login
A326565
Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.
8
1, 0, 1, 1, 4, 13, 91, 1318, 73581, 51913025
OFFSET
0,5
COMMENTS
An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.
EXAMPLE
The a(2) = 1 through a(5) = 13 antichains:
{{1,2}} {{1,2,3}} {{1,2,3,4}} {{1,2,3,4,5}}
{{1,4},{2,3}} {{1,2,5},{1,3,4}}
{{2,4},{1,2,3}} {{1,3,5},{2,3,4}}
{{3,4},{1,2,4}} {{1,4,5},{2,3,5}}
{{1,4,5},{1,2,3,4}}
{{2,3,5},{1,2,3,4}}
{{2,4,5},{1,2,3,5}}
{{3,4,5},{1,2,4,5}}
{{1,5},{2,4},{1,2,3}}
{{2,5},{3,4},{1,2,4}}
{{3,5},{1,2,5},{1,3,4}}
{{4,5},{1,3,5},{2,3,4}}
{{1,4,5},{2,3,5},{1,2,3,4}}
MATHEMATICA
stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n], {2, n}], SubsetQ[#1, #2]||Total[#1]!=Total[#2]&], Union@@#==Range[n]&];
Table[Length[cleq[n]], {n, 0, 5}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jul 13 2019
EXTENSIONS
a(9) from Andrew Howroyd, Aug 14 2019
STATUS
approved