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A326573 Number of connected antichains of subsets of {1..n}, all having different sums. 5
1, 1, 1, 5, 59, 2689, 787382 (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) = 59 antichains:

  {1234}  {12}{134}   {12}{13}{14}   {12}{13}{14}{24}   {12}{13}{14}{24}{34}

          {12}{234}   {12}{13}{24}   {12}{13}{14}{34}   {12}{13}{23}{24}{34}

          {13}{124}   {12}{13}{34}   {12}{13}{23}{24}

          {13}{234}   {12}{14}{34}   {12}{13}{23}{34}

          {14}{123}   {12}{23}{24}   {12}{13}{24}{34}

          {14}{234}   {12}{23}{34}   {12}{14}{24}{34}

          {23}{124}   {12}{24}{34}   {12}{23}{24}{34}

          {23}{134}   {13}{14}{24}   {13}{14}{24}{34}

          {24}{134}   {13}{23}{24}   {13}{23}{24}{34}

          {34}{123}   {13}{23}{34}   {12}{13}{14}{234}

          {123}{124}  {13}{24}{34}   {12}{23}{24}{134}

          {123}{134}  {14}{24}{34}   {123}{124}{134}{234}

          {123}{234}  {12}{13}{234}

          {124}{134}  {12}{14}{234}

          {124}{234}  {12}{23}{134}

          {134}{234}  {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.

Connected antichains are A048143.

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.

The non-connected case is A326572.

Cf. A000372, A293510, A307249, A321469, A323818, A326519, A326565, A326569, A326570, A326571.

Sequence in context: A249519 A001059 A290702 * A324240 A120608 A143766

Adjacent sequences:  A326570 A326571 A326572 * A326574 A326575 A326576

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 18 2019

STATUS

approved

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Last modified January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)