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The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings

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Abstract

Let \((\mathcal {P},\leqslant )\) be a finite poset. Define the numbers a1,a2,… (respectively, c1,c2,…) so that a1 + … + ak (respectively, c1 + … + ck) is the maximal number of elements of \(\mathcal {P}\) which may be covered by k antichains (respectively, k chains.) Then the number \(e(\mathcal {P})\) of linear extensions of poset \(\mathcal {P}\) is not less than \(\prod a_{i}!\) and not more than \(n!/\prod c_{i}!\). A corollary: if \(\mathcal {P}\) is partitioned onto disjoint antichains of sizes b1,b2,…, then \(e(\mathcal {P})\geqslant \prod b_{i}!\).

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Funding

The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.

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Authors and Affiliations

  1. St. Petersburg State University, Saint Petersburg, Russia

    I. A. Bochkov & F. V. Petrov

  2. St. Petersburg Department of V. A. Steklov Mathematical Institute RAS, St. Petersburg State University, Saint Petersburg, Russia

    F. V. Petrov

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  1. I. A. Bochkov
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  2. F. V. Petrov
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Correspondence to F. V. Petrov.

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Bochkov, I.A., Petrov, F.V. The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings. Order 38, 323–328 (2021). https://doi.org/10.1007/s11083-020-09542-3

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  • DOI: https://doi.org/10.1007/s11083-020-09542-3

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