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%I A143673 #12 Mar 19 2018 04:15:19
%S A143673 2,2,3,7,42,2361,37620704
%N A143673 Number of antichains in the poset of Dyck paths ordered by inclusion.
%C A143673 Also the number of order ideals (down-sets) for this poset.
%C A143673 This is the breakdown by size of (or number of elements in) the antichains beginning with antichains of size 0 and increasing:
%C A143673 n=0: 1, 1;
%C A143673 n=1: 1, 1;
%C A143673 n=2: 1, 2;
%C A143673 n=3: 1, 5, 1;
%C A143673 n=4: 1, 14, 21, 6;
%C A143673 n=5: 1, 42, 309, 793, 810, 348, 56, 2;
%C A143673 n=6: 1, 132, 4059, 54706, 390885, 1648100, 4380095, 7682096, 9172750, 7585779, 4370731, 1749626, 481189, 89055, 10676, 785, 38, 1;
%C A143673 Note that the number of maximum antichains (for each n) is given by the rightmost entry in each of these rows.
%D A143673 R. P. Stanley, Enumerative Combinatorics 1, Cambridge University Press, New York, 1997.
%H A143673 J. Woodcock, <a href="http://garsia.math.yorku.ca/~zabrocki/dyckpathposet.html">Properties of the poset of Dyck paths ordered by inclusion</a>
%e A143673 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}.
%Y A143673 Cf. A143672. Number of maximal antichains A143674.
%K A143673 nonn,more
%O A143673 0,1
%A A143673 Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Aug 28 2008
%E A143673 a(6) from _Alois P. Heinz_, Jul 28 2011