Singular Meanders
Yury Belousov
TL;DR
The paper generalizes the classical meander problem by introducing singular meanders, allowing tangential intersections, and develops a systematic combinatorial framework for their study. A key result relates counts of singular and closed singular meanders via $k\mathcal{M}_{n-1,k} = 2n\mathcal{M}_{n,k-1}$, and a generating-function relation $2\partial_x\phi_{(even)}(x,t) = \partial_t\phi_{(odd)}(x,t)$, enabling structural insights. It completely enumerates several natural families, including singular meanders with a single transverse intersection, iterated snakes, and irreducible meanders, with explicit generating functions and connections to OEIS sequences, as well as a decomposition framework into arborescent and B-meanders. These results advance understanding of meander factorization and linksingular variants to established combinatorial sequences, offering concrete enumeration tools and guiding future work on the broader singular-meander landscape.
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.