## Involve: A Journal of Mathematics

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

### Total positivity of a shuffle matrix

Audra McMillan

#### 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.

#### Article information

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

Dates
Revised: 17 July 2011
Accepted: 4 September 2011
First available in Project Euclid: 20 December 2017

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

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

#### References

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