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Linear extensions and directed clique counts via modular partitions

Daniela Egas Santander, Matteo Santoro, Jason P. Smith

TL;DR

The paper tackles the difficulty of counting linear extensions, a $\#P$-complete problem, by deriving recursive formulas for posets whose incomparability graphs admit modular partitions with skeletons that are trees, necklaces of cliques, or their combinations; a constructive pivot-based decomposition yields explicit enumeration via products over modular counts and structured binomial-sum terms. It connects LE counting to equivalent formulations in permutations and directed graphs, including edge-induced transitive tournaments and Hamiltonian paths, and extends these insights to applications in neuroscience and network design. The main technical contribution is a general theorem that formalizes how to partition the interval of a linear extension using pivots and to count local extensions within each interval, enabling closed forms in several structured cases (path, necklace, tree) and providing a pathway to constructive generation of all linear extensions. The work thus offers both theoretical advances in modular-decomposition-based counting and practical tools for constructing and analyzing complex directed networks with prescribed higher-order clique structures.

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.

Linear extensions and directed clique counts via modular partitions

TL;DR

The paper tackles the difficulty of counting linear extensions, a -complete problem, by deriving recursive formulas for posets whose incomparability graphs admit modular partitions with skeletons that are trees, necklaces of cliques, or their combinations; a constructive pivot-based decomposition yields explicit enumeration via products over modular counts and structured binomial-sum terms. It connects LE counting to equivalent formulations in permutations and directed graphs, including edge-induced transitive tournaments and Hamiltonian paths, and extends these insights to applications in neuroscience and network design. The main technical contribution is a general theorem that formalizes how to partition the interval of a linear extension using pivots and to count local extensions within each interval, enabling closed forms in several structured cases (path, necklace, tree) and providing a pathway to constructive generation of all linear extensions. The work thus offers both theoretical advances in modular-decomposition-based counting and practical tools for constructing and analyzing complex directed networks with prescribed higher-order clique structures.

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.
Paper Structure (11 sections, 12 theorems, 37 equations, 8 figures)

This paper contains 11 sections, 12 theorems, 37 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions, notation, and conventions
  4. Linear extensions of a poset and modular partitions
  5. Counting linear extensions in three special cases: necklaces, trees and paths
  6. Counting linear extensions via modular partitions of the incomparability graph
  7. Main result and proofs
  8. When is a partition of the incomparability graph not a poset partition
  9. Applications of linear extensions
  10. Equivalent counting problems
  11. Applications to the sciences

Key Result

Lemma 2.15

Sta11 Given any two posets $\mathcal{P}$ and $\mathcal{Q}$ we have

Figures (8)

  • Figure 1: Graphs associated to the chain poset $\mathbf{4}$. A: The graph $G_\mathbf{4}$. B: The comparability graph $\mathrm{CG}_\mathbf{4}$. C: The incomparability graph $\mathrm{IG}_\mathbf{4}$. D: The Hasse diagram of $\mathbf{4}$.
  • Figure 2: Modular partition and its skeleton.A: The Hasse diagram of a poset $\mathcal{P}$. B: The incomparability graph $\mathrm{IG}_\mathcal{P}$ with a modular partition $\mathcal{M}$. C: The skeleton graph of $\mathrm{IG}_\mathcal{P}/\mathcal{M}$. D: The formula for the number of linear extensions of $\mathcal{P}$.
  • Figure 3: Examples of the main results.A: The path, B: the necklace and C: the binary tree. For each: Left: the poset $\mathcal{P}$, Middle: the incomparability graph $\mathrm{IG}_\mathcal{P}$ with a modular partition, Right: the skeleton graph of the modular partition, Below: the formula for the number of linear extensions.
  • Figure 4: A: Incomparability graph of the poset and linear extension. Dash gray lines represent edges that may or may not be in the incomparability graph. All odd modules are non-empty while even modules are allowed to be empty. The red stars in the odd modules show the pivots giving the decomposition of $[n]$ into intervals. B: Possible structures of the poset $\mathcal{P}_{2d}$.
  • Figure 5: A: The transitive tournament $T_2$ and $T_2$ with a reverse edge added, and the values of $C_2$ the number edge-induced transitive 2-tournaments in each. B: The transitive tournament $T_3$ with two different reverse edges added, with the corresponding values of $C_3(R)$ and posets $P_3(R)$ beneath. C: The Hamiltonian path through the transitive tournament $T$ and corresponding permutation $\phi(T)$.
  • ...and 3 more figures

Theorems & Definitions (52)

  • Definition 2.2: Subgraphs
  • Definition 2.3: Transitive graphs
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7: Graphs associated to a poset
  • Definition 2.8
  • Definition 2.9: Modular partition
  • Definition 2.10: Modular partition of a digraph and poset
  • Definition 2.11: Skeleton of a modular partition
  • ...and 42 more