A bijection for loopless triangulations of a polygon with interior points
Abstract
Loopless triangulations of a polygon with $k$ vertices in $k+2n$ triangles (with interior points and possibly multiple edges) were enumerated by Mullin in 1965, using generating functions and calculations with the quadratic method. In this article we propose a simple bijective construction of Mullin's formula. The argument rests on \emph{conjugation of trees}, a variation of the cycle lemma designed for planar maps. In the much easier case of loopless triangulations of the sphere ($k=3$), we recover and prove correct an unpublished construction of the second author.