On the number of distributive lattices
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.
scientific article; zbMATH DE number 1757090