A proof of Xin-Zhang's tridiagonal determinant conjecture
Jiaqiang Hu, Chen Zhang
TL;DR
The paper resolves Xin and Zhang's conjecture by establishing a simple product formula for the determinant of a key tridiagonal matrix linked to the Ehrhart polynomial of the $n$th Birkhoff polytope. Central to the approach is a shift $C \to \widetilde{C}^{(n)} = C + (n-1)I$ and a explicit conjugation by an upper-triangular matrix $U$ with $U_{i,j}=\binom{n-i}{n-j}$ that reveals $U\widetilde{C}^{(n)}U^{-1}$ as (nearly) lower triangular, reducing det computation to a product of diagonal terms. The authors also provide an explicit formula for $U^{-1}$, demonstrate the similarity to a triangular form, and extend the method to a broader family of tridiagonal matrices, thereby offering a general tool for deriving exact enumeration formulas in combinatorics related to Ehrhart theory. These results connect determinant identities, combinatorial binomial identities, and polyhedral enumeration, with potential applications to other counting problems in similar polytopal settings.
Abstract
We confirm a recent conjecture by 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. The proof relies on an unexpected observation: shifting $C$ by a scalar multiple of the identity matrix yields a matrix similar to a lower triangular matrix. In triangular form, the characteristic polynomial reduces to the product of the diagonal entries, leading to the desired closed-form expression. Moreover, we extend this method to broader families of tridiagonal matrices. This provides a new approach for deriving exact enumeration formulas.