \(\tau = \tau _1\tau _2\ldots \tau _n \) of \(\{1,2,\ldots ,n\}\) and \(\sigma =\sigma _1\sigma _2\ldots \sigma _m \) of \(\{1,2,\ldots ,m\}\), the direct sum of \(\tau \) and \(\sigma \), denoted by \(\tau \oplus \sigma \), is the permutation \(\tau _1\tau _2\ldots \tau _n (\sigma _1+n)(\sigma _2+n)\ldots (\sigma _m+n)\). We establish an exterior peak set preserving bijection between \(\mathcal{S}\mathcal{T}_{\lambda }(321\oplus \tau )\) and \(\mathcal{S}\mathcal{T}_{\lambda }(213\oplus \tau )\) for any pattern \(\tau \) and any self-conjugate Young diagram \(\lambda \). Our result is a refinement of part of a result of Bousquet-Mélou–Steingrímsson for pattern-avoiding symmetric transversals. As applications, we derive several enumerative results concerning pattern-avoiding reverse alternating involutions, including two conjectured equalities posed by Barnabei–Bonetti–Castronuovo–Silimbani.