On the number of distributive lattices

Abstract

We investigate the numbers dk 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 vk of vertically indecomposable distributive lattices of size k. We present the explicit values of the numbers dk and vk for k < 50 and prove the following exponential bounds: 1.67k < vk lt; 2.33k and 1.84k < dk < 2.39k (k ≥ k 0). Important tools are (i) an algorithm coding all unlabeled distributive lattices of height n and size k by certain integer sequences 0 = z1 ≤ ⋯ ≤ zn ≤ k - 2, and (ii) a "canonical 2-decomposition" of ordinally indecomposable posets into "2-indecomposable" canonical summands.