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

0,18

Links

Table of n, a(n) for n=0..65.

Example

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}}

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}]

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.

Sequence in context: A331928 A228632 A229657 * A339861 A087936 A089804

Adjacent sequences: A306547 A306548 A306549 * A306551 A306552 A306553

Keyword

nonn,tabl,more

Author

Gus Wiseman, Feb 23 2019

Status

approved