A382829 Number of distinct rank vectors of distributive lattices of height n.

1, 1, 2, 5, 15, 51, 197, 864, 4325, 24922

Offset

0,3

Comments

Distributive lattices are ranked posets, and we define the rank vector of a ranked poset P as the vector whose k-th coordinate (starting at k = 0) is the number of elements of rank k in P.

By Birkhoff's representation theorem, elements of a finite distributive lattice L are in bijection with lower sets of the poset of join-irreducible elements of L, an element of rank k corresponding to a lower of set size k.

Links

Table of n, a(n) for n=0..9.

Example

The rank vectors corresponding to a(4) = 15 are:
  (1, 1, 1, 1, 1),   (1, 1, 1, 2, 1),   (1, 1, 2, 1, 1),
  (1, 1, 2, 2, 1),   (1, 1, 3, 3, 1),   (1, 2, 1, 1, 1),
  (1, 2, 1, 2, 1),   (1, 2, 2, 1, 1),   (1, 2, 2, 2, 1),
  (1, 2, 3, 2, 1),   (1, 2, 3, 3, 1),   (1, 3, 3, 1, 1),
  (1, 3, 3, 2, 1),   (1, 3, 4, 3, 1),   (1, 4, 6, 4, 1).
Two non-isomorphic distributive lattices have for rank vector (1, 2, 2, 2, 1).

Crossrefs

Cf. A000112, A006982.

Sequence in context: A001681 A343665 A192553 * A053553 A385760 A276721

Adjacent sequences: A382826 A382827 A382828 * A382830 A382831 A382832

Keyword

nonn,more

Author

Ludovic Schwob, Apr 06 2025

Status

approved