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

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

Offset: 0

Author

Ludovic Schwob, Apr 06 2025

Keywords

• nonn
• more

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.

Examples

			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.