Carries, shuffling, and symmetric functions

TL;DR

This paper explores the mathematical properties of carries in addition, riffle shuffles, and their connections to symmetric functions, Markov chains, and algebraic structures, providing proofs and convergence analysis.

Contribution

It introduces new proofs for the Markov chain of carries and shuffles, and links these processes to symmetric functions, Gaussian processes, and algebraic mappings.

Findings

Markov chain transition matrix for carries and shuffles is characterized

Convergence rates to stationarity are determined

Connections to Gaussian autoregressive processes and algebraic structures are established

Abstract

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.