Total positivity of a shuffle matrix

McMillan, Audra

doi:10.2140/involve.2012.5.61

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Total positivity of a shuffle matrix

Audra McMillan

Vol. 5 (2012), No. 1, 61–65

DOI: 10.2140/involve.2012.5.61

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.

Keywords

total positivity, shuffle, minors

Mathematical Subject Classification 2010

Primary: 15B48, 60C05

Milestones

Received: 24 February 2011

Revised: 17 July 2011

Accepted: 4 September 2011

Published: 28 April 2012

Communicated by John C. Wierman

Authors

Audra McMillan
School of Mathematics and Statistics University of Sydney Sydney, NSW 2006 Australia

Vol. 5, No. 1, 2012