Enumerating Matroids and Linear Spaces
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...
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| Language: | English |
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Académie des sciences
2023-02-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.423/ |
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| author | Kwan, Matthew Sah, Ashwin Sawhney, Mehtaab |
| author_facet | Kwan, Matthew Sah, Ashwin Sawhney, Mehtaab |
| author_sort | Kwan, Matthew |
| collection | DOAJ |
| description | 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: there are exact formulas for enumeration of rank-1 and rank-2 matroids, 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 $n$. |
| format | Article |
| id | doaj-art-167138ee6c0a42da98a433d2544332d4 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-02-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-167138ee6c0a42da98a433d2544332d42025-02-07T11:06:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-02-01361G256557510.5802/crmath.42310.5802/crmath.423Enumerating Matroids and Linear SpacesKwan, Matthew0Sah, Ashwin1Sawhney, Mehtaab2Institute of Science and Technology Austria, 3400 Klosterneuburg, AustriaDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USADepartment of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAWe 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: there are exact formulas for enumeration of rank-1 and rank-2 matroids, 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 $n$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.423/ |
| spellingShingle | Kwan, Matthew Sah, Ashwin Sawhney, Mehtaab Enumerating Matroids and Linear Spaces Comptes Rendus. Mathématique |
| title | Enumerating Matroids and Linear Spaces |
| title_full | Enumerating Matroids and Linear Spaces |
| title_fullStr | Enumerating Matroids and Linear Spaces |
| title_full_unstemmed | Enumerating Matroids and Linear Spaces |
| title_short | Enumerating Matroids and Linear Spaces |
| title_sort | enumerating matroids and linear spaces |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.423/ |
| work_keys_str_mv | AT kwanmatthew enumeratingmatroidsandlinearspaces AT sahashwin enumeratingmatroidsandlinearspaces AT sawhneymehtaab enumeratingmatroidsandlinearspaces |