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]).

Eigenvalues of the random-to-random operator acting on a simple module. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module [1].