A Generalization of a Theorem of Erné about the Number of Posets with a Fixed Antichain

Let X and Z be finite disjoint sets and let y be a point not contained in X Z . Marcel Erné showed in 1981, that the number of posets on X Z containing Z as an antichain equals the number of posets R on X Z y in which the points of Z { 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 Z containing Q as an induced sub-poset equals the number of posets R on X ∪ Z ∪{ y } which contain Q + y as an induced sub-poset and in which the maximal points of Q + y are exactly the maximal points of R . Here, Q + y denotes the direct sum of Q and the singleton-poset on y .