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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2502.09149 |
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| _version_ | 1866910824834007040 |
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| author | Taranenko, Anna A. |
| author_facet | Taranenko, Anna A. |
| contents | A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each of its lines equals $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $Ω_n^d$ known as the Birkhoff polytope. In this paper, we identify all vertices of the polytopes $Ω_4^3$ and $Ω_3^4$ correcting the results of Ke, Li, and Xiao (2016). Additionally, we describe constructions vertices of $Ω_n^d$ using multidimensional matrix products and find symmetric vertices of $Ω_3^d$ for all $d \geq 4$ with large support sizes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumeration and constructions of vertices of the polytope of polystochastic matrices Taranenko, Anna A. Combinatorics 15B51, 52B05, 15A15 A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each of its lines equals $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $Ω_n^d$ known as the Birkhoff polytope. In this paper, we identify all vertices of the polytopes $Ω_4^3$ and $Ω_3^4$ correcting the results of Ke, Li, and Xiao (2016). Additionally, we describe constructions vertices of $Ω_n^d$ using multidimensional matrix products and find symmetric vertices of $Ω_3^d$ for all $d \geq 4$ with large support sizes. |
| title | Enumeration and constructions of vertices of the polytope of polystochastic matrices |
| topic | Combinatorics 15B51, 52B05, 15A15 |
| url | https://arxiv.org/abs/2502.09149 |