Enumeration of planar bipartite tight irreducible maps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917800678785024 |
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| author | Bouttier, Jérémie Guitter, Emmanuel Manet, Hugo |
| author_facet | Bouttier, Jérémie Guitter, Emmanuel Manet, Hugo |
| contents | We consider planar bipartite maps which are both tight, i.e. without vertices of degree $1$, and $2b$-irreducible, i.e. such that each cycle has length at least $2b$ and such that any cycle of length exactly $2b$ is the contour of a face. It was shown by Budd that the number $\mathcal N_n^{(b)}$ of such maps made out of a fixed set of $n$ faces with prescribed even degrees is a polynomial in both $b$ and the face degrees. In this paper, we give an explicit expression for $\mathcal N_n^{(b)}$ by a direct bijective approach based on the so-called slice decomposition. More precisely, we decompose any of the maps at hand into a collection of $2b$-irreducible tight slices and a suitable two-face map. We show how to bijectively encode each $2b$-irreducible slice via a $b$-decorated tree drawn on its derived map, and how to enumerate collections thereof. We then discuss the polynomial counting of two-face maps, and show how to combine it with the former enumeration to obtain $\mathcal N_n^{(b)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08802 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Enumeration of planar bipartite tight irreducible maps Bouttier, Jérémie Guitter, Emmanuel Manet, Hugo Combinatorics We consider planar bipartite maps which are both tight, i.e. without vertices of degree $1$, and $2b$-irreducible, i.e. such that each cycle has length at least $2b$ and such that any cycle of length exactly $2b$ is the contour of a face. It was shown by Budd that the number $\mathcal N_n^{(b)}$ of such maps made out of a fixed set of $n$ faces with prescribed even degrees is a polynomial in both $b$ and the face degrees. In this paper, we give an explicit expression for $\mathcal N_n^{(b)}$ by a direct bijective approach based on the so-called slice decomposition. More precisely, we decompose any of the maps at hand into a collection of $2b$-irreducible tight slices and a suitable two-face map. We show how to bijectively encode each $2b$-irreducible slice via a $b$-decorated tree drawn on its derived map, and how to enumerate collections thereof. We then discuss the polynomial counting of two-face maps, and show how to combine it with the former enumeration to obtain $\mathcal N_n^{(b)}$. |
| title | Enumeration of planar bipartite tight irreducible maps |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.08802 |