Enumeration of plane hypermaps with a mixed boundary II
Abstract
A plane hypermap with a boundary is a connected graph embedded in the plane with a distinguished face, and whose inner faces are properly face-bicolored. We say that such hypermaps are k-alternating if the colors of the faces adjacent to the distinguished face alternate at most 2k times when going counterclockwise. Eynard and Orantin, in [EO2005], have found a universal expression for the generating functions of k-alternating hypermaps as a surprisingly simple function of the generating function of 1-alternating hypermaps. In this article, we will provide combinatorial formulas for k-alternating hypermaps using specific subclasses of hypermaps in which each vertex can either be connected to a previously marked vertex, called accessible, or connected to all other vertices in what we call a blob. In particular, this method will allow us to derive the results from [EO2005] from a combinatorial perspective, and we anticipate that it will lead to the computation of asymptotics for hypermaps in future work. This article is naturally a sequel to, and most importantly a generalization of, the results in [BEL26], which deals essentially with the case of 1- and 2-alternating hypermaps.
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BibTeX
@misc{lejeune2026enumeration,
title = {{Enumeration of plane hypermaps with a mixed boundary II}},
author = {Thomas Lejeune},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01148},
eprint = {2611.01148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01148},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Enumeration of plane hypermaps with a mixed boundary II
AU - Thomas Lejeune
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01148
UR - https://arxiv.org/abs/2611.01148
AB - A plane hypermap with a boundary is a connected graph embedded in the plane with a distinguished face, and whose inner faces are properly face-bicolored. We say that such hypermaps are k-alternating if the colors of the faces adjacent to the distinguished face alternate at most 2k times when going counterclockwise. Eynard and Orantin, in [EO2005], have found a universal expression for the generating functions of k-alternating hypermaps as a surprisingly simple function of the generating function of 1-alternating hypermaps. In this article, we will provide combinatorial formulas for k-alternating hypermaps using specific subclasses of hypermaps in which each vertex can either be connected to a previously marked vertex, called accessible, or connected to all other vertices in what we call a blob. In particular, this method will allow us to derive the results from [EO2005] from a combinatorial perspective, and we anticipate that it will lead to the computation of asymptotics for hypermaps in future work. This article is naturally a sequel to, and most importantly a generalization of, the results in [BEL26], which deals essentially with the case of 1- and 2-alternating hypermaps.
ER -