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The number of slim rectangular lattices

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Abstract

Slim rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. They are finite semimodular lattices L such that the ordered set Ji L of join-irreducible elements of L is the cardinal sum of two nontrivial chains. After describing these lattices of a given length n by permutations, we determine their number, |SRectL(n)|. Besides giving recursive formulas, which are effective up to about n = 1000, we also prove that |SRectL(n)| is asymptotically (n - 2)! · \({e^{2}/2}\). Similar results for patch lattices, which are special rectangular lattices introduced by G. Czédli and E. T. Schmidt in 2013, and for slim rectangular lattice diagrams are also given.

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Authors and Affiliations

  1. University of Szeged, Bolyai Institute, Szeged, Aradi vértanúk tere 1, Hungary, 6720

    Gábor Czédli, Tamás Dékány, Gergő Gyenizse & Júlia Kulin

Authors
  1. Gábor Czédli
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  2. Tamás Dékány
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  3. Gergő Gyenizse
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  4. Júlia Kulin
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Corresponding author

Correspondence to Gábor Czédli.

Additional information

Presented by J. Kung.

This research was supported by the European Union and co-funded by the European Social Fund under the project “Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences” of project number “TÁMOP-4.2.2.A-11/1/KONV-2012-0073”, and by NFSR of Hungary (OTKA), grant numbers K83219 and K104251.

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Czédli, G., Dékány, T., Gyenizse, G. et al. The number of slim rectangular lattices. Algebra Univers. 75, 33–50 (2016). https://doi.org/10.1007/s00012-015-0363-y

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  • DOI: https://doi.org/10.1007/s00012-015-0363-y

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