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 A326571 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having different sums. 5
 1, 0, 1, 5, 61, 2721, 788221 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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}. LINKS EXAMPLE The a(3) = 5 antichains:   {{1,2,3}}   {{1,3},{2,3}}   {{1,2},{2,3}}   {{1,2},{1,3}}   {{1,2},{1,3},{2,3}} The a(4) = 61 antichains:   {1234}  {12}{34}    {12}{13}{14}   {12}{13}{14}{24}   {12}{13}{14}{24}{34}           {13}{24}    {12}{13}{24}   {12}{13}{14}{34}   {12}{13}{23}{24}{34}           {12}{134}   {12}{13}{34}   {12}{13}{23}{24}           {12}{234}   {12}{14}{34}   {12}{13}{23}{34}           {13}{124}   {12}{23}{24}   {12}{13}{24}{34}           {13}{234}   {12}{23}{34}   {12}{14}{24}{34}           {14}{123}   {12}{24}{34}   {12}{23}{24}{34}           {14}{234}   {13}{14}{24}   {13}{14}{24}{34}           {23}{124}   {13}{23}{24}   {13}{23}{24}{34}           {23}{134}   {13}{23}{34}   {12}{13}{14}{234}           {24}{134}   {13}{24}{34}   {12}{23}{24}{134}           {34}{123}   {14}{24}{34}   {123}{124}{134}{234}           {123}{124}  {12}{13}{234}           {123}{134}  {12}{14}{234}           {123}{234}  {12}{23}{134}           {124}{134}  {12}{24}{134}           {124}{234}  {13}{14}{234}           {134}{234}  {13}{23}{124}                       {14}{34}{123}                       {23}{24}{134}                       {12}{134}{234}                       {13}{124}{234}                       {14}{123}{234}                       {23}{124}{134}                       {123}{124}{134}                       {123}{124}{234}                       {123}{134}{234}                       {124}{134}{234} 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}] CROSSREFS Antichain covers are A006126. Set partitions with different block-sums are A275780. MM-numbers of multiset partitions with different part-sums are A326535. Antichain covers with equal edge-sums and no singletons are A326565. Antichain covers with different edge-sizes and no singletons are A326569. The case with singletons allowed is A326572. Antichains with equal edge-sums are A326574. Cf. A000372, A003182, A035470, A307249, A321469, A326519, A326566, A326570, A326573. Sequence in context: A159316 A231798 A258672 * A201254 A116163 A092823 Adjacent sequences:  A326568 A326569 A326570 * A326572 A326573 A326574 KEYWORD nonn,more AUTHOR Gus Wiseman, Jul 18 2019 STATUS approved

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Last modified November 27 06:33 EST 2020. Contains 338678 sequences. (Running on oeis4.)