A326573 - OEIS

A326573

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

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:

{{1,3},{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: A046842 A348181 A374345 * A324240 A120608 A143766

Adjacent sequences: A326570 A326571 A326572 * A326574 A326575 A326576

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 18 2019

STATUS

approved