Annular Non-Crossing Matchings
Paul Drube, Puttipong Pongtanapaisan

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 half-circles in the half-plane with endpoints on the x-axis equals the Catalan number . 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 line segments embedded within an annulus. We prove that the number of such matchings with exterior endpoints and interior endpoints correspond to an entirely new, one-parameter generalization of the Catalan numbers with . 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)…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Data Management and Algorithms · Algorithms and Data Compression
