OFFSET
0,1
COMMENTS
An antichain is a finite set of finite sets, none of which is a subset of any other. 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}.
The antichain condition is implied by the condition on edge-sums. - Christian Sievers, Nov 04 2025
LINKS
Christian Sievers, Table of n, a(n) for n = 0..17
FORMULA
a(n) = 1 + Sum_{k=0..n*(n+1)/2} ( 2^T(n,k) - 1 ) where T(n,k) is the number of subsets of {1..n} with sum k (cf. A053632). - Christian Sievers, Nov 04 2025
EXAMPLE
The a(0) = 2 through a(4) = 22 antichains:
{} {} {} {} {}
{{}} {{}} {{}} {{}} {{}}
{{1}} {{1}} {{1}} {{1}}
{{2}} {{2}} {{2}}
{{1,2}} {{3}} {{3}}
{{1,2}} {{4}}
{{1,3}} {{1,2}}
{{2,3}} {{1,3}}
{{1,2,3}} {{1,4}}
{{3},{1,2}} {{2,3}}
{{2,4}}
{{3,4}}
{{1,2,3}}
{{1,2,4}}
{{1,3,4}}
{{2,3,4}}
{{1,2,3,4}}
{{3},{1,2}}
{{4},{1,3}}
{{1,4},{2,3}}
{{2,4},{1,2,3}}
{{3,4},{1,2,4}}
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]]]];
cleqset[set_]:=stableSets[Subsets[set], SubsetQ[#1, #2]||Total[#1]!=Total[#2]&];
Table[Length[cleqset[Range[n]]], {n, 0, 5}]
PROG
(PARI) a(n)=my(l=Vec(prod(i=1, n, 1+x^i))); 1+sum(k=1, #l, 2^l[k]-1) \\ Christian Sievers, Nov 04 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 18 2019
EXTENSIONS
a(9) from Andrew Howroyd, Aug 13 2019
More terms from Christian Sievers, Nov 04 2025
STATUS
approved
