Abstract
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.
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I am grateful to the anonymous reviewer #1 for his numerous valuable comments and suggestions.
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The author did not receive support from any organization for the submitted work. He has no relevant financial or non-financial interests to disclose, and he has no conflicts of interest to declare that are relevant to the content of this article. The author certifies that he has no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. The author has no financial or proprietary interests in any material discussed in this article. Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
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Campo, F.a. A Generalization of a Theorem of Erné about the Number of Posets with a Fixed Antichain. Order 39, 421–434 (2022). https://doi.org/10.1007/s11083-021-09585-0
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DOI: https://doi.org/10.1007/s11083-021-09585-0