Antimatroids and Balanced Pairs

TL;DR

This paper extends the 1/3-2/3 conjecture from partial orders to antimatroids, proving it for specific classes and verifying it computationally for small cases, advancing understanding of element ordering probabilities.

Contribution

It generalizes the 1/3-2/3 conjecture to antimatroids and proves it for multiple classes, including convex dimension two and height two, with computational verification for small cases.

Findings

Conjecture holds for convex dimension two antimatroids.

Conjecture verified for antimatroids of height two.

Computer search confirms the conjecture for all antimatroids with up to six elements.

Abstract

We generalize the 1/3-2/3 conjecture from partially ordered sets to antimatroids: we conjecture that any antimatroid has a pair of elements x,y such that x has probability between 1/3 and 2/3 of appearing earlier than y in a uniformly random basic word of the antimatroid. We prove the conjecture for antimatroids of convex dimension two (the antimatroid-theoretic analogue of partial orders of width two), for antimatroids of height two, for antimatroids with an independent element, and for the perfect elimination antimatroids and node search antimatroids of several classes of graphs. A computer search shows that the conjecture is true for all antimatroids with at most six elements.