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A370640
Number of maximal subsets of {1..n} such that it is possible to choose a different binary index of each element.
12
1, 1, 1, 3, 3, 8, 17, 32, 32, 77, 144, 242, 383, 580, 843, 1201, 1201, 2694, 4614, 7096, 10219, 14186, 19070, 25207, 32791, 42160
OFFSET
0,4
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.
Also choices of A029837(n) elements of {1..n} such that it is possible to choose a different binary index of each.
EXAMPLE
The a(0) = 1 through a(6) = 17 subsets:
{} {1} {1,2} {1,2} {1,2,4} {1,2,4} {1,2,4}
{1,3} {1,3,4} {1,2,5} {1,2,5}
{2,3} {2,3,4} {1,3,4} {1,2,6}
{1,3,5} {1,3,4}
{2,3,4} {1,3,5}
{2,3,5} {1,3,6}
{2,4,5} {1,4,6}
{3,4,5} {1,5,6}
{2,3,4}
{2,3,5}
{2,3,6}
{2,4,5}
{2,5,6}
{3,4,5}
{3,4,6}
{3,5,6}
{4,5,6}
The a(0) = 1 through a(6) = 17 set-systems:
{1} {1}{2} {1}{2} {1}{2}{3} {1}{2}{3} {1}{2}{3}
{1}{12} {1}{12}{3} {1}{12}{3} {1}{12}{3}
{2}{12} {2}{12}{3} {1}{2}{13} {1}{2}{13}
{2}{12}{3} {1}{2}{23}
{2}{3}{13} {1}{3}{23}
{1}{12}{13} {2}{12}{3}
{12}{3}{13} {2}{3}{13}
{2}{12}{13} {1}{12}{13}
{1}{12}{23}
{1}{13}{23}
{12}{3}{13}
{12}{3}{23}
{2}{12}{13}
{2}{12}{23}
{2}{13}{23}
{3}{13}{23}
{12}{13}{23}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Select[Subsets[Range[n], {IntegerLength[n, 2]}], Select[Tuples[bpe/@#], UnsameQ@@#&]!={}&]], {n, 0, 10}]
CROSSREFS
Dominated by A357812.
The version for set-systems is A368601, max of A367902 (complement A367903).
For prime indices we have A370585, with n A370590, see also A370591.
This is the maximal case of A370636 (complement A370637).
The case of a unique choice is A370638.
The case containing n is A370641, non-maximal A370639.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Sequence in context: A296106 A329094 A327327 * A328976 A059197 A049974
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 10 2024
STATUS
approved