Total positivity of a shuffle matrix
type: Item
instance of: scholarly article
title: Total positivity of a shuffle matrix
zbMATH Open document ID: 1252.15037
author: Audra McMillan
published in: Involve
publication date: 2012-08-15T00:00:00Z
full work available at URL: p07.xhtml
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 6066179
zbMATH Keywords: total positivity
zbMATH Keywords: shuffle matrix
zbMATH Keywords: minors
zbMATH Keywords: Vandermonde convolution
MaRDI profile type: Publication
OpenAlex ID: W2042188352
DOI: 10.2140/INVOLVE.2012.5.61
review text: \textit{J. M. Holte} [Am. Math. Mon. 104, No. 2, 138--149 (1997; Zbl 0889.15021)] introduced an \(n \times n\) matrix \(P\) with entries NEWLINE\[NEWLINEP(i,j)=\frac{1}{b^n}\sum_{r=0}^{j- \lfloor i/b \rfloor}(-1)^r {{n + 1} \choose r} {{n-1-i+(j+1-r)b} \choose n}, NEWLINE\]NEWLINE where the \(P(i, j)\) entry gives the probability that when adding \(n\) random numbers base \(b\), the next carry will be \(j\), given that the previous carry was \(i\). \textit{P. Diaconis} and \textit{J. Fulman} [Am. Math. Mon. 116, No. 9, 788--803 (2009; Zbl 1229.60011)] conjectured that the matrix \(P\) is totally nonnegative for all positive integers \(n\) and \(b\). They proved that for all \(n\) and \(b\), \(P\) is totally nonnegative of order 2, that is all the \(2 \times 2\) minors are nonnegative, and that when \(b\) is a power of 2, \(P\) is totally nonnegative.NEWLINENEWLINE In this paper, the matrix \(P\) is further analyzed and results are generalized to other \(b\). Using the Vandermonde convolution the matrix \(P\) is factorized into a product of an upper unitriangular matrix and a totally nonnegative matrix. It is thus proved the positivity of all the leading principal minors for general \(n\) and \(b\).
description: scientific article; zbMATH DE number 6066179