Combinatorics of Exterior Peaks on Pattern-Avoiding Symmetric Transversals

Let S T λ ( τ ) denote the set of symmetric transversals of a self-conjugate Young diagram λ which avoid the permutation pattern τ . Given two permutations τ = τ 1 τ 2 … τ n of { 1 , 2 , … , n } and σ = σ 1 σ 2 … σ m of { 1 , 2 , … , m } , the direct sum of τ and σ , denoted by τ ⊕ σ , is the permutation τ 1 τ 2 … τ n ( σ 1 + n ) ( σ 2 + n ) … ( σ m + n ) . We establish an exterior peak set preserving bijection between S T λ ( 321 ⊕ τ ) and S T λ ( 213 ⊕ τ ) for any pattern τ and any self-conjugate Young diagram λ . 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.