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The asymptotic number of planar, slim, semimodular lattice diagrams

Gábor Czédli

Abstract

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. We prove that there exists a positive constant C such that, up to similarity, the number of planar diagrams of these lattices of size n is asymptotically C times 2 to the n.

The asymptotic number of planar, slim, semimodular lattice diagrams

Abstract

A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. We prove that there exists a positive constant C such that, up to similarity, the number of planar diagrams of these lattices of size n is asymptotically C times 2 to the n.

Paper Structure

This paper contains 3 sections, 8 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction and the result
  2. Lattice theoretic lemmas
  3. Tools from Analysis at work

Key Result

Theorem 1.1

There exists a constant $C$ such that $0<C<1$ and $N_{\textup{ssd}}(n)$ is asymptotically $C\cdot 2^n$, that is, $\lim_{n\to\infty}\bigl(N_{\textup{ssd}}(n)/ 2^n\bigr)=C$.

Figures (2)

  • Figure 1: Left and right ranks
  • Figure 2: An illustration to Lemma \ref{['loglemma']}

Theorems & Definitions (15)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 5 more