Planar maps and continued fractions

Understand

We present an unexpected connection between two map enumeration problems.

• The first one consists in counting planar maps with a boundary of prescribed length.
• The second one consists in counting planar maps with two points at a prescribed distance.
• We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion.