A formula for the Euler characteristic of a poset through the determinant of the order-complement matrix

2025/11/28 by Pedro J. Chocano, Chocano, Pedro J., Luis Felipe Prieto–Martínez +1
Mathematics · #06A07 #06A11 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2512.00217

openalex publication_date 2025/11/28 · openalex created_date 2025/12/03 · openalex updated_date 2026/07/28

Abstract

Given a finite poset P, its zeta matrix \mathbf Z encode fundamental incidence-theoretic information about the order structure. In this paper we introduce and study the order-complement matrix \mathbf Z = \mathbf J - \mathbf Z, where \mathbf J is the all-ones matrix. We prove a closed formula for its characteristic polynomial and for its determinant, showing that det(\mathbf Z) = (-1)ⁿ χ(P), where n = |P| and χ(P) is the reduced Euler characteristic of P. This provides a new, unexpectedly simple linear-algebraic expression for the Euler characteristic of a poset, complementing existing determinant formulas for matrices derived from incidence relations.

Citations

• Matrix Invariants as Homotopy Invariants in Finite T₀-spaces
• Zeros of the zeta series of a poset and iterated barycentric subdivision
• The Euler characteristic of a category as the sum of a divergent series
• Determinants Associated to Zeta Matrices of Posets

Cited by

• Matrix Invariants as Homotopy Invariants in Finite T₀-spaces

Related

• Determinants Associated to Zeta Matrices of Posets
• A poset version of Ramanujan results on Eulerian numbers and zeta values
• A New Approach to Order Polynomials of Labeled Posets and Their Generalizations
• The uncrossing partial order on matchings is Eulerian
• Unusual formulae for the Euler characteristic