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 A326572 Number of covering antichains of subsets of {1..n}, all having different sums. 7
 2, 1, 2, 8, 80, 3015, 803898 (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-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(0) = 2 through a(3) = 8 antichains:   {}    {{1}}  {{1,2}}    {{1,2,3}}   {{}}         {{1},{2}}  {{1},{2,3}}                           {{2},{1,3}}                           {{1,2},{1,3}}                           {{1,2},{2,3}}                           {{1},{2},{3}}                           {{1,3},{2,3}}                           {{1,2},{1,3},{2,3}} The a(4) = 80 antichains:   {1234}  {1}{234}    {1}{2}{34}     {1}{2}{3}{4}       {12}{13}{14}{24}{34}           {12}{34}    {1}{3}{24}     {1}{23}{24}{34}    {12}{13}{23}{24}{34}           {13}{24}    {1}{4}{23}     {2}{13}{14}{34}           {2}{134}    {2}{3}{14}     {12}{13}{14}{24}           {3}{124}    {1}{23}{24}    {12}{13}{14}{34}           {4}{123}    {1}{23}{34}    {12}{13}{23}{24}           {12}{134}   {1}{24}{34}    {12}{13}{23}{34}           {12}{234}   {2}{13}{14}    {12}{13}{24}{34}           {13}{124}   {2}{13}{34}    {12}{14}{24}{34}           {13}{234}   {2}{14}{34}    {12}{23}{24}{34}           {14}{123}   {3}{14}{24}    {13}{14}{24}{34}           {14}{234}   {4}{12}{23}    {13}{23}{24}{34}           {23}{124}   {12}{13}{14}   {12}{13}{14}{234}           {23}{134}   {12}{13}{24}   {12}{23}{24}{134}           {24}{134}   {12}{13}{34}   {123}{124}{134}{234}           {34}{123}   {12}{14}{34}           {123}{124}  {12}{23}{24}           {123}{134}  {12}{23}{34}           {123}{234}  {12}{24}{34}           {124}{134}  {13}{14}{24}           {124}{234}  {13}{23}{24}           {134}{234}  {13}{23}{34}                       {13}{24}{34}                       {14}{24}{34}                       {12}{13}{234}                       {12}{14}{234}                       {12}{23}{134}                       {12}{24}{134}                       {13}{14}{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]], 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 are A326566. Antichain covers with different edge-sizes are A326570. The case without singletons is A326571. Antichains with equal edge-sums are A326574. Cf. A000372, A035470, A307249, A321469, A326519, A326565, A326569, A326573. Sequence in context: A242841 A143208 A188664 * A119419 A109529 A022694 Adjacent sequences:  A326569 A326570 A326571 * A326573 A326574 A326575 KEYWORD nonn,more AUTHOR Gus Wiseman, Jul 18 2019 STATUS approved

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Last modified November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)