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 in a 1997 paper of Holte. This same Markov chain occurs in following the number of descents 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.