A generalization of a theorem of Ern\'{e}

TL;DR

This paper generalizes Erne9's theorem by establishing a correspondence between posets containing a sub-poset and those with a related induced sub-poset on an extended set, broadening the understanding of poset enumeration.

Contribution

It extends Erne9's theorem to a broader class of posets by relating counts of posets containing a sub-poset to those with a specific induced sub-poset on an augmented set.

Findings

Established a bijection between posets with a sub-poset and those with an induced sub-poset on an extended set.

Generalized Erne9's theorem to include arbitrary posets $Q$ as induced sub-posets.

Provided a new combinatorial framework for counting posets with specified sub-structures.

Abstract

Let $X$ be a finite set, $Z \subseteq X$ and $y \notin X$. Marcel Ern\'{e} showed in 1981, that the number of posets on $X$ containing $Z$ as an antichain equals the number of posets $R$ on $X \cup \{ y \}$ in which the points of $Z \cup \{ y \}$ are exactly the maximal points of $R$. We prove the following generalization: For every poset $Q$ with carrier $Z$, the number of posets on $X$ containing $Q$ as an induced sub-poset equals the number of posets $R$ on $X \cup \{ y \}$ which contain $Q^d + A_y$ as an induced sub-poset and in which the maximal points of $Q^d + A_y$ are exactly the maximal points of $R$. Here, $Q^d$ is the dual of $Q$, $A_y$ is the singleton-poset on $y$, and $Q^d + A_y$ denotes the direct sum of $Q^d$ and $A_y$.