#### Vol. 5, No. 1, 2012

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
Total positivity of a shuffle matrix

### Audra McMillan

Vol. 5 (2012), No. 1, 61–65
##### Abstract

Holte introduced a $n×n$ 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.

##### Keywords
total positivity, shuffle, minors
##### Mathematical Subject Classification 2010
Primary: 15B48, 60C05