Slit-slide-sew bijections for oriented planar maps

TL;DR

The paper introduces slit-slide-sew growth bijections for three classes of oriented planar maps—bipolar oriented quasi-triangulations, bipolar oriented maps, and Schnyder woods—and uses a novel rerooting/orbit analysis to obtain bijective proofs of key counting identities. Central to the approach are the slit-slide-sew operations and a detailed treatment of boundary-reaching probabilities obtained by averaging over rerooting orbits, linking vertex- and edge-marked configurations to boundary-marked ones. The authors provide explicit counting formulas for the three families, derive growth identities, and connect the results to tableaux via hook-length and hook-content interpretations, while also discussing base cases and random generation strategies. These techniques yield combinatorial insight into enumeration, catalyzing potential extensions to tableaux and random sampling for planar maps with oriented structures. The methods have potential implications for exact counting, probabilistic growth processes, and efficient uniform generation within these map classes.

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.

Figures (14)

• Figure 1: Left. A bipolar oriented map. The poles are in purple; the root is in red; the internal vertices are in green. Right. A bipolar oriented quasi-triangulation: all the internal faces have degree $3$.
• Figure 2: The local rules around the vertices and faces of a bipolar oriented map. Here, the right length of the face is $2$.
• Figure 3: Left. A Schnyder wood on $9$ vertices. The edges of three trees are oriented toward the respective tree roots. Right. The local rule around an internal vertex.
• Figure 4: Left. To obtain the bipolar oriented map from the Schnyder wood, remove the edges of $\mathbf{t}_1$, the vertex $\rho_1$ and the two external edges incident to $\rho_1$, revert the edges of $\mathbf{t}_2$, set $\mathrm{S}\mathrel{\mathop:}\space=\rho_2$, $\mathrm{N}\mathrel{\mathop:}\space=\rho_3$, and the root as the remaining external edge, oriented from $\mathrm{S}$ to $\mathrm{N}$. Observe that the degree of $\rho_1$ becomes the degree of the external face. Right. The internal vertices of the Schnyder wood naturally correspond to the faces of the bipolar oriented map and these all have right length $2$.
• Figure 5: The slit-slide-sew bijections $\Phi$, $\Psi$ between bipolar oriented maps with a marked internal vertex and bipolar oriented maps with a marked right-internal boundary-reaching edge. The little arrows show at which corner to enter in the slitting step. The marked face is highlighted in red and sees its left length decrease or increase by one.

Theorems & Definitions (27)

• Proposition 1: Bipolar oriented quasi-triangulations
• Proposition 2: Bipolar oriented maps
• Proposition 3: Schnyder woods
• Proposition 4
• proof
• Proposition 5
• proof
• Theorem 6
• proof
• Corollary 7