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Differential equations for bipartite maps with bounded face degrees — arXiv:2608.10772

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.10772
License (as archived): CC0-1.0
Archived: 2026-10-19T06:27:27.106Z
SHA-256 of the archived text: 0ea0267539a1cd2d5f81dd5714f928646b079ec53a0f3f5fdf83a04547a04e82
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How to cite

Differential equations for bipartite maps with bounded face degrees — arXiv:2608.10772 — arXiv (en), https://arxiv.org/abs/2608.10772 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_dcdad3fa704a (Metraipolis archive dc_dcdad3fa704a, sha256 0ea0267539a1cd2d5f81dd5714f928646b079ec53a0f3f5fdf83a04547a04e82)

BibTeX

@misc{dcdcdad3fa704a,
  title = {Differential equations for bipartite maps with bounded face degrees — arXiv:2608.10772},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.10772},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_dcdad3fa704a; sha256 0ea0267539a1cd2d5f81dd5714f928646b079ec53a0f3f5fdf83a04547a04e82},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.10772. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In recent years, integrable hierarchies have been used to great advantage for the enumeration of combinatorial maps. They have led to recurrence formulas with respect to the size and genus of the maps, e.g. for triangulations, bipartite quadrangulations and bipartite maps, and for constellations. These formulas are not only remarkably simple but also provide the fastest way of calculating these numbers of maps. With the exception of Louf's work on constellations, it has however remained a challenge to obtain recurrence formulas that control the degrees of the faces of the maps. Here we show how to achieve this for bipartite maps with bounded face degrees. By combining equations from the KP hierarchy and from the Virasoro constraints, a differentially algebraic system is obtained. It couples the generating functions of bipartite maps with bounded root face degrees while controlling the numbers of edges, black vertices, white vertices and number of faces of each degree (and in particular the genus). Finally, this system of ODEs is shown to give recurrence formulas that allows to calculate all the corresponding numbers of maps.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-12 in math.CO).
Categorie arXiv: Combinatorics (math.CO) ; Exactly Solvable and Integrable Systems (nlin.SI).
Pagina del lavoro: https://arxiv.org/abs/2608.10772 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-19T06:20:30.689Z dalla pagina https://arxiv.org/catchup/math.CO/2026-08-12?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).