A330231 - OEIS

%I #5 Dec 10 2019 20:01:15

%S 1,1,1,1,1,2,2,1,1,1,1,1,3,6,6,3,1,2,3,6,3,3,6,6,2,1,6,3,6,6,3,3,1,3,

%T 2,6,3,6,3,6,2,6,1,3,6,3,6,3,3,6,6,3,1,3,3,3,3,6,6,3,3,3,3,1,1,3,3,3,

%U 3,6,6,3,3,3,3,1,3,6,6,3,3,6,3,6,3,3,6

%N Number of distinct set-systems that can be obtained by permuting the vertices of the set-system with BII-number n.

%C A set-system is a finite set of finite nonempty sets.

%C 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.

%F a(n) is a divisor of A326702(n)!.

%e 30 is the MM-number of {{2},{3},{1,2},{1,3}}, with vertex permutations

%e   {{1},{2},{1,3},{2,3}}

%e   {{1},{3},{1,2},{2,3}}

%e   {{2},{3},{1,2},{1,3}}

%e so a(30) = 3.

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

%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];

%t Table[Length[graprms[bpe/@bpe[n]]],{n,0,100}]

%Y Positions of 1's are A330217.

%Y Positions of first appearances are A330218.

%Y The version for MM-numbers is A330098.

%Y Achiral set-systems are counted by A083323.

%Y BII-numbers of fully chiral set-systems are A330226.

%Y Cf. A000120, A003238, A007716, A016031, A048793, A055621, A070939, A214577, A326031, A326702, A330101, A330195, A330229, A330230, A330233.

%K nonn

%O 0,6

%A _Gus Wiseman_, Dec 09 2019