Voiculescu's free probability theory { which was introduced in an operator algebraic context, but has since then developed into an exciting theory with a lot of links to other elds { has an interesting combinatorial facet: it can be described by the combinatorial concept of multiplicative functions on the lattice of non-crossing partitions. In this survey I want to explain this connection { wit...

Four statistics, ls, rb, rs, and lb, previously studied on all partitions of { 1, 2, ..., n }, are applied to non-crossing partitions. We consider single and joint distributions of these statistics and prove equidistribution results. We obtain qand p, q-analogues of Catalan and Narayana numbers which refine the rank symmetry and unimodality of the lattice of non-crossing partitions. Two unimoda...

In this paper we present an algorithm for enumerating without repetitions all non-crossing geometric spanning trees on a given set of n points in the plane under edge constraints (i.e., some edges are required to be included in spanning trees). We will first prove that a set of all edge-constrained non-crossing spanning trees is connected via remove-add flips, based on the constrained smallest ...

We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R1, . . . , Rn in the plane and finite point sets Pi ⊂ Ri for i = 1, . . . , n, are there non-crossing connectors γi for (Ri, Pi), i.e., arc-connected sets γi with Pi ⊂ γi ⊂ Ri for every i = 1, . . . , n, such that γi ∩ γj = ∅ for all i 6= j? We prove that non-crossing connectors do alw...

Let Pn be a set of n m points that are the vertices of a convex polygon and let Mm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross and edges joining two perfect matchings M and M if M M a b c d a d b c for some points a b c d of Pn We prove the following results about Mm its diameter is m it is bipartite for e...

We introduce analogues of the lattice of non-crossing set partitions for the classical reeection groups of type B and D. The type B analogues ((rst considered by Montenegro in a diierent guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved. In both cases, they are EL-labellable ranked lattices with symmetric chain dec...

This paper describes a systematic approach to the enumeration of \non-crossing" geometric conngurations built on vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. A consequence is exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Limit la...

Iterated hairpin completion is an operation on formal languages that is inspired by the hairpin formation in DNA biochemistry. Iterated hairpin completion of a word (or more precisely a singleton language) is always a context-sensitive language and for some words it is known to be non-context-free. However, it is unknown whether regularity of iterated hairpin completion of a given word is decid...

I. INTRODUCTION Let Q = [0, 1] × [0, 1] denote the unit square and let L n be a set of n line segments in Q. Two line segments are said to be crossing if they intersect at any point. A subset of line segments is called non-crossing if no two segments in the subset are crossing. Consider the scenario where the endpoints of the n line segments are randomly distributed, independently and uniformly...

Here we consider two parameters for random non-crossing trees: i the number of random cuts to destroy a sizen non-crossing tree and ii the spanning subtree-size of p randomly chosen nodes in a size-n non-crossing tree. For both quantities, we are able to characterise for n ∞ the limiting distributions. Non-crossing trees are almost conditioned Galton-Watson trees, and it has been already shown,...