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arXiv · 2607.26900

Lehmer Codes and the Reverse-Complement Mapping from (32-1)-Avoiding Permutations to (3-21)-Avoiding Permutations

Andrew Beveridge·Yufan Hu·Yucheng Liu
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Abstract

Let $S_n(32\text{-}1)$ and $S_n(3\text{-}21)$ denote the sets of $n$-permutations avoiding the vincular patterns $32\text{-}1$ and $3\text{-}21$, respectively. Using Lehmer codes, we realize these families as weighted posets $L_n(32\text{-}1)$ and $L_n(3\text{-}21)$, where the weight of a code is the inversion number of its permutation. We show that the maximal elements of each of these posets, $\operatorname{Max} L_n(32\text{-}1)$ and $\operatorname{Max} L_n(3\text{-}21)$, are enumerated by the Fibonacci numbers. We demonstrate that the classical reverse-complement map on permutations restricts to a natural bijection between these two sets of maximal elements, revealing a deep symmetry between their underlying poset structures.

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Andrew Beveridge, Yufan Hu, Yucheng Liu. 2026-07-29. Lehmer Codes and the Reverse-Complement Mapping from (32-1)-Avoiding Permutations to (3-21)-Avoiding Permutations. https://arxiv.org/abs/2607.26900

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