Joseph E. Bonin

Some q-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths

Abstract We present enumerative results concerning plane lattice paths starting at the origin, with steps (1,0), (1,1) and (0,1). Such paths with a specified endpoint are counted by the Delannoy numbers, while those paths which in addition do not run above the line y=x are counted by the Schroder numbers. We develop q-analogues of the Delannoy and Schroder numbers derived from several combinatorial statistics: the number of diagonal steps, the area under the path, and the major index. We investigate the symmetry and unimodality of the resulting polynomials, and determine the asymptotic behavior of the expected number of diagonal steps and area under a path. Using the number of diagonal steps statistic, we describe the ƒ-vector of the associahedron in terms of lattice paths counted by the Schroder numbers.

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.

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.

Characterizing Combinatorial Geometries by Numerical Invariants

We show that the projective geometry PG(r? 1,q ) for r& 3 is the only rank- r(combinatorial) geometry with (qr? 1)/(q? 1) points in which all lines have at least q+ 1 points. For r= 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice.

Automorphisms of Dowling Lattices and Related Geometries

Dowling lattices are a class of geometric lattices, based on groups, which have been shown to share many properties with projective geometries. In this paper we show that the automorphisms of Dowling lattices are analogs of the automorphisms of projective geometries. We also treat similar results for several related geometric lattices.

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.

Lattice path matroids: The excluded minors

A lattice path matroid is a transversal matroid for which some antichain of intervals in some linear order on the ground set is a presentation. We characterize the minor-closed class of lattice path matroids by its excluded minors.

Multi-Path Matroids

We introduce the minor-closed, dual-closed class of multi-path matroids. We give a polynomial-time algorithm for computing the Tutte polynomial of a multi-path matroid, we describe their basis activities, and we prove some basic structural properties. Key elements of this work are two complementary perspectives we develop for these matroids: on the one hand, multi-path matroids are transversal matroids that have special types of presentations; on the other hand, the bases of multi-path matroids can be viewed as sets of lattice paths in certain planar diagrams.

Tutte polynomials of q —cones

Abstract We derive a formula for the Tutte polynomial t ( G ′; x , y ) of a q-cone G ′ of a GF( q )-representable geometry G in terms of t ( G ; x , y ). We use this to construct collections of infinite sequences of GF( q )-representable geometries in which corresponding geometries are not isomorphic and yet have the same Tutte polynomial. We also use this to construct, for each positive integer k, sets of non-isomorphic GF( q )-representable geometries all of which have the same Tutte polynomial and vertical (or Whitney) connectivity at least k.

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.