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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A327129 Number of connected set-systems covering n vertices with at least one edge whose removal (along with any non-covered vertices) disconnects the set-system (non-spanning edge-connectivity 1). 9
0, 1, 2, 35, 2804 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

LINKS

Table of n, a(n) for n=0..4.

FORMULA

Inverse binomial transform of A327196.

EXAMPLE

The a(3) = 35 set-systems:

  {123}  {1}{12}{23}   {1}{2}{12}{13}   {1}{2}{3}{12}{13}

         {1}{13}{23}   {1}{2}{12}{23}   {1}{2}{3}{12}{23}

         {1}{2}{123}   {1}{2}{13}{23}   {1}{2}{3}{13}{23}

         {1}{3}{123}   {1}{2}{3}{123}   {1}{2}{3}{12}{123}

         {2}{12}{13}   {1}{3}{12}{13}   {1}{2}{3}{13}{123}

         {2}{13}{23}   {1}{3}{12}{23}   {1}{2}{3}{23}{123}

         {2}{3}{123}   {1}{3}{13}{23}

         {3}{12}{13}   {2}{3}{12}{13}

         {3}{12}{23}   {2}{3}{12}{23}

         {1}{23}{123}  {2}{3}{13}{23}

         {2}{13}{123}  {1}{2}{13}{123}

         {3}{12}{123}  {1}{2}{23}{123}

                       {1}{3}{12}{123}

                       {1}{3}{23}{123}

                       {2}{3}{12}{123}

                       {2}{3}{13}{123}

MATHEMATICA

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];

eConn[sys_]:=If[Length[csm[sys]]!=1, 0, Length[sys]-Max@@Length/@Select[Union[Subsets[sys]], Length[csm[#]]!=1&]];

Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&eConn[#]==1&]], {n, 0, 3}]

CROSSREFS

The restriction to simple graphs is A327079, with non-covering version A327231.

The version for spanning edge-connectivity is A327145, with BII-numbers A327111.

The BII-numbers of these set-systems are A327099.

The non-covering version is A327196.

Cf. A003465, A006129, A263296, A322395, A323818, A327071, A327149.

Sequence in context: A110697 A322514 A132100 * A291809 A182070 A325630

Adjacent sequences:  A327126 A327127 A327128 * A327130 A327131 A327132

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 27 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 July 15 02:31 EDT 2020. Contains 335762 sequences. (Running on oeis4.)