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

TL;DR

This paper introduces a novel bijective proof for counting bipartite planar maps with prescribed face degrees, revealing simpler underlying formulas and novel 'slit-slide-sew' operations, connecting combinatorics with integrable hierarchies.

Contribution

It provides the first bijective proof for a formula from an integrable hierarchy with infinitely many parameters, using new operations on maps and trees.

Findings

Established a bijective proof for Louf's formula.

Revealed two simpler formulas related to the original.

Introduced 'slit-slide-sew' operations on maps.

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.