Let c k = cr k (G) denote the minimum number of edge crossings when a graph G is drawn on an orientable surface of genus k. The (orientable) crossing sequence c 0 ; c 1 ; c 2 ; : : : encodes the trade-oo between adding handles and decreasing crossings. We focus on sequences of the type c 0 > c 1 > c 2 = 0; equivalently, we study the planar and toroidal crossing number of doubly-toroidal graphs....

We address two problems. In the first, we refine the analysis of a lower bound construction of a point set having many non-crossing spanning trees. We also give more precise upper bounds on the maximum number of non-crossing spanning trees, perfect matchings and simple polygons of a planar point set. In the second, we give an improved lower bound construction for the d-interval problem. 1 Non-c...

In this paper a new nonparametric estimate of conditional quantiles is proposed, that avoids the problem of crossing quantile curves [calculated for various p ∈ (0, 1)]. The method uses an initial estimate of the conditional distribution function in a first step and solves the problem of inversion and monotonization with respect to p ∈ (0, 1) simultaneously. It is demonstrated that the new esti...

For any finite Coxeter group W of rank n we show that the order complex of the lattice of non-crossing partitions NC(W ) embeds as a chamber subcomplex into a spherical building of type An−1. We use this to give a new proof of the fact that the non-crossing partition lattice in type An is supersolvable for all n. Moreover, we show that in case Bn, this is only the case if n < 4. We also obtain ...

This paper describes a systematic approach to the enumeration of 'non-crossing' geometric configurations built on vertices of a convex n-gon in the plane. It relies on generating functions, symbolic methods, singularity analysis, and singularity perturbation. Consequences are both exact and asymptotic counting results for trees, forests, graphs, connected graphs, dissections, and partitions. Li...

Let R and B be sets of n red and n blue points in the plane, respectively, with P = R∪ B. Let M be a perfect matching between points from R and B, using n straight line segments to match the points, that is, each point is an endpoint of exactly one line segment, and each line segment has one red and one blue endpoint. We forbid line segments to cross. Denote the length of a longest line segment...

In a previous paper El], we proved results -about the enumer;ation of certain types of chains in the non-crossing partition lattice T, and its, generalizations. In this paper we present bijections to certain classes of trees which reprove one theorem [l, Corollary 3.41 and provide a combinatoridi proof for the other [I, Theorem 5.31. We begin with a review of the definitions. A set partition X ...

We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the noncrossing cumulants of R. Speicher. Here we introduce the concept of no...