Theoretical analysis demonstrates growth bijections for bipolar oriented planar maps and Schnyder woods, providing direct combinatorial counting proofs.
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
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Bettinelli et al. (2026) studied this question.
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