Pairs of eventually constant maps and nilpotent pairs
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915650946990080 |
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| author | Chen, Weixi Im, Mee Seong Khovanov, Mikhail Lillja, Catherine Rugo, Nicolas |
| author_facet | Chen, Weixi Im, Mee Seong Khovanov, Mikhail Lillja, Catherine Rugo, Nicolas |
| contents | Tom Leinster gave a bijective correspondence between the set of operators on a finite-dimensional vector space $V$ and the set of pairs consisting of a nilpotent operator and a vector in $V$. Over a finite field this bijection implies that the probability that an operator be nilpotent is the reciprocal of the number of vectors in $V$. We generalize this correspondence to pairs of operators between pairs of vector spaces and determine the probability that a random pair of operators be nilpotent. We also determine the set-theoretical counterpart of this construction and compute the number of eventually constant pairs of maps between two finite sets, closely related to the number of spanning trees in a complete bipartite graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pairs of eventually constant maps and nilpotent pairs Chen, Weixi Im, Mee Seong Khovanov, Mikhail Lillja, Catherine Rugo, Nicolas Representation Theory Combinatorics Rings and Algebras Primary: 17B08, 20F18, 20F19, 20D15, 16N40, Secondary: 05C10, 05C20, 05C85, 05C05 Tom Leinster gave a bijective correspondence between the set of operators on a finite-dimensional vector space $V$ and the set of pairs consisting of a nilpotent operator and a vector in $V$. Over a finite field this bijection implies that the probability that an operator be nilpotent is the reciprocal of the number of vectors in $V$. We generalize this correspondence to pairs of operators between pairs of vector spaces and determine the probability that a random pair of operators be nilpotent. We also determine the set-theoretical counterpart of this construction and compute the number of eventually constant pairs of maps between two finite sets, closely related to the number of spanning trees in a complete bipartite graph. |
| title | Pairs of eventually constant maps and nilpotent pairs |
| topic | Representation Theory Combinatorics Rings and Algebras Primary: 17B08, 20F18, 20F19, 20D15, 16N40, Secondary: 05C10, 05C20, 05C85, 05C05 |
| url | https://arxiv.org/abs/2512.03367 |