33 research outputs found
The Catalan matroid
We show how the set of Dyck paths of length 2n naturally gives rise to a
matroid, which we call the "Catalan matroid" C_n. We describe this matroid in
detail; among several other results, we show that C_n is self-dual, it is
representable over the rationals but not over finite fields F_q with q < n-1,
and it has a nice Tutte polynomial.
We then generalize our construction to obtain a family of matroids, which we
call "shifted matroids". They arose independently and almost simultaneously in
the work of Klivans, who showed that they are precisely the matroids whose
independence complex is a shifted complex.Comment: 17 pages; submitted to the Journal of Combinatorial Theory - Series
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 beta 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 beta invariant of certain lattice path matroids.Comment: 28 pages, 11 figure
Toric Ideals of Lattice Path Matroids and Polymatroids
We show that the toric ideal of a lattice path polymatroid is generated by
quadrics corresponding to symmetric exchanges, and give a monomial order under
which these quadrics form a Gr\"obner basis. We then obtain an analogous result
for lattice path matroids.Comment: 9 pages, 4 figure