Linear Extensions and Comparable Pairs in Partial Orders

TL;DR

This paper investigates the extremal counts of linear extensions in partial orders with specified comparable pairs and analyzes the typical number of linear extensions in random interval orders, revealing exponential decay related to the number of elements.

Contribution

It provides bounds on the number of linear extensions for partial orders with a given proportion of comparable pairs and characterizes the typical number in random interval orders.

Findings

Maximum and minimum linear extensions estimated for given comparable pairs.

In random interval orders, the number of linear extensions is approximately n! divided by an exponential factor.

High-probability bounds on linear extensions in random interval orders.

Abstract

We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on $n$ elements, which has close to a third of the pairs comparable with high probability: we show that the number of linear extensions is $n! \, 2^{-\Theta(n)}$ with high probability.