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

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 $A_y$.