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On the number of slim, semimodular lattices

  • Gábor Czédli EMAIL logo , Tamás Dékány , László Ozsvárt , Nóra Szakács and Balázs Udvari
Published/Copyright: April 26, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 1

Abstract

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Czédli and E. T. Schmidt as the duals of the lattices consisting of the intersections of the members of two composition series in a group. Our main result determines the number of (isomorphism classes of) these lattices of a given size in a recursive way. The corresponding planar Hasse diagrams, up to similarity, are also enumerated. We prove that the number of diagrams of slim, distributive lattices of a given length n is the nth Catalan number. Besides lattice theory, the paper includes some combinatorial arguments on permutations and their inversions.

Keywords: composition series; Jordan-Hölder theorem; counting lattices; number of inversions; number of permutations; slim lattice; planar lattice; semimodular lattice
MSC: Primary 06C10

This research was supported by the NFSR of Hungary (OTKA), grant numbers K77432 and K83219, and by TÁMOP-4.2.1/B-09/1/KONV-2010-0005 and TÁMOP-4.2.2/B-10/1-2010-0012.

  1. This paper has been communicated by Miroslav Ploščica.

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Received: 2012-6-5
Accepted: 2013-10-9
Published Online: 2016-4-26
Published in Print: 2016-2-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0111
Keywords for this article
composition series; Jordan-Hölder theorem; counting lattices; number of inversions; number of permutations; slim lattice; planar lattice; semimodular lattice

Articles in the same Issue

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  2. Prof. RNDr. Ján Jakubík, DrSc. passed away
  3. Research Article
  4. On the number of slim, semimodular lattices
  5. Research Article
  6. Quantum computational algebra with a non-commutative generalization
  7. Research Article
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  9. Research Article
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  13. Research Article
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