On the number of slim, semimodular lattices

TL;DR

This paper counts and characterizes slim, semimodular lattices, providing a recursive formula for their enumeration and linking their planar diagrams to Catalan numbers, with combinatorial insights.

Contribution

It introduces a recursive method to determine the number of slim, semimodular lattices of a given size and enumerates their planar diagrams, connecting to Catalan numbers.

Findings

Number of such lattices of size n determined recursively

Planar diagrams of these lattices enumerated up to similarity

Number of diagrams of slim, distributive lattices of length b equals the Catalan number

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. Cz\'edli 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 diagrams, up to similarity, are also enumerated. We prove that the number of diagrams of slim, distributive lattices of a given length b is the n-th Catalan number. Besides lattice theory, the paper includes some combinatorial arguments on permutations and their inversions.