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

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, 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, 76482, 70370, 2116, 1, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, Feb 23 2019

Keywords

• nonn
• tabl
• more

Examples

			Array begins:
    n=0: n=1: n=2: n=3: n=4: n=5:
---------------------------------
k=0:  1    0    0    0    0    0
k=1:  1    1    1    1    1    1
k=2:  0    0    1    6   25   90
k=3:  0    0    0    2   56  790
k=4:  0    0    0    0   25 1895
k=5:  0    0    0    0    6 2116
Column n = 3 counts the following antichains:
  {{123}}  {{1}{23}}   {{1}{2}{3}}
           {{2}{13}}   {{12}{13}{23}}
           {{3}{12}}
           {{12}{13}}
           {{12}{23}}
           {{13}{23}}
		

Crossrefs

Column sums are A006126. Row k = 2 is A000392. Rows k = 3-9 are A056046-A056049, A056052, A056101, A056104.

Cf. A000372, A006126, A006602, A014466, A261005, A305000, A305844, A317674.

Programs

Mathematica

nn=8;
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]]]]];
ae[n_,k_]:=Length[Select[stableSets[Subsets[Range[n]],SubsetQ,k],Union@@#==Range[n]&]];
Table[ae[k,n-k],{n,0,nn},{k,0,n}]