The asymptotic number of planar, slim, semimodular lattice diagrams

TL;DR

This paper establishes the asymptotic count of planar diagrams of slim, semimodular lattices, showing it grows exponentially with size, and identifies a constant factor in this growth.

Contribution

It proves the asymptotic number of such lattice diagrams is proportional to 2^n, providing a precise growth rate and a constant factor.

Findings

Number of diagrams grows as C * 2^n for large n

Identifies a positive constant C in the asymptotic formula

Provides a detailed enumeration of 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.