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A generalization of a theorem of Erné (2105.00711v1)
Published 3 May 2021 in math.CO
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 $Qd + A_y$ as an induced sub-poset and in which the maximal points of $Qd + A_y$ are exactly the maximal points of $R$. Here, $Qd$ is the dual of $Q$, $A_y$ is the singleton-poset on $y$, and $Qd + A_y$ denotes the direct sum of $Qd$ and $A_y$.