On the number of distributive lattices
type: Item
instance of: scholarly article
title: On the number of distributive lattices
zbMATH Open document ID: 0989.05005
author: Marcel Erné
author: Jobst Heitzig
author: Jürgen Reinhold
published in: The Electronic Journal of Combinatorics
publication date: 2002-07-01T00:00:00Z
full work available at URL: https://eudml.org/doc/121886
full work available at URL: v9i1r24.html
review text: Summary: We investigate the numbers \(d_k\) of all (isomorphism classes of) distributive lattices with \(k\) elements, or, equivalently, of (unlabeled) posets with \(k\) antichains. Closely related and useful for combinatorial identities and inequalities are the numbers \(v_k\) of vertically indecomposable distributive lattices of size \(k\). We present the explicit values of the numbers \(d_k\) and \(v_k\) for \(k<50\) and prove the following exponential bounds: \(1.67^k<v_k<2.33^k\) and \(1.84^k<d_k<2.39^k\) \((k\geq k_0)\). Important tools are (i) an algorithm coding all unlabeled distributive lattices of height \(n\) and size \(k\) by certain integer sequences \(0=z_1\leq\cdots\leq z_n\leq k-2\), and (ii) a ``canonical 2-decomposition'' of ordinally indecomposable posets into ``2-indecomposable'' canonical summands.
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 05A16
Mathematics Subject Classification ID: 06A07
zbMATH DE Number: 1757090
zbMATH Keywords: canonical poset
zbMATH Keywords: ordinal (vertical) decomposition
zbMATH Keywords: combinatorial identities
zbMATH Keywords: distributive lattices
zbMATH Keywords: indecomposable posets
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1757090