

A327081


BIInumbers of maximal uniform setsystems covering an initial interval of positive integers.


3



1, 3, 4, 11, 52, 64, 139, 2868, 13376, 16384, 32907
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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 setsystem with BIInumber n to be obtained by taking the binary indices of each binary index of n. Every setsystem (finite set of finite nonempty sets) has a different BIInumber. 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 BIInumber of {{2},{1,3}} is 18. Elements of a setsystem are sometimes called edges.
A setsystem is uniform if all edges have the same size.


LINKS

Table of n, a(n) for n=1..11.


EXAMPLE

The sequence of all maximal uniform setsystems covering an initial interval together with their BIInumbers 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}}


MATHEMATICA

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[#]}, #==0normQ[Union@@sys]&&SameQ@@Length/@sys&&Length[sys]==Binomial[Length[Union@@sys], Length[First[sys]]]]&]


CROSSREFS

BIInumbers of uniform setsystems are A326783.
BIInumbers of maximal uniform setsystems are A327080.
Cf. A000120, A048793, A070939, A326031, A326784, A326785, A327041.
Sequence in context: A296256 A101982 A041947 * A201970 A102013 A192223
Adjacent sequences: A327078 A327079 A327080 * A327082 A327083 A327084


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Aug 20 2019


STATUS

approved



