Singular Meanders

TL;DR

This paper introduces singular meanders, a broader class of meander-like objects allowing tangential intersections, and develops their combinatorial enumeration and connections to known sequences.

Contribution

It systematically studies singular meanders, providing a combinatorial framework, enumerations, and links to existing mathematical objects and sequences.

Findings

Developed a combinatorial framework for singular meanders.

Enumerated several natural families of singular meanders.

Connected singular meanders to known integer sequences.

Abstract

The problem of enumerating meanders -- pairs of simple plane curves with transverse intersections -- was formulated about forty years ago and is still far from solved. Recently, it was discovered that meanders admit a factorization into prime components. This factorization naturally leads to a broader class of objects, which we call singular meanders, in which tangential intersections between the curves are also allowed. In the present paper we initiate a systematic study of singular meanders: we develop a basic combinatorial framework, point out connections with other combinatorial objects and known integer sequences, and completely enumerate several natural families of singular meanders.