Marc Noy

Analytic combinatorics of non-crossing configurations

This paper describes a systematic approach to the enumeration of ‘non-crossing’ geometric congurations 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. Limit laws of the Gaussian type are also established in this framework; they concern a variety of parameters like number of leaves in trees, number of components or edges in graphs, etc. c 1999 Elsevier Science B.V. All rights reserved

Consecutive patterns in permutations

In this paper we study the distribution of the number of occurrences of a permutation σ as a subword among all permutations in Sn. We solve the problem in several cases depending on the shape of σ by obtaining the corresponding bivariate exponential generating functions as solutions of certain linear differential equations with polynomial coefficients. Our method is based on the representation of permutations as increasing binary trees and on symbolic methods.

Lattice path matroids: enumerative aspects and Tutte polynomials

Fix two lattice paths P and Q from (0,0) to (m, r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0,0) to (m, r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the β invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the β invariant of certain lattice path matroids.

Bipartite embedding of trees in the plane

Given a tree T on n vertices and a set P of n points in the plane in general position, it is known that T can be straight line embedded in P without crossings. Now imagine the set P is partitioned into two disjoint subsets R and B, and we ask for an embedding of T in P without crossings and with the property that all edges join a point in R (red) and a point in B (blue). In this case we say that T admits a bipartite embedding with respect to the bipartition (R, B). Examples show that the problem in its full generality is not solvable. In view of this fact we consider several embedding problems and study for which bipartitions they can be solved. We present several results that are valid for any bipartition (R, B) in general position, and some other results that hold for particular configurations of points.

Graph of triangulations of a convex polygon and tree of triangulations

Abstract Define a graph G T ( n ) with one node for each triangulation of a convex n-gon. Place an edge between each pair of nodes that differ by a single flip: two triangles forming a quadrilateral are exchanged for the other pair of triangles forming the same quadrilateral. In this paper we introduce a tree of all triangulations of polygons with any number of vertices which gives a unified framework in which several results on G T ( n ) admit new and simple proofs.

Enumeration of noncrossing trees on a circle

We consider several enumerative problems concerning labelled trees whose vertices lie on a circle and whose edges are rectilinear and do not cross.

A lower bound on the number of triangulations of planar point sets

We show that the number of straight-edge triangulations exhibited by any set of n points in general position in the plane is bounded from below by Ω(2.33n).

Separating objects in the plane by wedges and strips

In this paper we study the separability of two disjoint sets of objects in the plane according to two criteria: wedge separability and strip separability. We give algorithms for computing all the separating wedges and strips, the wedges with the maximum and minimum angle, and the narrowest and the widest strip. The objects we consider are points, segments, polygons and circles. As applications, we improve the computation of all the largest circles separating two sets of line segments by a logn factor, and we generalize the algorithm for computing the minimum polygonal separator of two sets of points to two sets of line segments with the same running time. ? 2001 Elsevier Science B.V. All rights reserved.

Computing the Tutte Polynomial on Graphs of Bounded Clique-Width

The Tutte polynomial is a notoriously hard graph invariant, and efficient algorithms for it are known only for a few special graph classes, like for those of bounded tree-width. The notion of clique-width extends the definition of cograhs (graphs without induced P4), and it is a more general notion than that of tree-width. We show a subexponential algorithm (running in time expO(n2/3)) for computing the Tutte polynomial on cographs. The algorithm can be extended to a subexponential algorithm computing the Tutte polynomial on on all graphs of bounded clique-width. In fact, our algorithm computes the more general U-polynomial. 2000 Math Subjects Classification: 05C85, 68R10.

Graphs determined by polynomial invariants

Many polynomials have been defined associated to graphs, like the characteristic, matchings, chromatic and Tutte polynomials. Besides their intrinsic interest, they encode useful combinatorial information about the given graph. It is natural then to ask to what extent any of these polynomials determines a graph and, in particular, whether one can find graphs that can be uniquely determined by a given polynomial. In this paper we survey known results in this area and, at the same time, we present some new results.