Total positivity of a shuffle matrix (Q444608)

\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 \(b\).