A matrix approach to the structure, enumeration, and applications of partially ordered sets

TL;DR

This paper introduces a matrix-theoretic framework using poset matrices and Pascal matrices to study, enumerate, and analyze finite partially ordered sets, offering new structural insights and unified treatment of classical problems.

Contribution

It develops a systematic matrix approach for representing and enumerating finite posets, connecting poset theory with matrix analysis and Pascal matrices.

Findings

Poset matrices can be represented as principal submatrices of the Pascal matrix.

The matrix approach unifies the treatment of Birkhoff and Dedekind problems.

Structural insights are gained through permutation similarity and domination relations.

Abstract

We present a matrix-theoretic approach for studying and enumerating finite posets through their incidence representations, referred to as poset matrices. Naturally labelled posets are encoded as Boolean lower triangular matrices, allowing a unified treatment of Birkhoff problem on non-isomorphic posets and Dedekind problem on antichains. A key idea is a systematic construction and indexing of poset matrices as principal submatrices of the binary Pascal matrix, leading to new structural insights through permutation similarity and domination relations. This approach provides a consistent matrix-based perspective on classical enumeration problems in poset theory.