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%I #7 Feb 24 2019 01:54:37
%S 1,0,1,0,1,0,0,1,0,0,0,1,1,0,0,0,1,6,0,0,0,0,1,25,2,0,0,0,0,1,90,56,0,
%T 0,0,0,0,1,301,790,25,0,0,0,0,0,1,966,8380,1895,6,0,0,0,0,0,1,3025,
%U 76482,70370,2116,1,0,0,0,0
%N Array read by antidiagonals where A(n,k) is the number of labeled k-antichains covering n vertices.
%e Array begins:
%e n=0: n=1: n=2: n=3: n=4: n=5:
%e ---------------------------------
%e k=0: 1 0 0 0 0 0
%e k=1: 1 1 1 1 1 1
%e k=2: 0 0 1 6 25 90
%e k=3: 0 0 0 2 56 790
%e k=4: 0 0 0 0 25 1895
%e k=5: 0 0 0 0 6 2116
%e Column n = 3 counts the following antichains:
%e {{123}} {{1}{23}} {{1}{2}{3}}
%e {{2}{13}} {{12}{13}{23}}
%e {{3}{12}}
%e {{12}{13}}
%e {{12}{23}}
%e {{13}{23}}
%t nn=8;
%t stableSets[u_,Q_,k_]:=If[k==0,{{}},If[Length[u]==0,{},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q,k],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q,k-1]]]]];
%t ae[n_,k_]:=Length[Select[stableSets[Subsets[Range[n]],SubsetQ,k],Union@@#==Range[n]&]];
%t Table[ae[k,n-k],{n,0,nn},{k,0,n}]
%Y Column sums are A006126. Row k = 2 is A000392. Rows k = 3-9 are A056046-A056049, A056052, A056101, A056104.
%Y Cf. A000372, A006126, A006602, A014466, A261005, A305000, A305844, A317674.
%K nonn,tabl,more
%O 0,18
%A _Gus Wiseman_, Feb 23 2019
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