A327358 - OEIS

A327358

Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity >= k.

1, 1, 0, 2, 1, 0, 5, 3, 2, 0, 20, 14, 10, 6, 0, 180, 157, 128, 91, 54, 0

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OFFSET

0,4

COMMENTS

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

If empty edges are allowed, we have T(0,0) = 2.

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

Triangle begins:

1 0

2 1 0

5 3 2 0

20 14 10 6 0

180 157 128 91 54 0

Non-isomorphic representatives of the antichains counted in row n = 4:

{1234} {1234} {1234} {1234}

{1}{234} {12}{134} {123}{124} {12}{134}{234}

{12}{34} {123}{124} {12}{13}{234} {123}{124}{134}

{12}{134} {12}{13}{14} {12}{134}{234} {12}{13}{14}{234}

{123}{124} {12}{13}{24} {123}{124}{134} {123}{124}{134}{234}

{1}{2}{34} {12}{13}{234} {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}

{2}{13}{14} {12}{134}{234} {12}{13}{14}{234}

{12}{13}{14} {123}{124}{134} {12}{13}{14}{23}{24}

{12}{13}{24} {12}{13}{14}{23} {123}{124}{134}{234}

{1}{2}{3}{4} {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}

{12}{13}{234} {12}{13}{14}{234}

{12}{134}{234} {12}{13}{14}{23}{24}

{123}{124}{134} {123}{124}{134}{234}

{4}{12}{13}{23} {12}{13}{14}{23}{24}{34}

{12}{13}{14}{23}

{12}{13}{24}{34}

{12}{13}{14}{234}

{12}{13}{14}{23}{24}

{123}{124}{134}{234}

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

CROSSREFS

Column k = 0 is A261005, or A006602 if empty edges are allowed.

Column k = 1 is A261006 (clutters), if we assume A261006(0) = A261006(1) = 0.

Column k = 2 is A305028 (blobs), if we assume A305028(0) = A305028(2) = 0.

Column k = n - 1 is A327425 (cointersecting).

The labeled version is A327350.

Negated first differences of rows are A327359.

Cf. A006126, A055621, A120338, A293606, A293993, A327334, A327351, A327356.

Sequence in context: A073583 A324162 A060136 * A256664 A226783 A245972

Adjacent sequences: A327355 A327356 A327357 * A327359 A327360 A327361

KEYWORD

nonn,tabl,more

AUTHOR

Gus Wiseman, Sep 09 2019

STATUS

approved