Counting linear extensions of restricted posets

TL;DR

This paper extends the #P-completeness result for counting linear extensions to restricted classes of posets, including height two, incidence posets of graphs, and dimension two, highlighting computational complexity in these cases.

Contribution

It proves #P-completeness for counting linear extensions in new restricted classes of posets, such as height two and dimension two, expanding prior complexity results.

Findings

Counting linear extensions is #P-complete for height two posets.

Counting linear extensions is #P-complete for incidence posets of graphs.

Counting linear extensions is #P-complete for dimension two posets.

Abstract

The classical 1991 result by Brightwell and Winkler states that the number of linear extensions of a poset is #P-complete. We extend this result to posets with certain restrictions. First, we prove that the number of linear extension for posets of height two is #P-complete. Furthermore, we prove that this holds for incidence posets of graphs. Finally, we prove that the number of linear extensions for posets of dimension two is #P-complete.