A matrix approach to the structure, enumeration, and applications of partially ordered sets
Combinatorics
2026-02-05 v1
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.
Cite
@article{arxiv.2602.04533,
title = {A matrix approach to the structure, enumeration, and applications of partially ordered sets},
author = {Gi-Sang Cheon and Hong Joon Choi and Gukwon Kwon and Hojoon Lee and Yaling Wang},
journal= {arXiv preprint arXiv:2602.04533},
year = {2026}
}