Bulletin of the Australian Mathematical Society· 2026Q2
A PROOF OF XIN–ZHANG’S TRIDIAGONAL DETERMINANT CONJECTURE
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- 2026year
Short summary
A simple product formula for the characteristic polynomial of an (n-1)x(n-1) tridiagonal matrix has been proven, confirming a conjecture by Xin and Zhang.
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Key points
- The conjecture by Xin and Zhang regarding a product formula for a specific tridiagonal matrix characteristic polynomial is confirmed.
- The characteristic polynomial is linked to a recurrence relation for enumerating n x n matrices with equal row and column sums.
- These matrices are also related to the Ehrhart polynomial of the nth Birkhoff polytope.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract 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 ) $(n-1) \times (n-1)$ 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 $n \times n$ 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.
The authors' abstract, as published at the source. Bulletin of the Australian Mathematical Society, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics