On the purity of maximal weakly separated set families

TL;DR

This paper provides a simple, elementary combinatorial proof that all maximal weakly separated set families within a certain size range have equal cardinality, highlighting a uniformity property in their structure.

Contribution

It introduces a direct, elementary proof establishing the size uniformity of maximal weakly separated set families within specified bounds.

Findings

Maximal weakly separated set families have the same size within given bounds.

Elementary combinatorics of lattice paths suffices for the proof.

The result simplifies understanding of the structure of these set families.

Abstract

We present a short proof that every maximal family of weakly separated subsets of $[n]$ of cardinality between $[a,b]$ have the same size. Our proof is direct and only uses elementary combinatorics of lattice paths.