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 A326569 Number of covering antichains of subsets of {1..n} with no singletons and different edge-sizes. 4
 1, 0, 1, 1, 13, 121, 2566, 121199, 13254529 (list; graph; refs; listen; history; text; internal format)
 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-sizes are the numbers of vertices in each edge, so for example the edge sizes of {{1,3},{2,5},{3,4,5}} are {2,2,3}. LINKS FORMULA a(n) = A326570(n) - n*a(n-1) for n > 0. - Andrew Howroyd, Aug 13 2019 EXAMPLE The a(2) = 1 through a(4) = 13 antichains:   {{1,2}}  {{1,2,3}}  {{1,2,3,4}}                       {{1,2},{1,3,4}}                       {{1,2},{2,3,4}}                       {{1,3},{1,2,4}}                       {{1,3},{2,3,4}}                       {{1,4},{1,2,3}}                       {{1,4},{2,3,4}}                       {{2,3},{1,2,4}}                       {{2,3},{1,3,4}}                       {{2,4},{1,2,3}}                       {{2,4},{1,3,4}}                       {{3,4},{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]]]]; cleq[n_]:=Select[stableSets[Subsets[Range[n], {2, n}], SubsetQ[#1, #2]||Length[#1]==Length[#2]&], Union@@#==Range[n]&]; Table[Length[cleq[n]], {n, 0, 6}] CROSSREFS Antichain covers are A006126. Set partitions with different block sizes are A007837. The case with singletons is A326570. Cf. A000372, A003182, A306021, A307249, A326565, A326571, A326572, A326573. Sequence in context: A096053 A033470 A297594 * A016230 A327961 A278276 Adjacent sequences:  A326566 A326567 A326568 * A326570 A326571 A326572 KEYWORD nonn,more AUTHOR Gus Wiseman, Jul 18 2019 EXTENSIONS a(8) from Andrew Howroyd, Aug 13 2019 STATUS approved

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Last modified October 27 01:02 EDT 2020. Contains 338035 sequences. (Running on oeis4.)