A052446 -id:A052446 - cp's OEIS frontend

A327437 Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

1, 1, 3, 6, 15, 52, 410, 32697

Offset: 0

Gus Wiseman, Sep 11 2019

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 spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 antichains:
  {}  {}         {}             {}
      {{1}}      {{1}}          {{1}}
      {{1},{2}}  {{1,2}}        {{1,2}}
                 {{1},{2}}      {{1},{2}}
                 {{1},{2,3}}    {{1,2,3}}
                 {{1},{2},{3}}  {{1},{2,3}}
                                {{1,2},{1,3}}
                                {{1},{2},{3}}
                                {{1},{2,3,4}}
                                {{1,2},{3,4}}
                                {{1},{2},{3,4}}
                                {{1},{2},{3},{4}}
                                {{2},{1,3},{1,4}}
                                {{1,2},{1,3},{2,3}}
                                {{4},{1,2},{1,3},{2,3}}

Column k = 0 of A327438.
The labeled version is A327355.
The covering case is A327426.
Cf. A014466 A052446, A120338, A261005, A293606, A327201, A327236, A327352, A327353, A327354, A327438.

a(n > 0) = A306505(n) - A261006(n).

A263914 Number of (not necessarily connected) simple bridgeless graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 16, 77, 582, 8002, 205538, 10010657, 912838330, 154634281045, 48597689465264, 28412286324844316, 31024936551325074359, 63533058735488301141874, 244916078109873267213212830, 1783406527132994841804241539063, 24605674622456537969150523621546114

Offset: 1

Eric W. Weisstein, Oct 29 2015

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..30
Eric Weisstein's World of Mathematics, Bridgeless Graph
Eric Weisstein's World of Mathematics, Simple Graph

Cf. A000088 (number of simple graphs).
Cf. A007146 (number of simple connected bridgeless graphs).
Cf. A052446 (number of simple connected bridged graphs).
Cf. A263915 (number of simple bridged graphs).

a(n) = A000088(n) - A263915(n).

Euler transform of A007146. - Falk Hüffner, Jan 18 2016

More terms from A007146 by Falk Hüffner, Jan 18 2016

A263915 Number of (not necessarily connected) simple bridged graphs with n nodes.

Original entry on oeis.org

0, 1, 2, 6, 18, 79, 462, 4344, 69130, 1994511, 106159534, 10456891547, 1904341902688, 641869332391172, 401549418479234409, 467956969039256753054, 1019786043659665470506946, 4171198012616858743636651785, 32134630668466555232483869886654

Offset: 1

Eric W. Weisstein, Oct 29 2015

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..30
Eric Weisstein's World of Mathematics, Bridged Graph
Eric Weisstein's World of Mathematics, Simple Graph

Cf. A000088 (number of simple graphs).
Cf. A007146 (number of simple connected bridgeless graphs).
Cf. A052446 (number of simple connected bridged graphs).
Cf. A263914 (number of simple bridgeless graph).

a(n) = A000088(n) - A263914(n).

More terms using formula by Falk Hüffner, Jan 18 2016

A327438 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 2, 1, 15, 7, 5, 2, 52, 53, 62, 31, 9, 1, 1

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Triangle begins:
   1
   1  1
   3  1
   6  2  1
  15  7  5  2
  52 53 62 31  9  1  1
The antichains counted in row n = 4 are the following:
  0             {1234}         {12}{134}{234}     {123}{124}{134}{234}
  {1}           {12}{134}      {123}{124}{134}    {12}{13}{14}{23}{24}{34}
  {12}          {123}{124}     {12}{13}{24}{34}
  {123}         {12}{13}{14}   {12}{13}{14}{234}
  {1}{2}        {12}{13}{24}   {12}{13}{14}{23}{24}
  {1}{23}       {12}{13}{234}
  {12}{13}      {12}{13}{14}{23}
  {1}{234}
  {12}{34}
  {1}{2}{3}
  {1}{2}{34}
  {2}{13}{14}
  {12}{13}{23}
  {1}{2}{3}{4}
  {4}{12}{13}{23}

Row sums are A306505.
Column k = 0 is A327437.
The labeled version is A327352.
Cf. A014466, A052446, A327062, A327071, A327103, A327111, A327144, A327352, A327353.

cp's OEIS Frontend

A327437 Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

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A263914 Number of (not necessarily connected) simple bridgeless graphs with n nodes.

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A263915 Number of (not necessarily connected) simple bridged graphs with n nodes.

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A327438 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

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