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

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 with infinitely many parameters arising from an integrable hierarchy.