login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=0..6.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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