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

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.