0,1

Also the number of order ideals (down-sets) for this poset.

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

n=0 1,1

n=1 1,1

n=2 1,2

n=3 1,5,1

n=4 1,14,21,6

n=5 1,42,309,793,810,348,56,2

Note that the number of maximum antichains (for each n) is given by the rightmost entry in each of these rows.

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

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

For n = 3 there are 7 antichains. Assume that the five elements in the D_3 poset are depicted using a Hasse diagram and labelled A through E from bottom to top. Then the 7 antichains are: { }, {A}, {B}, {C}, {D}, {E}, {B,C}.

Cf. A143672. Number of maximal antichains A143674.

Sequence in context: A092970 A052449 A053413 this_sequence A089543 A058023 A139073

Adjacent sequences: A143670 A143671 A143672 this_sequence A143674 A143675 A143676

more,nonn

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