A matrix approach to the structure, enumeration, and applications of partially ordered sets
Gi-Sang Cheon, Hong Joon Choi, Gukwon Kwon, Hojoon Lee, Yaling Wang
TL;DR
The paper develops a matrix-centric framework for finite posets by encoding naturally labelled posets as Boolean, unit lower triangular poset matrices and showing every NL poset embeds as an induced submatrix of the binary Pascal matrix $P_{2^n}$. It introduces domination and Pascal-equivalence to classify poset matrices up to isomorphism, providing an algorithmic route to enumerate non-isomorphic NL posets via domination operations. Duality is developed through a flip-transpose relation, yielding conditions for self-duality and linking duals to paired index vectors. A novel matrix formulation of Dedekind’s problem is given through the Dedekind-Pascal numbers $D_P(n)$, equating antichain counts with solutions to the Boolean fixed-point equation $x{f P}_n=x$ and recovering the classical $M(k)$ values at $n=2^k$. Overall, the work bridges poset enumeration with Pascal-matrix structure, offering a constructive, matrix-based toolkit for Birkhoff- and Dedekind-type problems and enabling algorithmic, scalable analysis.
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.