OFFSET
1,3
COMMENTS
The set of maximal elements of an ideal is an antichain; conversely, the down-set of a nonempty antichain is an ideal. The down-set of the top element of the n-dimensional Boolean lattice contains all 2^n elements of the lattice, and thus is not a proper ideal.
Empirically, the rows are unimodal.
By the Markowsky paper, T(n, k) = T(n, 2^n - k).
Also, T(n,k) is the number of n-dimensional Ferrers diagrams with k nodes (i.e., (n-1)-dimensional partitions) that fit into an n-dimensional hypercube of side 2 (i.e., a Boolean or binary hupercube). T(n, k) = T(n, 2^n - k) follows from the map that takes a Ferrers diagram to its complement in the box. - Suresh Govindarajan, Apr 10 2016
LINKS
Danny Rorabaugh and Suresh Govindarajan, Table of n, a(n) for n = 1..279
George Markowsky, The level polynomials of the free distributive lattices, Discrete Math., 29 (1980), 275-285.
Wikipedia, Boolean algebra and Ideal.
EXAMPLE
For row n = 3, the k-element proper ideals are the down-sets of the following antichains:
T(3, 1) = 1: [{}];
T(3, 2) = 3: [{0}], [{1}], [{2}];
T(3, 3) = 3: [{0},{1}], [{0},{2}], [{1},{2}];
T(3, 4) = 4: [{0,1}], [{0,2}], [{1,2}], [{0},{1},{2}];
T(3, 5) = 3: [{0,1},{2}], [{0,2},{1}], [{1,2},{0}];
T(3, 6) = 3: [{0,1},{0,2}], [{0,1},{1,2}], [{0,2},{1,2}];
T(3, 7) = 1: [{0,1},{0,2},{1,2}].
E.g., the 5-element down-set of [{0,1},{2}] is [{},{0},{1},{2},{0,1}].
The table begins:
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 1
2 1 2 1
3 1 3 3 4 3 3 1
4 1 4 6 10 13 18 19 24 19 18 13 10 6 4 1
5 1 5 10 20 35 61 95 155 215 310 387 470 530 580 605 621 605 ...
PROG
(Sage) # Returns row n.
def T(n):
B = posets.BooleanLattice(n)
t = [0]*(2^n + 1)
for A in B.antichains():
t[len(B.order_ideal(A))] += 1
return t[1:-1]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Danny Rorabaugh, Mar 03 2016
STATUS
approved