proposed

approved

editing

proposed

A binary index of n is any position of a 1 in its reversed binary digitsexpansion. 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 digits 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.

allocated for Gus WisemanBII-numbers of maximal uniform set-systems covering an initial interval of positive integers.

1, 3, 4, 11, 52, 64, 139, 2868, 13376, 16384, 32907

1,2

A binary index of n is any position of a 1 in its reversed binary digits. 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 digits (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.

A set-system is uniform if all edges have the same size.

The sequence of all maximal uniform set-systems covering an initial interval together with their BII-numbers begins:

0: {}

1: {{1}}

3: {{1},{2}}

4: {{1,2}}

11: {{1},{2},{3}}

52: {{1,2},{1,3},{2,3}}

64: {{1,2,3}}

139: {{1},{2},{3},{4}}

2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

13376: {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

16384: {{1,2,3,4}}

32907: {{1},{2},{3},{4},{5}}

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

Select[Range[1000], With[{sys=bpe/@bpe[#]}, #==0||normQ[Union@@sys]&&SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys], Length[First[sys]]]]&]

BII-numbers of uniform set-systems are A326783.

BII-numbers of maximal uniform set-systems are A327080.

Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.

allocated

nonn,more

Gus Wiseman, Aug 20 2019

approved

editing

allocated for Gus Wiseman

allocated

approved