Slit-slide-sew bijections for oriented planar maps

TL;DR

This paper introduces slit-slide-sew bijections for bipolar oriented planar maps and Schnyder woods, providing combinatorial proofs of counting identities through a novel sliding operation and rerooting analysis.

Contribution

It presents a new bijective method using slit-slide-sew operations and orbit analysis to prove enumeration formulas for specific classes of oriented planar maps.

Findings

Established bijections for bipolar oriented maps and Schnyder woods

Provided combinatorial proofs of counting identities

Developed a new sliding operation technique

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.