Carries, shuffling, and symmetric functions

A shuffling theorem for lozenge tilings of doubly-dented hexagons

MacMahon's theorem on plane partitions yields a simple product formula for tiling number of a hexagon, and Cohn, Larsen and Propp's theorem provides an explicit enumeration for tilings of a dented semihexagon via semi-strict Gelfand--Tsetlin patterns. In this paper, we prove a natural hybrid of the two theorems for hexagons with an arbitrary set of unit triangles removed along a horizontal axis. In particular, we show that the `shuffling' of removed unit triangles only changes the tiling number of the region by a simple multiplicative factor. Our main result generalizes a number of known enume…

通过Jucys-Murphy元素之和进行洗牌

Shuffling via sums of Jucys--Murphy Elements

本文研究了一种通过对称群S_m的Jucys-Murphy元素对应的置换进行洗牌的家族，确定了这些洗牌的转移矩阵的本征值，并研究了特定情况下的混合时间，证明了k-星置换洗牌在总变差截止时具有特定的渐进行为。

Applications of Symmetric Functions to Cycle and Increasing Subsequence Structure after Shuffles (Part 2)

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: cut a deck into roughly two equal piles, thoroughly mix the first pile and then riffle it with the second pile. Conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. An upper bound on the convergence rate is given. Connections are made with extended Schur functions and with work of Baik and Rain…

Applications of Symmetric Functions to Cycle and Subsequence Structure after Shuffles

Using symmetric function theory, we study the cycle structure and increasing subsequence structure of permutations after various shuffling methods, emphasizing the role of Cauchy type identities and the Robinson-Schensted-Knuth correspondence. One consequence is a cycle index for riffle shuffles when one deals from the bottom of the deck; conclusions are drawn concerning the distribution of fixed points and the asymptotic distribution of cycle structure. In studying cycle structure of unimodal permutations and their generalizations we are led to an interesting variation of the RSK corresponden…

Valeurs propres des opérateurs de mélanges symétrisés

English title: Eigenvalues of Symmetrized Shuffling Operators The random-to-random shuffling operator explains, for example, the evolution of a deck of cards subject to the following random process: draw a card randomly from the deck and reinsert it at a random position. If one instead draws more than one card at a time before reinserting, then the resulting operator is an example of a family of symmetrized shuffling operators studied by Victor Reiner, Franco Saliola and Volkmar Welker. This thesis describes a way to obtain the eigenvalues of these operators. We build on the work of Anton Diek…

对称群代数中的单侧洗牌

The one-sided cycle shuffles in the symmetric group algebra

研究对称群代数中的单侧洗牌算子，计算其特征值与重数，证明可对角化条件，并给出随机到下方洗牌的强平稳时间。

辫排列面代数的不变量理论

Invariant theory for the face algebra of the braid arrangement

本文从不变量理论视角研究对称群在辫排列面代数上的作用，揭示了其作为对称群和 Solomon 降代数联合表示的结构，并重现了降代数的 Cartan 不变量等经典结果。

Analysis of Carries in Signed Digit Expansions

The number of positive and negative carries in the addition of two independent random signed digit expansions of given length is analyzed asymptotically for the $(q, d)$-system and the symmetric signed digit expansion. The results include expectation, variance, covariance between the positive and negative carries and a central limit theorem. Dependencies between the digits require determining suitable transition probabilities to obtain equidistribution on all expansions of given length. A general procedure is described to obtain such transition probabilities for arbitrary regular languages. Th…

对图分类的闭合对称单子结构

Closed symmetric monoidal structures on the category of graphs

本文研究了图分类的闭合对称单子结构，证明其存在盒积和范畴积两种闭合对称单子积。

基于旗流形几何的 Weyl 群上的 Markov 链

Markov chains on Weyl groups from the geometry of the flag variety

本文研究了旗空间 $G/B$ 上的 Burnside 过程，通过 Bruhat 分解在对称群上构建 Markov 链，给出实用算法并确定其极限行为，最终推广至有限 Chevalley 群，在 Weyl 群上提出一类新的 Markov 链。

Polynomial Expressions of Carries in p-ary Arithmetics

It is known that any $n$-variable function on a finite prime field of characteristic $p$ can be expressed as a polynomial over the same field with at most $p^n$ monomials. However, it is not obvious to determine the polynomial for a given concrete function. In this paper, we study the concrete polynomial expressions of the carries in addition and multiplication of $p$-ary integers. For the case of addition, our result gives a new family of symmetric polynomials, which generalizes the known result for the binary case $p = 2$ where the carries are given by elementary symmetric polynomials. On th…

The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices

Horn's problem, i.e., the study of the eigenvalues of the sum $C=A+B$ of two matrices, given the spectrum of $A$ and of $B$, is re-examined, comparing the case of real symmetric, complex Hermitian and self-dual quaternionic $3\times 3$ matrices. In particular, what can be said on the probability distribution function (PDF) of the eigenvalues of $C$ if $A$ and $B$ are independently and uniformly distributed on their orbit under the action of, respectively, the orthogonal, unitary and symplectic group? While the two latter cases (Hermitian and quaternionic) may be studied by use of explicit form…

Cylindric Reverse Plane Partitions and 2D TQFT

The ring of symmetric functions carries the structure of a Hopf algebra. When computing the coproduct of complete symmetric functions $h_λ$ one arrives at weighted sums over reverse plane partitions (RPP) involving binomial coefficients. Employing the action of the extended affine symmetric group at fixed level $n$ we generalise these weighted sums to cylindric RPP and define cylindric complete symmetric functions. The latter are shown to be $h$-positive, that is, their expansions coefficients in the basis of complete symmetric functions are non-negative integers. We state an explicit formula …