A327108 BII-numbers of set-systems with spanning edge-connectivity 2.

52, 53, 54, 55, 60, 61, 62, 63, 84, 85, 86, 87, 92, 93, 94, 95, 100, 101, 102, 103, 108, 109, 110, 111, 112, 113, 114, 115, 120, 121, 122, 123, 772, 773, 774, 775, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 848, 849, 850

Offset

1,1

Comments

Differs from A327109 in lacking 116, 117, 118, 119, 124, 125, 126, 127, ...

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..55.

Example

The sequence of all set-systems with spanning edge-connectivity 2 together with their BII-numbers begins:
   52: {{1,2},{1,3},{2,3}}
   53: {{1},{1,2},{1,3},{2,3}}
   54: {{2},{1,2},{1,3},{2,3}}
   55: {{1},{2},{1,2},{1,3},{2,3}}
   60: {{1,2},{3},{1,3},{2,3}}
   61: {{1},{1,2},{3},{1,3},{2,3}}
   62: {{2},{1,2},{3},{1,3},{2,3}}
   63: {{1},{2},{1,2},{3},{1,3},{2,3}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   86: {{2},{1,2},{1,3},{1,2,3}}
   87: {{1},{2},{1,2},{1,3},{1,2,3}}
   92: {{1,2},{3},{1,3},{1,2,3}}
   93: {{1},{1,2},{3},{1,3},{1,2,3}}
   94: {{2},{1,2},{3},{1,3},{1,2,3}}
   95: {{1},{2},{1,2},{3},{1,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  101: {{1},{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}
  103: {{1},{2},{1,2},{2,3},{1,2,3}}

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[#]]==2&]

Crossrefs

Positions of 2's in A327144.

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

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

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

BII-numbers for non-spanning edge-connectivity 2 are A327097.

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

BII-numbers for spanning edge-connectivity 1 are A327111.

Cf. A326749, A326787, A327041, A327069, A327075, A327102, A327103, A327104.

Sequence in context: A249404 A327374 A327109 * A295156 A181461 A346455

Adjacent sequences: A327105 A327106 A327107 * A327109 A327110 A327111

Keyword

nonn

Author

Gus Wiseman, Aug 23 2019

Status

approved