Vol. 5, No. 1, 2012

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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
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