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.

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 26 2019

STATUS

approved