Related papers: The one-sided cycle shuffles in the symmetric grou…

We study sorting by queues that can rearrange their content by applying permutations from a predefined set. These new sorting devices are called shuffle queues and we investigate those of them corresponding to sets of permutations defining…

Using the Berele/Remmel/Kerov/Vershik variation of the Robinson-Schensted-Knuth correspondence, we study the cycle and increasing subsequence structure after various methods of shuffling. One consequence is a cycle index for shuffles like:…

Standard perfect shuffles involve splitting a deck of $2n$ cards into two stacks and interlacing the cards from the stacks. There are two ways that this interlacing can be done, commonly referred to as an in shuffle and an out shuffle,…

We define and study the shuffle algebra $Sh_{m|n}$ of the quantum toroidal algebra $\mathcal E_{m|n}$ associated to Lie superalgebra $\mathfrak{gl}_{m|n}$. We show that $Sh_{m|n}$ contains a family of commutative subalgebras $\mathcal…

Let $B_t(n)$ be the number of set partitions of a set of size~$t$ into at most $n$ parts and let $B'_t(n)$ be the number of set partitions of $\{1,\ldots, t\}$ into at most $n$ parts such that no part contains both $1$ and~$t$ or both $i$…

A common algorithm for the computation of eigenvalues of real symmetric tridiagonal matrices is the iteration of certain special maps $F_\sigma$ called shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is ${\cal…

These are detailed lecture notes of the crash-course on shuffle algebras delivered by the author at Tokyo University of Marine Science and Technology during the second week of March 2019. These notes consist of three chapters, providing a…

We investigate the $k$-cycle shuffle on repeated cards, namely on a deck consisting of $l$ identical copies of each of $m$ card types, with total size $n=ml$. We establish asymptotic results for the total variation mixing of this shuffle,…

In the cyclic-to-random shuffle, we are given n cards arranged in a circle. At step k, we exchange the k'th card along the circle with a uniformly chosen random card. The problem of determining the mixing time of the cyclic-to-random…

We develop a general theory for Markov chains whose transition probabilities are the coefficients of descent operators on combinatorial Hopf algebras. These model the breaking-then-recombining of combinational objects. Examples include the…

We use representation theory of $S_n$ to analyze the mixing of permutation cycle type statistics $a_j(\sigma) = ${# of $j$-cycles of $\sigma$} for any fixed $j$ and $\sigma$ resulting from a random $i$-cycle walk on $S_n$. We also derive…

We study the overlaps between right and left eigenvectors for random matrices of the spherical and truncated unitary ensembles. Conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent…

Let R be a commutative ring and let n,m be two positive integers. Let be the polynomial ring in m x n commuting independent variables R. The symmetric group on n letters acts diagonally on A(n,m). We give generators and relations of the…

Drinfeld orbifold algebras deform skew group algebras in polynomial degree at most one and hence encompass graded Hecke algebras, and in particular symplectic reflection algebras and rational Cherednik algebras. We introduce parametrized…

Fix positive numbers $\alpha$ and $\beta$. For the family of doubly cyclic matrices of the form $diag(a_1, a_2, ... ,a_n) - diag(b_1, b_2, ... ,b_n) \Sigma_*$, where $\Sigma_*$ is a permutation matrix for the $n$-cycle $1 \to 2$, $2 \to 3$,…

For a smooth affine algebra $R$ of dimension $d \geq 3$ over a field $k$ and an invertible alternating matrix $\chi$ of rank $2n$, the group $Sp(\chi)$ of invertible matrices of rank $2n$ over $R$ which are symplectic with respect to $\chi$…

In this paper, we study the discrete-time quantum walks on 1D Chain with the moving and swapping shift operators. We derive analytical solutions for the eigenvalues and eigenstates of the evolution operator $\hat{U}$ using the Chebyshev…

This note will give an enumeration of $n$-cycles in the symmetric group ${\mathcal S}_n$ by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of $n$-cycles under the…

We consider a problem of shuffling a deck of cards with ordered labels. Namely we split the deck of N=k^tq cards (where t>=1 is maximal) into k equally sized stacks and then take the top card off of each stack and sort them by the order of…

We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up…