Enumerating Matroids and Linear Spaces
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866916265303474176 |
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| author | Kwan, Matthew Sah, Ashwin Sawhney, Mehtaab |
| author_facet | Kwan, Matthew Sah, Ashwin Sawhney, Mehtaab |
| contents | We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the puzzle for enumerating fixed-rank matroids at this level of accuracy: the numbers of rank-1 and rank-2 matroids on a ground set of size $n$ have exact representations in terms of well-known combinatorial functions, and it was recently proved by van der Hofstad, Pendavingh, and van der Pol that for constant $r\ge 4$ there are $(e^{1-r}n+o(n))^{n^{r-1}/r!}$ rank-$r$ matroids on a ground set of size $n$. In our proof, we introduce a new approach for bounding the number of clique decompositions of a complete graph, using quasirandomness instead of the so-called entropy method that is common in this area. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_03788 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Enumerating Matroids and Linear Spaces Kwan, Matthew Sah, Ashwin Sawhney, Mehtaab Combinatorics We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the puzzle for enumerating fixed-rank matroids at this level of accuracy: the numbers of rank-1 and rank-2 matroids on a ground set of size $n$ have exact representations in terms of well-known combinatorial functions, and it was recently proved by van der Hofstad, Pendavingh, and van der Pol that for constant $r\ge 4$ there are $(e^{1-r}n+o(n))^{n^{r-1}/r!}$ rank-$r$ matroids on a ground set of size $n$. In our proof, we introduce a new approach for bounding the number of clique decompositions of a complete graph, using quasirandomness instead of the so-called entropy method that is common in this area. |
| title | Enumerating Matroids and Linear Spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2112.03788 |