A236756 Number of (unlabeled) partially ordered sets on n elements which are Cohen-Macaulay over the integers.

1, 1, 2, 4, 9, 24, 73, 261, 1103

Offset: 0

Author

Sam DeHority, Jan 30 2014

Keywords

• nonn
• hard
• more

Examples

			For n=1, there is one poset up to isomorphism which corresponds to the simplicial complex with one facet of a single element. The homology groups over Z vanish for this, and it is therefore Cohen-Macaulay, thus a(1) = 1.
For n=2, there are two posets up to isomorphism. The first of which, the sum of two singletons, also has order complex with vanishing homology groups over Z. The second has an order complex with a single facet, and any interval has a contractible order complex, which therefore has vanishing homology groups. They are both Cohen-Macaulay thus a(2) = 2.
		

References

• R. P. Stanley, Enumerative Combinatorics, Volume 1, Second edition, Cambridge, page 273.

Crossrefs

Cf. A000112.

Programs

Sage

for n in range(66):
    j=0;
    for p in Posets(n):
        if p.order_complex().is_cohen_macaulay():
            j = j+1;
    print(j)