Expected rank in antimatroids ✩ Gary

We consider a probabilistic antimatroid A on the ground set E, where each element e ∈ E may succeed with probability pe . We focus on the expected rank ER(A) of a subset of E as a polynomial in the pe . General formulas hold for arbitrary antimatroids, and simpler expressions are valid for certain well-studied classes, including trees, rooted trees, posets, and finite subsets of the plane. We connect the Tutte polynomial of an antimatroid to ER(A). When S is a finite subset of the plane with no three points collinear, we derive an expression for the expected rank that has surprising symmetry properties. Corollaries include new formulas involving the beta invariant of subsets of S and new proofs of some known formulas.  2003 Elsevier Inc. All rights reserved.