Linear extensions and directed clique counts via modular partitions

TL;DR

This paper introduces recursive formulas for counting linear extensions of certain structured posets, enabling explicit generation and analysis of these extensions with applications to neural circuit structures.

Contribution

It presents new recursive formulas for posets with modular partitions having tree or necklace structures, expanding computational methods for linear extension counting.

Findings

Recursive formulas for specific poset classes

Explicit generation of linear extensions

Applications to neural circuit analysis

Abstract

Counting linear extensions is a fundamental problem in poset theory. It is known to be #P-complete, with polynomial-time formulas available in special cases. In this work, we develop new recursive formulas for counting linear extensions of posets whose modular partitions have particular structure. Specifically, we focus on posets whose incomparability graph has a modular partition with a skeleton that is a tree, a necklace of cliques, or a combination of both. The proofs are constructive and allow for the explicit generation of all linear extensions. We also discuss equivalent formulations of the problem in terms of permutations and directed graphs. The directed graph perspective is related to counting directed simplices in the directed flag complex of a digraph, with applications to understanding higher-order structure in neural circuits.