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.

中文翻译：

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排列中长度为 5 的模式的双射

保留五个经典集值排列统计量的双射$（31245,32145,31254,32154）$-避免排列和$（31425,32415,31524,32514）$-构建避免排列。将此双射与 Baril-Vajnovszki 和 Martinez-Savage 分别引入的两种排列编码相结合，我们证明了 Gau 和 Kitaev 提出的枚举猜想。此外，还证明了公共计数序列的生成函数是代数的。