Shuffling via sums of Jucys--Murphy Elements

We consider a family of card shuffles of $n$ cards in which the allowed moves involve transpositions corresponding to the Jucys–Murphy elements of the symmetric group $\{S_m{\}}_{m\le n}$. We determine the eigenvalues of the corresponding $n!\times n!$ transition matrices of these shuffles and study the mixing times for a special case, the $k$–star transpositions shuffle, a natural interpolation between the random transpositions shuffle, introduced and studied by Diaconis and Shahshahani, and the star transpositions shuffle, introduced and studied by Diaconis. We prove that the $k$–star transpositions shuffle exhibits total variation cutoff at $\frac{2n-(k+1)}{2(n-1)}n\log n$ with a window of $\frac{2n-(k+1)}{2(n-1)}n$. Furthermore, in the regimes $k\mathrm{/}n\to 0$ or $k\mathrm{/}n\to 1$, this shuffle has the same limit profile as random transpositions, which has been fully determined by Teyssier.