On quasi-polynomials counting planar tight maps
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913416568897536 |
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| author | Bouttier, Jérémie Guitter, Emmanuel Miermont, Grégory |
| author_facet | Bouttier, Jérémie Guitter, Emmanuel Miermont, Grégory |
| contents | A tight map is a map with some of its vertices marked, such that every vertex of degree $1$ is marked. We give an explicit formula for the number $N_{0,n}(d_1,\ldots,d_n)$ of planar tight maps with $n$ labeled faces of prescribed degrees $d_1,\ldots,d_n$, where a marked vertex is seen as a face of degree $0$. It is a quasi-polynomial in $(d_1,\ldots,d_n)$, as shown previously by Norbury. Our derivation is bijective and based on the slice decomposition of planar maps. In the non-bipartite case, we also rely on enumeration results for two-type forests. We discuss the connection with the enumeration of non necessarily tight maps. In particular, we provide a generalization of Tutte's classical slicings formula to all non-bipartite maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_14796 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On quasi-polynomials counting planar tight maps Bouttier, Jérémie Guitter, Emmanuel Miermont, Grégory Combinatorics A tight map is a map with some of its vertices marked, such that every vertex of degree $1$ is marked. We give an explicit formula for the number $N_{0,n}(d_1,\ldots,d_n)$ of planar tight maps with $n$ labeled faces of prescribed degrees $d_1,\ldots,d_n$, where a marked vertex is seen as a face of degree $0$. It is a quasi-polynomial in $(d_1,\ldots,d_n)$, as shown previously by Norbury. Our derivation is bijective and based on the slice decomposition of planar maps. In the non-bipartite case, we also rely on enumeration results for two-type forests. We discuss the connection with the enumeration of non necessarily tight maps. In particular, we provide a generalization of Tutte's classical slicings formula to all non-bipartite maps. |
| title | On quasi-polynomials counting planar tight maps |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2203.14796 |