Parking function varieties for combinatorial tree models

Parking on a Random Tree

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Rényi’s parking problem, alternatively called blocking RSA (random sequential adsorption): at every vertex of the tree a particle (or “car”) arrives with rate one. The particle sticks to the vertex whenever the ver...

Varieties of Combinatorial Geometries

A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class 'S of geometries is a sequence (T„ ) of geometries in ?T with rank Tn = n, and satisfying the universal property: if G is a geometry in 5" of rank n, then G is a subgeometry of T„. A variety of geometries...

Combinatorial Secant Varieties

The construction of joins and secant varieties is studied in the combinatorial context of monomial ideals. For ideals generated by quadratic monomials, the generators of the secant ideals are obstructions to graph colorings, and this leads to a commutative algebra version of the Strong Perfect Graph Theorem. Given any projective variety and any term order, we explore whether the initial ideal o...

Hyperplane Arrangements, Parking Functions and Tree Inversions

Tree algebras and varieties of tree languages

We consider several aspects of Wilke’s [T. Wilke, An algebraic characterization of frontier testable tree languages, Theoret. Comput. Sci. 154 (1996) 85–106] tree algebra formalism for representing binary labelled trees and compare it with approaches that represent trees as terms in the traditional way. A convergent term rewriting system yields normal form representations of binary trees and co...