A327109 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

A327109

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, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 772, 773, 774, 775, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Differs from A327108 in having 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, 1000], spanEdgeConn[Union@@bpe/@bpe[#], bpe/@bpe[#]]>=2&]

CROSSREFS

Positions of terms >= 2 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 non-spanning edge-connectivity >= 2 are A327102.

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

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

Cf. A326749, A326753, A326787, A327041, A327069, A327071, A327075.