A326572 Number of covering antichains of subsets of {1..n}, all having different sums.

2, 1, 2, 8, 80, 3015, 803898

Offset: 0

Author

Gus Wiseman, Jul 18 2019

Keywords

• nonn
• more

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}.

Examples

			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}
		

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.

Programs

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}]