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 XZ. Marcel Erné showed in 1981, that the number of posets on XZ containing Z as an antichain equals the number of posets R on XZy 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 XZ 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.