A proof of Xin-Zhang's tridiagonal determinant conjecture (extended version)

TL;DR

This paper proves Xin-Zhang's conjecture on the characteristic polynomial of a specific tridiagonal matrix, linking it to combinatorial enumeration and extending the method to broader matrix families.

Contribution

It confirms a recent conjecture and extends the proof technique to more general classes of tridiagonal matrices.

Findings

Confirmed Xin-Zhang's conjecture on the product formula

Established a link between the characteristic polynomial and Ehrhart polynomials

Extended the method to broader tridiagonal matrix families

Abstract

We confirm a recent conjecture of Xin and Zhang, which establishes a simple product formula for the characteristic polynomial of an $(n-1) \times (n-1)$ tridiagonal matrix $C$. This characteristic polynomial arises from a recurrence relation that enumerates $n \times n$ nonnegative integer matrices with all row and column sums equal to $t$, also called the Ehrhart polynomial of the $n$th Birkhoff polytope. Moreover, we extend our method to broader families of tridiagonal matrices.