Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting

Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group S_n are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter $q$ satisfies $gcd(n,q-1)=1$, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. It is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness. Generating functions for the first pile size in patience sorting from decks with repeated values are derived. This relates to random matrices.Comment: Galley version for J. Alg.; minor revisions in Sec.

Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type $A$ are found. This leads to new models of card shuffling and has interesting combinatorial and number theoretic consequences. An analysis of type C gives another solution to a problem of Rogers in dynamical systems: the enumeration of unimodal permutations by cycle structure. The proof uses the factorization theory of palindromic polynomials over finite fields. Contact is made with symmetric function theory.Comment: One change: we fix a typo in definition of f(m,k,i,d) on page 1

Colored Permutation Statistics by Conjugacy Class

In this paper, we consider the moments of permutation statistics on conjugacy classes of colored permutation groups. We first show that when the cycle lengths are sufficiently large, the moments of arbitrary permutation statistics are independent of the conjugacy class. For permutation statistics that can be realized via $\textit{symmetric}$ constraints, we show that for a fixed number of colors, each moment is a polynomial in the degree $n$ of the $r$-colored permutation group $\mathfrak{S}_{n,r}$. Hamaker & Rhoades (arXiv 2022) established analogous results for the symmetric group as part of their far-reaching representation-theoretic framework. Independently, Campion Loth, Levet, Liu, Stucky, Sundaram, & Yin (arXiv, 2023) arrived at independence and polynomiality results for the symmetric group using instead an elementary combinatorial framework. Our techniques in this paper build on this latter elementary approach