Affiliation:
1. Bolyai Institute University of Szeged Aradi vértanúk tere 1 6720 Szeged Hungary,
Abstract
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.
Reference28 articles.
1. Bóna, M.: Combinatorics of Permutations. Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004.
2. Czédli, G.: The matrix of a slim semimodular lattice, Order 29 (2012), 85–103.
3. Czédli, G.-Grätzer, G.: Planar semimodular lattices and their diagrams. In: Lattice Theory: Special Topics and Applications (G. Grätzer, F. Wehrung, eds.), Birkhäuser, Basel, 2014.
4. Czédli, G.-Ozsvárt, L.-Udvari, B.: How many ways can two composition series intersect?, Discrete Math. 312 (2012), 3523–3536.
5. Czédli, G.-Schmidt, E. T.: Some results on semimodular lattices. In: Contributions to General Algebra 19 (Proc. Conf. Olomouc 2010), Johannes Hein verlag, Klagenfurt, 2010, pp. 45–56.
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献