Four Variations on Graded Posets

TL;DR

This paper studies the enumeration and structure of various classes of graded posets, providing new theorems and generalizations, and connecting classical results with modern combinatorial questions.

Contribution

It introduces new enumerative and structural results for classes of graded posets, generalizes existing theorems, and addresses open questions in the field.

Findings

Enumerative formulas for graded posets

Structural theorems for (2+2)- and (3+1)-avoiding posets

Connections between labeled and unlabeled poset enumeration

Abstract

We explore the enumeration of some natural classes of graded posets, including all graded posets, (2+2)- and (3+1)-avoiding graded posets, (2+2)-avoiding graded posets, and (3+1)-avoiding graded posets. We obtain enumerative and structural theorems for all of them. Along the way, we discuss a situation when we can switch between enumeration of labeled and unlabeled objects with ease, generalize a result of Postnikov and Stanley from the theory of hyperplane arrangements, answer a question posed by Stanley, and see an old result of Klarner in a new light.