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 A143673 Number of antichains in the poset of Dyck paths ordered by inclusion. 4
 2, 2, 3, 7, 42, 2361, 37620704 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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; n=6: 1, 132, 4059, 54706, 390885, 1648100, 4380095, 7682096, 9172750, 7585779, 4370731, 1749626, 481189, 89055, 10676, 785, 38, 1; Note that the number of maximum antichains (for each n) is given by the rightmost entry in each of these rows. REFERENCES R. P. Stanley, Enumerative Combinatorics 1, Cambridge University Press, New York, 1997. LINKS Table of n, a(n) for n=0..6. J. Woodcock, Properties of the poset of Dyck paths ordered by inclusion EXAMPLE For n = 3 there are 7 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 7 antichains are: { }, {A}, {B}, {C}, {D}, {E}, {B,C}. CROSSREFS Cf. A143672. Number of maximal antichains A143674. Sequence in context: A052449 A053413 A232542 * A089543 A139073 A329955 Adjacent sequences: A143670 A143671 A143672 * A143674 A143675 A143676 KEYWORD nonn,more AUTHOR Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Aug 28 2008 EXTENSIONS a(6) from Alois P. Heinz, Jul 28 2011 STATUS approved

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