Knuth and dual Knuth equivalence for permutations

Fix $n\ge 1$. Use $S_n=\mathrm{Sym}(\{1,\dots ,n\})$ and its one-line notation from Permutation Weyl group and inversion length. The zero-based realization in The finite symmetric group $S_n$, one-line notation, and cycle notation is identified with this one by the order-preserving relabelling $i\mapsto i-1$ on inputs and values; this relabelling preserves the comparisons below.

For a permutation $x\in S_n$, write its one-line word as $x_1{x}_2\cdots x_n$, where $x_i=x(i)$. For $a<b<c$, an elementary Knuth move replaces a contiguous three-letter factor $bca$ by $bac$, or $cab$ by $acb$, with all letters before and after that factor unchanged; either replacement may be reversed. These moves keep the word a permutation of $\{1,\dots ,n\}$.

The insertion tableau $P(x)$ is obtained by starting with the empty tableau and successively row-inserting $x_1,\dots ,x_n$ as in Row insertion and the bumping route. Two permutations $x,y\in S_n$ are Knuth equivalent, written $x{\sim }_{K}y$, if a finite sequence of elementary Knuth moves transforms $x$ into $y$; a sequence of length zero is allowed. They are dual Knuth equivalent, written $x{\sim }_{dK}y$, if and only if $x^{-1}{\sim }_K{y}^{-1}$. The relation ${\sim }_K$ is an equivalence relation because length-zero sequences give reflexivity, each move is reversible, and move sequences concatenate. Since inversion is a bijection of $S_n$, ${\sim }_{dK}$ is also an equivalence relation. Each elementary Knuth move preserves $P$; more precisely, two permutations are Knuth equivalent if and only if their insertion tableaux agree, as proved in Knuth classes are the fibers of the insertion tableau ↗.