Proof of a conjecture on the total positivity of amazing matrices

Abstract

Let n and b be positive integers. Define the amazing matrix Pn,b=[P(i,j)]i,j=0n−1 to be an n×n matrix with entriesP(i,j)=1bn∑r≥0(−1)r(n+1r)(n−1−i+(j+1−r)bn). Diaconis and Fulman conjectured that the amazing matrix is totally positive. We give an affirmative answer to this conjecture.