## Involve: A Journal of Mathematics

- Involve
- Volume 5, Number 1 (2012), 61-65.

### 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.

#### Article information

**Source**

Involve, Volume 5, Number 1 (2012), 61-65.

**Dates**

Received: 24 February 2011

Revised: 17 July 2011

Accepted: 4 September 2011

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513733449

**Digital Object Identifier**

doi:10.2140/involve.2012.5.61

**Mathematical Reviews number (MathSciNet)**

MR2924314

**Zentralblatt MATH identifier**

1252.15037

**Subjects**

Primary: 15B48: Positive matrices and their generalizations; cones of matrices 60C05: Combinatorial probability

**Keywords**

total positivity shuffle minors

#### Citation

McMillan, Audra. Total positivity of a shuffle matrix. Involve 5 (2012), no. 1, 61--65. doi:10.2140/involve.2012.5.61. https://projecteuclid.org/euclid.involve/1513733449