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

Let X be a finite set, Z ⊆ X and y / ∈ X. Marcel Erné showed in 1981, that the number of posets on X containing Z as an antichain equals the number of posets R on X ∪{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 containing Q as an induced sub-poset equals the number of posets R on X ∪ {y} which contain Q+Ay as an induced sub-poset and in which the maximal points of Q + Ay are exactly the maximal points of R. Here, Q d is the dual of Q, Ay is the singleton-poset on y, and Q d +Ay denotes the direct sum of Q and Ay. Mathematics Subject Classification: Primary: 06A07. Secondary: 06A06.