Bijections for Maps with Boundaries: Krikun’s Formula for Triangulations, and a Quadrangulation Analogue

We present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized formula obtained by Krikun using a (technically involved) generating function approach. The analogous formula for quadrangulations is new. Our method is to show that these maps with boundaries can be endowed with certain “canonical” orientations, making them amenable to the master bijection approach we developed in previous articles. As an application of our formulas, we note that they provide an exact solution of the dimer model on rooted triangulations and quadrangulations.