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!)
A327111 BII-numbers of set-systems with spanning edge-connectivity 1. 22
1, 2, 4, 5, 6, 7, 8, 16, 17, 20, 21, 22, 23, 24, 25, 28, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 88, 89, 90, 91, 96, 97, 98, 99 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

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 disconnected or empty set-system.

LINKS

Table of n, a(n) for n=1..67.

EXAMPLE

The sequence of all set-systems with spanning edge-connectivity 1 together with their BII-numbers begins:

   1: {{1}}

   2: {{2}}

   4: {{1,2}}

   5: {{1},{1,2}}

   6: {{2},{1,2}}

   7: {{1},{2},{1,2}}

   8: {{3}}

  16: {{1,3}}

  17: {{1},{1,3}}

  20: {{1,2},{1,3}}

  21: {{1},{1,2},{1,3}}

  22: {{2},{1,2},{1,3}}

  23: {{1},{2},{1,2},{1,3}}

  24: {{3},{1,3}}

  25: {{1},{3},{1,3}}

  28: {{1,2},{3},{1,3}}

  29: {{1},{1,2},{3},{1,3}}

  30: {{2},{1,2},{3},{1,3}}

  31: {{1},{2},{1,2},{3},{1,3}}

  32: {{2,3}}

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

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]]]]]]]]];

spanEdgeConn[vts_, eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds], Union@@#!=vts||Length[csm[#]]!=1&];

Select[Range[0, 100], spanEdgeConn[Union@@bpe/@bpe[#], bpe/@bpe[#]]==1&]

CROSSREFS

Graphs with spanning edge-connectivity >= 2 are counted by A095983.

BII-numbers for vertex-connectivity 1 are A327098.

BII-numbers for non-spanning edge-connectivity 1 are A327099.

BII-numbers for spanning edge-connectivity 2 are A327108.

BII-numbers for spanning edge-connectivity >= 2 are A327109.

Set-systems with spanning edge-connectivity 2 are counted by A327130.

Graphs with spanning edge-connectivity 1 are counted by A327145.

Graphs with spanning edge-connectivity 2 are counted by A327146.

Cf. A013922, A322395, A326749, A327041, A327069, A327071, A327097, A327144, A327145.

Sequence in context: A039085 A302433 A326749 * A326853 A326879 A326875

Adjacent sequences:  A327108 A327109 A327110 * A327112 A327113 A327114

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 25 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 February 24 17:43 EST 2021. Contains 341577 sequences. (Running on oeis4.)