A Proof of Xin–zhang’s Tridiagonal Determinant Conjecture

We confirm a recent conjecture of Xin and Zhang [‘A variation of the Morris constant term’, Preprint, 2024, arXiv:2409.14356v2], which establishes a simple product formula for the characteristic polynomial of an ( n − 1 ) × ( n − 1 ) (n-1) × (n-1) left parenthesis n minus 1 right parenthesis times left parenthesis n minus 1 right parenthesis tridiagonal matrix. This characteristic polynomial arises from a recurrence relation that enumerates n × n n × n n times n nonnegative integer matrices with all row and column sums equal to t , also called the Ehrhart polynomial of the n th Birkhoff polytope.