Total positivity of a shuffle matrix

Abstract

Holte introduced a $n\times n$ matrix $P$ as a transition matrix related to the carries obtained when summing $n$ numbers base $b$. Since then Diaconis and Fulman have further studied this matrix proving it to also be a transition matrix related to the process of $b$-riffle shuffling $n$ cards. They also conjectured that the matrix $P$ is totally nonnegative. In this paper, the matrix $P$ is written as a product of a totally nonnegative matrix and an upper triangular matrix. The positivity of the leading principal minors for general $n$ and $b$ is proven as well as the nonnegativity of minors composed from initial columns and arbitrary rows.