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%I #11 Jul 27 2019 14:57:51
%S 0,1,1,2,2,3,3,4,1,2,2,3,3,4,4,5,2,3,3,4,4,5,5,6,3,4,4,5,5,6,6,7,2,3,
%T 3,4,4,5,5,6,3,4,4,5,5,6,6,7,4,5,5,6,6,7,7,8,5,6,6,7,7,8,8,9,3,4,4,5,
%U 5,6,6,7,4,5,5,6,6,7,7,8,5,6,6,7,7,8,8,9
%N Weight of the set-system with BII-number n.
%C A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets of positive integers 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, it follows that the BII-number of {{2},{1,3}} is 18. The weight of a set-system is the sum of sizes of its elements (sometimes called its edges).
%F a(2^x + ... + 2^z) = w(x + 1) + ... + w(z + 1), where x...z are distinct nonnegative integers and w = A000120. For example, a(6) = a(2^2 + 2^1) = w(3) + w(2) = 3.
%e The sequence of set-systems together with their BII-numbers begins:
%e 0: {}
%e 1: {{1}}
%e 2: {{2}}
%e 3: {{1},{2}}
%e 4: {{1,2}}
%e 5: {{1},{1,2}}
%e 6: {{2},{1,2}}
%e 7: {{1},{2},{1,2}}
%e 8: {{3}}
%e 9: {{1},{3}}
%e 10: {{2},{3}}
%e 11: {{1},{2},{3}}
%e 12: {{1,2},{3}}
%e 13: {{1},{1,2},{3}}
%e 14: {{2},{1,2},{3}}
%e 15: {{1},{2},{1,2},{3}}
%e 16: {{1,3}}
%e 17: {{1},{1,3}}
%e 18: {{2},{1,3}}
%e 19: {{1},{2},{1,3}}
%e 20: {{1,2},{1,3}}
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Table[Length[Join@@bpe/@bpe[n]],{n,0,100}]
%Y Cf. A000120, A029931, A048793, A061775, A070939, A072639, A116549, A302242, A305830, A326701, A326702, A326703, A326704.
%K nonn
%O 0,4
%A _Gus Wiseman_, Jul 20 2019
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