Eventually constant maps for two sets and nilpotent pairs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918233329631232 |
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| author | Chen, Weixi Im, Mee Seong Lillja, Catherine Rugo, Nicolas |
| author_facet | Chen, Weixi Im, Mee Seong Lillja, Catherine Rugo, Nicolas |
| contents | We give a bijective correspondence between the number of nilpotent matrices over a Boolean semiring and the number of directed acyclic graphs on ordered vertices. We then enumerate pairs of maps between two finite sets whose composites are eventually constant by forming a bijection that relates a pair of such maps with a spanning tree in a complete bipartite graph, and an edge of said tree. This generalizes the main principle of A. Joyal's proof of Cayley's formula. Finally, we generalize T. Leinster's work by considering a pair of finite-dimensional vector spaces and show a bijectivity between a nilpotent pair of maps and a balanced vector with the hom spaces between them. This leads us to an elegant formula for the number of nilpotent pairs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_05269 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eventually constant maps for two sets and nilpotent pairs Chen, Weixi Im, Mee Seong Lillja, Catherine Rugo, Nicolas Combinatorics Rings and Algebras Representation Theory Primary: 20F18, 17B08, 20F19, 20D15, Secondary: 05C05, 05C10, 05C85, 05C20 We give a bijective correspondence between the number of nilpotent matrices over a Boolean semiring and the number of directed acyclic graphs on ordered vertices. We then enumerate pairs of maps between two finite sets whose composites are eventually constant by forming a bijection that relates a pair of such maps with a spanning tree in a complete bipartite graph, and an edge of said tree. This generalizes the main principle of A. Joyal's proof of Cayley's formula. Finally, we generalize T. Leinster's work by considering a pair of finite-dimensional vector spaces and show a bijectivity between a nilpotent pair of maps and a balanced vector with the hom spaces between them. This leads us to an elegant formula for the number of nilpotent pairs. |
| title | Eventually constant maps for two sets and nilpotent pairs |
| topic | Combinatorics Rings and Algebras Representation Theory Primary: 20F18, 17B08, 20F19, 20D15, Secondary: 05C05, 05C10, 05C85, 05C20 |
| url | https://arxiv.org/abs/2512.05269 |