Holte introduced a matrix as a transition matrix related to the carries obtained when summing numbers base . Since then Diaconis and Fulman have further studied this matrix proving it to also be a transition matrix related to the process of -riffle shuffling cards. They also conjectured that the matrix is totally nonnegative. In this paper, the matrix is written as a product of a totally nonnegative matrix and an upper triangular matrix. The positivity of the leading principal minors for general and is proven as well as the nonnegativity of minors composed from initial columns and arbitrary rows.
"Total positivity of a shuffle matrix." Involve 5 (1) 61 - 65, 2012. https://doi.org/10.2140/involve.2012.5.61