A bijection for length-5 patterns in permutations

Abstract

A bijection which preserves five classical set-valued permutation statistics between $(31245,32145,31254,32154)$-avoiding permutations and $(31425,32415,31524,32514)$-avoiding permutations is constructed. Combining this bijection with two codings of permutations introduced respectively by Baril–Vajnovszki and Martinez–Savage, we prove an enumerative conjecture posed by Gao and Kitaev. Moreover, the generating function for the common counting sequence is proved to be algebraic.