OFFSET
1,3
COMMENTS
A set-system is a finite set of finite nonempty sets. It is achiral if it is not changed by any permutation of the vertices.
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 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.
EXAMPLE
The sequence of all achiral set-systems together with their BII-numbers begins:
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
7: {{1},{2},{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
16: {{1,3}}
25: {{1},{3},{1,3}}
32: {{2,3}}
42: {{2},{3},{2,3}}
52: {{1,2},{1,3},{2,3}}
63: {{1},{2},{3},{1,2},{1,3},{2,3}}
64: {{1,2,3}}
75: {{1},{2},{3},{1,2,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Select[Range[0, 1000], Length[graprms[bpe/@bpe[#]]]==1&]
CROSSREFS
These are numbers n such that A330231(n) = 1.
Achiral set-systems are counted by A083323.
MG-numbers of planted achiral trees are A214577.
Non-isomorphic achiral multiset partitions are A330223.
Achiral integer partitions are counted by A330224.
BII-numbers of fully chiral set-systems are A330226.
MM-numbers of achiral multisets of multisets are A330232.
Achiral factorizations are A330234.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 06 2019
STATUS
approved