A matrix approach to the enumeration of naturally labeled posets

TL;DR

The paper presents a matrix-based framework to enumerate naturally labeled posets by representing each poset with a poset matrix and exploring v-extensions that preserve transitivity when the added element corresponds to an order ideal. It establishes a tight link between poset vectors, order ideals, and distributive lattices, enabling recursive counting via automorphism group actions and Burnside’s lemma, and provides a constructive generation scheme by relating extensions to topological growth in distributive lattices per Birkhoff’s theorem. The twin-class decomposition offers structural insights into automorphism groups, yielding criteria for trivial automorphisms and aiding symmetry-aware enumeration. Together, these results yield both theoretical and algorithmic tools for the constructive enumeration of NL posets and their distribution lattices.

Abstract

We propose a matrix approach for enumerating naturally labeled posets by representing each poset $P$ on $[n]$ as a Boolean poset matrix $A$. This algebraic representation enables a systematic handling of partial orderings through $v$-extensions of the form $A^v=\bigl[\begin{smallmatrix}A&0\\ v&1\end{smallmatrix}\bigr]$. We show that $A^v$ defines a valid poset matrix if and only if the Boolean vector $v$ represents an order ideal of the poset $P$ associated to $A$, equivalently satisfying the fixed-point equation $vA=v$. Furthermore, we explore the twin-class decomposition of $A$, which partitions the elements of $P$ according to identical down- and up-sets. Finally, we present an algorithmic generation scheme for the posets based on the topological growth of their distribution lattices, offering a new approach to constructive enumeration of poset families.

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Theorems & Definitions (65)

• Proposition 1.1
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Theorem 2.4
• proof
• Corollary 2.5
• proof