Meanders in a Cayley graph

TL;DR

This paper explores the combinatorial structure of meanders and their correspondence with maximal intervals in Cayley graphs of symmetric groups, linking geometric and algebraic concepts.

Contribution

It establishes a novel connection between meanders and the lattice of noncrossing partitions via Cayley graph intervals of symmetric groups.

Findings

Meanders correspond to pairs of maximally separated vertices in the Cayley graph.

The lattice of noncrossing partitions is represented as a maximal interval in the Cayley graph.

A new combinatorial interpretation of meanders in algebraic terms.

Abstract

A meander of order n is a simple closed curve in the plane which intersects a horizontal line transversely at 2n points. (Meanders which differ by an isotopy of the line and plane are considered equivalent.) Let Gamma_n be the Cayley graph of the symmetric group S_n as generated by all (n choose 2) transpositions. Let Lambda_n be any interval of maximal length in Gamma_n; this graph is the Hasse diagram of the lattice of noncrossing partitions. The meanders of order n are in one-to-one correspondence with ordered pairs of maximally separated vertices of Lambda_n.