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Slit-Slide-Sew bijections for planar bipartite maps with prescribed degree

Juliette Schabanel

TL;DR

The paper provides a bijective proof for the planar case of Louf's formula for bipartite maps with prescribed face degrees, tying the result to the Toda hierarchy. By dualizing to Eulerian maps and invoking Schaeffer's bijection between balanced Eulerian trees and Eulerian maps, the authors reduce the problem to two simpler tree identities and interpret them as Slit-Sew and Cut-Close operations on maps. They show that the two mapping families act identically under duality and opening/closing, establishing the desired bijections first on trees and then on bipartite maps. This work delivers the first bijection for a formula arising from an integrable hierarchy with infinitely many parameters, providing structural insight into map enumeration via tree-based frameworks and dualities, and broadens the toolkit for combinatorial interpretations of integrable systems.

Abstract

We present a bijective proof for the planar case of Louf's counting formula on bipartite planar maps with prescribed face degree, that arises from the Toda hierarchy. We actually show that his formula hides two simpler formulas, both of which can be rewritten as equations on trees using duality and Schaeffer's bijection for eulerian maps. We prove them bijectively and show that the constructions we provide for trees can also be interpreted as "slit-slide-sew" operations on maps. As far as we know, this is the first bijection for a formula arising from an integrable hierarchy with infinitely many parameters.

Slit-Slide-Sew bijections for planar bipartite maps with prescribed degree

TL;DR

The paper provides a bijective proof for the planar case of Louf's formula for bipartite maps with prescribed face degrees, tying the result to the Toda hierarchy. By dualizing to Eulerian maps and invoking Schaeffer's bijection between balanced Eulerian trees and Eulerian maps, the authors reduce the problem to two simpler tree identities and interpret them as Slit-Sew and Cut-Close operations on maps. They show that the two mapping families act identically under duality and opening/closing, establishing the desired bijections first on trees and then on bipartite maps. This work delivers the first bijection for a formula arising from an integrable hierarchy with infinitely many parameters, providing structural insight into map enumeration via tree-based frameworks and dualities, and broadens the toolkit for combinatorial interpretations of integrable systems.

Abstract

We present a bijective proof for the planar case of Louf's counting formula on bipartite planar maps with prescribed face degree, that arises from the Toda hierarchy. We actually show that his formula hides two simpler formulas, both of which can be rewritten as equations on trees using duality and Schaeffer's bijection for eulerian maps. We prove them bijectively and show that the constructions we provide for trees can also be interpreted as "slit-slide-sew" operations on maps. As far as we know, this is the first bijection for a formula arising from an integrable hierarchy with infinitely many parameters.

Paper Structure

This paper contains 16 sections, 21 theorems, 15 equations, 11 figures.

Table of Contents

  1. Setting and presentation of the results
  2. Definitions and notations
  3. Recursive formulas
  4. Bipartite maps and eulerian trees
  5. Duality
  6. Eulerian trees
  7. A bijection for the marked vertex formula
  8. Interpretation in terms of trees
  9. Translation to bipartite maps
  10. The two pairs of mappings have the same action
  11. The spanning trees are dual of each other
  12. The Merging process and the Slit and Sew process act the same
  13. The Separation process and the Cut and Close process act the same
  14. A bijection for the marked face formula
  15. Interpretation in trees
  16. ...and 1 more sections

Key Result

Theorem 1

For every planar map $M$, $v(M)+f(M) = n(M)+2$.

Figures (11)

  • Figure 1: The blue map is the dual of the purple map.
  • Figure 2: A eulerian tree, the associated bracketing word and two possible root choices, the square one being unbalanced and the circle balanced.
  • Figure 3: A eulerian tree and its closure.
  • Figure 4: The successor of the purple half-edge is the green one.
  • Figure 5: The bijection for the marked vertex on trees.
  • ...and 6 more figures

Theorems & Definitions (33)

  • Theorem 1: Euler's formula
  • Theorem 2
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5
  • Theorem 6: Sch97
  • Lemma 7
  • proof
  • Lemma 8: Rerooting
  • ...and 23 more