proposed

approved

editing

proposed

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.

A326031 gives weight of the set-system with BII-number n.

A368100 ranks choosable multisets, complement A355529.

Cf. A072639, A140637, A326031, A355529, A367905, A368109, `A370583, A370589, A370591, `A370593, A370636, `A370638, `A370639, A370640.

0, 0, 0, 0, 0, 1, 4, 13, 13, 26, 56, 126, 243, 471, 812, 1438

allocated for Gus WisemanNumber of minimal subsets of {2..n} such that it is not possible to choose a different binary index of each element.

0, 0, 0, 0, 0, 1, 4, 13, 13, 26, 56, 126, 243

0,7

The a(0) = 0 through a(7) = 13 subsets:

. . . . . {2,3,4,5} {2,4,6} {2,4,6}

{2,3,4,5} {2,3,4,5}

{2,3,5,6} {2,3,4,7}

{3,4,5,6} {2,3,5,6}

{2,3,5,7}

{2,3,6,7}

{2,4,5,7}

{2,5,6,7}

{3,4,5,6}

{3,4,5,7}

{3,4,6,7}

{3,5,6,7}

{4,5,6,7}

The a(0) = 0 through a(7) = 13 set-systems:

. . . . . {2}{12}{3}{13} {2}{3}{23} {2}{3}{23}

{2}{12}{3}{13} {2}{12}{3}{13}

{12}{3}{13}{23} {12}{3}{13}{23}

{2}{12}{13}{23} {2}{12}{13}{23}

{2}{12}{3}{123}

{2}{3}{13}{123}

{12}{3}{13}{123}

{12}{3}{23}{123}

{2}{12}{13}{123}

{2}{12}{23}{123}

{2}{13}{23}{123}

{3}{13}{23}{123}

{12}{13}{23}{123}

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

fasmin[y_]:=Complement[y, Union@@Table[Union[s, #]& /@ Rest[Subsets[Complement[Union@@y, s]]], {s, y}]];

Table[Length[fasmin[Select[Subsets[Range[2, n]], Select[Tuples[bpe/@#], UnsameQ@@#&]=={}&]]], {n, 0, 10}]

The version with ones allowed is A370642, minimal case of A370637.

This is the minimal case of A370643.

A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

A326031 gives weight of the set-system with BII-number n.

A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.

A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.

A368100 ranks choosable multisets, complement A355529.

A370585 counts maximal choosable sets.

Cf. A072639, A140637, A367905, A368109, `A370583, A370589, A370591, `A370593, A370636, `A370638, `A370639, A370640.

allocated

nonn,more

Gus Wiseman, Mar 11 2024

approved

editing

allocated for Gus Wiseman

allocated

approved