A formula for the Euler characteristic of a poset through the determinant of the order-complement matrix
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909945437356032 |
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| author | Chocano, Pedro J. Prieto-Martínez, Luis Felipe |
| author_facet | Chocano, Pedro J. Prieto-Martínez, Luis Felipe |
| contents | 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 \emph{order-complement matrix} $\overline{\mathbf Z} = \mathbf J - \mathbf Z$, where $\mathbf J$ is the all-ones matrix. We prove a closed formula for its characteristic polynomial and for its determinant, showing that $\det(\overline{\mathbf Z}) = (-1)^n \tildeχ(P)$, where $n = |P|$ and $\tildeχ(P)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_00217 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A formula for the Euler characteristic of a poset through the determinant of the order-complement matrix Chocano, Pedro J. Prieto-Martínez, Luis Felipe Combinatorics 06A11, 06A07 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 \emph{order-complement matrix} $\overline{\mathbf Z} = \mathbf J - \mathbf Z$, where $\mathbf J$ is the all-ones matrix. We prove a closed formula for its characteristic polynomial and for its determinant, showing that $\det(\overline{\mathbf Z}) = (-1)^n \tildeχ(P)$, where $n = |P|$ and $\tildeχ(P)$ 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. |
| title | A formula for the Euler characteristic of a poset through the determinant of the order-complement matrix |
| topic | Combinatorics 06A11, 06A07 |
| url | https://arxiv.org/abs/2512.00217 |