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 A326570 Number of covering antichains of subsets of {1..n} with different edge-sizes. 4
 2, 1, 1, 4, 17, 186, 3292, 139161, 14224121 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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 EXAMPLE The a(0) = 2 through a(4) = 17 antichains:   {}    {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}   {{}}                  {{1},{2,3}}  {{1},{2,3,4}}                         {{2},{1,3}}  {{2},{1,3,4}}                         {{3},{1,2}}  {{3},{1,2,4}}                                      {{4},{1,2,3}}                                      {{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]], 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 without singletons is A326569. (Antichain) covers with equal edge-sizes are A306021. Cf. A000372, A003182, A307249, A326026, A326514, A326566, A326572. Sequence in context: A083677 A075803 A127966 * A184880 A257859 A231622 Adjacent sequences:  A326567 A326568 A326569 * A326571 A326572 A326573 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 November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)