23 research outputs found
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
Borel generators
We use the notion of Borel generators to give alternative methods for
computing standard invariants, such as associated primes, Hilbert series, and
Betti numbers, of Borel ideals. Because there are generally few Borel
generators relative to ordinary generators, this enables one to do manual
computations much more easily. Moreover, this perspective allows us to find new
connections to combinatorics involving Catalan numbers and their
generalizations. We conclude with a surprising result relating the Betti
numbers of certain principal Borel ideals to the number of pointed
pseudo-triangulations of particular planar point sets.Comment: 23 pages, 2 figures; very minor changes in v2. To appear in J.
Algebr
Generalizing the Borel property
We introduce the notion of Q-Borel ideals: ideals which are closed under the
Borel moves arising from a poset Q. We study decompositions and homological
properties of these ideals, and offer evidence that they interpolate between
Borel ideals and arbitrary monomial ideals.Comment: 19 pages, 1 figur