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A306550 Array read by antidiagonals where A(n,k) is the number of labeled k-antichains covering n vertices.

%I A306550 #7 Feb 24 2019 01:54:37
%S A306550 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 A306550 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 A306550 76482,70370,2116,1,0,0,0,0
%N A306550 Array read by antidiagonals where A(n,k) is the number of labeled k-antichains covering n vertices.
%e A306550 Array begins:
%e A306550     n=0: n=1: n=2: n=3: n=4: n=5:
%e A306550 ---------------------------------
%e A306550 k=0:  1    0    0    0    0    0
%e A306550 k=1:  1    1    1    1    1    1
%e A306550 k=2:  0    0    1    6   25   90
%e A306550 k=3:  0    0    0    2   56  790
%e A306550 k=4:  0    0    0    0   25 1895
%e A306550 k=5:  0    0    0    0    6 2116
%e A306550 Column n = 3 counts the following antichains:
%e A306550   {{123}}  {{1}{23}}   {{1}{2}{3}}
%e A306550            {{2}{13}}   {{12}{13}{23}}
%e A306550            {{3}{12}}
%e A306550            {{12}{13}}
%e A306550            {{12}{23}}
%e A306550            {{13}{23}}
%t A306550 nn=8;
%t A306550 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 A306550 ae[n_,k_]:=Length[Select[stableSets[Subsets[Range[n]],SubsetQ,k],Union@@#==Range[n]&]];
%t A306550 Table[ae[k,n-k],{n,0,nn},{k,0,n}]
%Y A306550 Column sums are A006126. Row k = 2 is A000392. Rows k = 3-9 are A056046-A056049, A056052, A056101, A056104.
%Y A306550 Cf. A000372, A006126, A006602, A014466, A261005, A305000, A305844, A317674.
%K A306550 nonn,tabl,more
%O A306550 0,18
%A A306550 _Gus Wiseman_, Feb 23 2019