A327105 BII-numbers of set-systems with minimum degree 1.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 48, 49, 50, 56, 57, 58, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 88, 89, 96, 98, 104, 106, 128

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.

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

Links

Table of n, a(n) for n=1..66.

Example

The sequence of all set-systems with minimum degree 1 together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  18: {{2},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[0, 100], If[#==0, 0, Min@@Length/@Split[Sort[Join@@bpe/@bpe[#]]]]==1&]

Crossrefs

Positions of 1's in A327103.

BII-numbers for minimum degree > 1 are A327107.

Graphs with minimum degree 1 are counted by A245797, with covering case A327227.

Set-systems with minimum degree 1 are counted by A327228, with covering case A327229.

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

Sequence in context: A345254 A004726 A298747 * A356451 A129618 A349536

Adjacent sequences: A327102 A327103 A327104 * A327106 A327107 A327108

Keyword

nonn

Author

Gus Wiseman, Aug 26 2019

Status

approved