A327107 BII-numbers of set-systems with minimum vertex-degree > 1.

7, 25, 30, 31, 42, 45, 47, 51, 52, 53, 54, 55, 59, 60, 61, 62, 63, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 99, 100, 101, 102, 103, 105, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

Offset: 1

Author

Gus Wiseman, Aug 26 2019

Keywords

• nonn

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.

In a set-system, the degree of a vertex is the number of edges containing it.

Examples

			The sequence of all set-systems with maximum degree > 1 together with their BII-numbers begins:
   7: {{1},{2},{1,2}}
  25: {{1},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}
  31: {{1},{2},{1,2},{3},{1,3}}
  42: {{2},{3},{2,3}}
  45: {{1},{1,2},{3},{2,3}}
  47: {{1},{2},{1,2},{3},{2,3}}
  51: {{1},{2},{1,3},{2,3}}
  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}}
  59: {{1},{2},{3},{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}}
  75: {{1},{2},{3},{1,2,3}}
  76: {{1,2},{3},{1,2,3}}
		

Crossrefs

Positions of terms > 1 in A327103.

BII-numbers for minimum degree 1 are A327105.

Graphs with minimum degree > 1 are counted by A059167.

Cf. A000120, A029931, A048793, A058891, A070939, A245797, A326031, A326701, A326786, A327041, A327104, A327227-A327230.

Programs

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]>1&]