A143674 Number of maximal antichains in the poset of Dyck paths ordered by inclusion.

1, 1, 2, 4, 17, 379, 526913

Offset: 0

Author

Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Aug 28 2008

Keywords

• nonn
• more

Comments

Maximal antichains are those which cannot be extended without violating the antichain condition.

This is the breakdown by size of (or number of elements in) the antichains beginning with antichains of size 0 and increasing:

n=0: 0, 1;

n=1: 0, 1;

n=2: 0, 2;

n=3: 0, 3, 1;

n=4: 0, 3, 8, 6;

n=5: 0, 3, 14, 62, 132, 124, 42, 2;

n=6: 0, 3, 21, 157, 983, 4438, 15454, 41827, 79454, 112344, 117259, 88915, 47295, 14909, 3498, 334, 21, 1

Examples

			For n = 3 there are 4 maximal antichains. Assume that the five elements in the D_3 poset are depicted using a Hasse diagram and labeled A through E from bottom to top. Then the 4 maximal antichains are {A}, {B,C}, {D}, {E}.
		

References

• R. P. Stanley, Enumerative Combinatorics 1, Cambridge University Press, New York, 1997.

Links

• J. Woodcock, Properties of the poset of Dyck paths ordered by inclusion

Crossrefs

Cf. A143672. Total number of antichains A143673.

Extensions

a(6) from Alois P. Heinz, Jul 31 2011