We present a general bijective approach to planar hypermaps with two main results. First we obtain unified bijections for classes of maps or hypermaps defined by face-degree constraints and girth constraints. To any such class we associate bijectively a class of plane trees characterized by local constraints. This unifies and greatly generalizes several bijections for maps and hypermaps. Second, we present yet another level of generalization of the bijective approach by considering classes of maps with non-uniform girth constraints. More precisely, we consider well-charged maps, which are maps with an assignment of charges (real numbers) to vertices and faces, with the constraints that the length of any cycle of the map is at least equal to the sum of the charges of the vertices and faces enclosed by the cycle. We obtain a bijection between charged hypermaps and a class of plane trees characterized by local constraints.
Publié le :
DOI : 10.4171/aihpd/82
Mots-clés : Planar hypermaps, constellations, bijections, mobiles, girth, generating functions, Ising model
@article{AIHPD_2020__7_1_75_0,
author = {Bernardi, Olivier and Fusy, \'Eric},
title = {Unified bijections for planar hypermaps with general cycle-length constraints},
journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
pages = {75--164},
volume = {7},
number = {1},
year = {2020},
doi = {10.4171/aihpd/82},
zbl = {1439.05059},
language = {en},
url = {http://www.numdam.org/articles/10.4171/aihpd/82/}
}
TY - JOUR AU - Bernardi, Olivier AU - Fusy, Éric TI - Unified bijections for planar hypermaps with general cycle-length constraints JO - Annales de l’Institut Henri Poincaré D PY - 2020 SP - 75 EP - 164 VL - 7 IS - 1 UR - http://www.numdam.org/articles/10.4171/aihpd/82/ DO - 10.4171/aihpd/82 LA - en ID - AIHPD_2020__7_1_75_0 ER -
%0 Journal Article %A Bernardi, Olivier %A Fusy, Éric %T Unified bijections for planar hypermaps with general cycle-length constraints %J Annales de l’Institut Henri Poincaré D %D 2020 %P 75-164 %V 7 %N 1 %U http://www.numdam.org/articles/10.4171/aihpd/82/ %R 10.4171/aihpd/82 %G en %F AIHPD_2020__7_1_75_0
Bernardi, Olivier; Fusy, Éric. Unified bijections for planar hypermaps with general cycle-length constraints. Annales de l’Institut Henri Poincaré D, Tome 7 (2020) no. 1, pp. 75-164. doi : 10.4171/aihpd/82. http://www.numdam.org/articles/10.4171/aihpd/82/
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