Annular Non-Crossing Matchings

TL;DR

This paper introduces a new class of annular non-crossing matchings, generalizing Catalan numbers, and provides explicit formulas and bijections to other combinatorial structures.

Contribution

It defines a new one-parameter generalization of Catalan numbers for annular matchings and establishes explicit formulas and combinatorial bijections.

Findings

Number of annular matchings generalizes Catalan numbers

Explicit formula derived using Burnside's Lemma

Bijections with binary necklaces and planar graphs

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 $n,m \geq 0$.