Eigenvalues of the random-to-random operator acting on the regular representation.
This statistic is defined for a permutation $w$ as:
$$\left[\binom{\ell(w) + 1}{2} + \operatorname{diag}\left(Q(w)\right)\right] - \left[\binom{\ell(u) + 1}{2} + \operatorname{diag}\left(Q(u)\right)\right]$$
where:

• $u$ is the longest suffix of $w$ (viewed as a word) whose first ascent is even;
• $\ell(w)$ is the size of the permutation $w$ (equivalently, the length of the word $w$);
• $Q(w), Q(u)$ denote the recording tableaux of $w, u$ under the RSK correspondence;
• $\operatorname{diag}(\lambda)$ denotes the diagonal index (or content) of an integer partition $\lambda$;
• and $\operatorname{diag}(T)$ of a tableau $T$ denotes the diagonal index of the partition given by the shape of $T$.

The regular representation of the symmetric group of degree n has dimension n!, so any linear operator acting on this vector space has n! eigenvalues (counting multiplicities). Hence, the eigenvalues of the random-to-random operator can be indexed by permutations; and the values of this statistic give all the eigenvalues of the operator (Theorem 12 of [1]).

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.

Sends a permutation to its inverse.