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Slit-slide-sew bijections for oriented planar maps

Year: 2026
Type: article
Abstract: We construct growth bijections for bipolar oriented planar maps and for Schnyder woods. These give direct combinatorial proofs of several counting identities for these objects. Our method mainly uses two ingredients. First, a slit-slide-sew operation, which consists in slightly sliding a map along a well-chosen path. Second, the study of the orbits of natural rerooting operations on the considered classes of oriented maps. Mathematics Subject Classifications: 05A15, 05A19 Keywords: Bipolar orientations, Schnyder woods, combinatorial maps, enumeration, bijection, homomesy
DOI: 10.5070/c6.68700
Institutions: Centre National de la Recherche Scientifique, Laboratoire d'Informatique de l'École Polytechnique, Institut Polytechnique de Paris, Laboratoire d'Informatique Gaspard-Monge, Université Gustave Eiffel, Institut de Mathématiques de Bordeaux
Language: English
Open Access status: diamond
Awards: High genus geometry, A combinatorial study of Ising model on maps, Combinatorics of maps, meanders and multicurves, and its interactions, G5296751580, Large Objects Under Combinatorial Constraints and Outside Uniform Models, G7080048889, Combinatorics of 3D Maps
OpenAlex ID: W4405627074
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