Carries, Shuffling and An Amazing Matrix

Carries, shuffling, and symmetric functions

The "carries" when n random numbers are added base b form a Markov chain with an "amazing" transition matrix determined by Holte. This same Markov chain occurs in following the number of descents or rising sequences when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Similar results are given for type B shuffles. We also develop connections with Gaussian autoregressive processes and the Veronese mapping of commutative algebra.

Foulkes Characters, Eulerian Idempotents, and an Amazing Matrix

John Holte [16] introduced a family of "amazing matrices" which give the transition probabilities of "carries" when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra [4,6,7] and in the analysis of riffle shuffling [6,7]. We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday [20] in work on Hochschild homology. The connections give new closed formulae for Foulkes c…

A generalization of carries process and Eulerian numbers

We study a generalization of Holte's amazing matrix, the transition probability matrix of the Markov chains of the 'carries' in a non-standard numeration system. The stationary distributions are explicitly described by the numbers which can be regarded as a generalization of the Eulerian numbers and the MacMahon numbers. We also show that similar properties hold even for the numeration systems with the negative bases.

Combinatorics of balanced carries

We study the combinatorics of addition using balanced digits, deriving an analog of Holte's "amazing matrix" for carries in usual addition. The eigenvalues of this matrix for base b balanced addition of n numbers are found to be 1,1/b,...,1/b^n, and formulas are given for its left and right eigenvectors. It is shown that the left eigenvectors can be identified with hyperoctahedral Foulkes characters, and that the right eigenvectors can be identified with hyperoctahedral Eulerian idempotents. We also examine the carries that occur when a column of balanced digits is added, showing this process …

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。