Anna de Mier

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.

k -noncrossing and k -nonnesting graphs and fillings of ferrers diagrams

We give a correspondence between graphs with a given degree sequence and fillings of Ferrers diagrams by nonnegative integers with prescribed row and column sums. In this setting, k-crossings and k-nestings of the graph become occurrences of the identity and the antiidentity matrices in the filling. We use this to show the equality of the numbers of k-noncrossing and k-nonnesting graphs with a given degree sequence. This generalizes the analogous result for matchings and partition graphs of Chen, Deng, Du, Stanley, and Yan, and extends results of Klazar to k > 2. Moreover, this correspondence reinforces the links recently discovered by Krattenthaler between fillings of diagrams and the results of Chen et al.

Lattice path matroids: structural properties

This paper studies structural aspects of lattice path matroids. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One of the main results is a characterization of lattice path matroids in terms of fundamental flats, which are special connected flats from which one can recover the paths that define the matroid. We examine some aspects related to key topics in the literature of transversal matroids and we determine the connectivity of lattice path matroids. We also introduce notch matroids, a minor-closed, dual-closed subclass of lattice path matroids, and we find their excluded minors.

A solution to the tennis ball problem

We present a complete solution to the so-called tennis ball problem, which is equivalent to counting the number of lattice paths in the plane that use North and East steps and lie between certain boundaries. The solution takes the form of explicit expressions for the corresponding generating functions.Our method is based on the properties of Tutte polynomials of matroids associated to lattice paths. We also show how the same method provides a solution to a wide generalization of the problem.

Tutte polynomials of generalized parallel connections

We use weighted characteristic polynomials to compute Tutte polynomials of generalized parallel connections in the case in which the simplification of the maximal common restriction of the two constituent matroids is a modular flat of the simplifications of both matroids. In particular, this includes cycle matroids of graphs that are identified along complete subgraphs. We also develop formulas for Tutte polynomials of the k-sums that are obtained from such generalized parallel connections.

T-uniqueness of some families of k-chordal matroids

We define k-chordal matroids as a generalization of chordal matroids, and develop tools for proving that some k-chordal matroids are T-unique, that is, determined up to isomorphism by their Tutte polynomials. We apply this theory to wheels, whirls, free spikes, binary spikes, and certain generalizations.

Lower bounds on the maximum number of non-crossing acyclic graphs

This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has ? ( 12.5 2 N ) non-crossing spanning trees and ? ( 13.6 1 N ) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Toth (2013). Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O ( 22.1 2 N ) for the number of non-crossing spanning trees of the double chain is also obtained.

Tutte uniqueness of line graphs

We prove that if a graph H has the same Tutte polynomial as the line graph of a d-regular, d-edge-connected graph, then H is the line graph of a d-regular graph. Using this result, we prove that the line graph of a regular complete t-partite graph is uniquely determined by its Tutte polynomial. We prove the same result for the line graph of any complete bipartite graph.

On the number of self-dual rooted maps

We compute the number of self-dual rooted maps with n edges. We also compute the number of 2-connected and 3-connected self-dual rooted maps, and show that the latter are counted by the Fine numbers.

Extremal statistics on non-crossing configurations

We obtain several properties of extremal statistics in non-crossing configurations with n vertices. We prove that the maximum degree and the largest component are of logarithmic order, and the diameter is of order