Enumerative combinatorics
Constructive combinatorics
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
A free subalgebra of the algebra of matroids
European Journal of Combinatorics
A unique factorization theorem for matroids
Journal of Combinatorial Theory Series A
Lattice path matroids: structural properties
European Journal of Combinatorics
A solution to the tennis ball problem
Theoretical Computer Science - In memoriam: Alberto Del Lungo (1965-2003)
Combinatorics, Probability and Computing
Note: Simple formulas for lattice paths avoiding certain periodic staircase boundaries
Journal of Combinatorial Theory Series A
The number of lattice paths below a cyclically shifting boundary
Journal of Combinatorial Theory Series A
Lattice path matroids: The excluded minors
Journal of Combinatorial Theory Series B
Some inequalities for the Tutte polynomial
European Journal of Combinatorics
European Journal of Combinatorics
Positroids and Schubert matroids
Journal of Combinatorial Theory Series A
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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.