Annular Non-Crossing Matchings

Abstract

It is well known that the number of distinct non-crossing matchings of $n$ half-circles in the half-plane with endpoints on the x-axis equals the $n^{th}$ Catalan number $C_n$. This paper generalizes that notion of linear non-crossing matchings, as well as the circular non-crossings matchings of Goldbach and Tijdeman, to non-crossings matchings of $n$ line segments embedded within an annulus. We prove that the number of such matchings $\vert Ann(n,m) \vert$ with $n$ exterior endpoints and $m$ interior endpoints correspond to an entirely new, one-parameter generalization of the Catalan numbers with $C_n = \vert Ann(1,m) \vert$. We also develop bijections between specific classes of annular non-crossing matchings and other combinatorial objects such as binary combinatorial necklaces and planar graphs. Finally, we use Burnside's Lemma to obtain an explicit formula for $\vert Ann(n,m) \vert$ for all $n,m \geq 0$.