proposed

approved

editing

proposed

Choosing For multisets instead of sequences gives we have A367915, firsts of A367912, unsorted A367913, firsts A367912.

Cf. A072639, A253317, `A309326, A326031, A326702, `A326753, `A355731, ~A355739, A355741, `A355744, A367771, A367905, `A367906, A367911, A368184.

allocated for Gus WisemanSorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).

1, 4, 20, 52, 64, 68, 84, 116, 308, 372, 820, 884, 1088, 1092, 1108, 1140, 1396, 1908, 2868, 2932, 3956, 5184, 5188, 5204, 5236, 5492, 6004, 8052, 13376, 13380, 13396, 13428, 13684, 14196, 16244, 17204, 17268, 18292, 19252, 19316, 20340, 22388, 24436, 30580

1,2

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

The terms together with the corresponding set-systems begin:

1: {{1}}

4: {{1,2}}

20: {{1,2},{1,3}}

52: {{1,2},{1,3},{2,3}}

64: {{1,2,3}}

68: {{1,2},{1,2,3}}

84: {{1,2},{1,3},{1,2,3}}

116: {{1,2},{1,3},{2,3},{1,2,3}}

308: {{1,2},{1,3},{2,3},{1,4}}

372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}

820: {{1,2},{1,3},{2,3},{1,4},{2,4}}

884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

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

c=Table[Length[Tuples[bpe/@bpe[n]]], {n, 1000}];

Select[Range[Length[c]], FreeQ[Take[c, #-1], c[[#]]]&]

Choosing multisets instead of sequences gives A367915, firsts of A367912, unsorted A367913.

Sorted positions of first appearances in A368109.

The unsorted version is A368111.

A048793 lists binary indices, length A000120, sum A029931.

A058891 counts set-systems, covering A003465, connected A323818.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A072639, A253317, `A309326, A326031, A326702, `A326753, `A355731, ~A355739, A355741, `A355744, A367771, A367905, `A367906, A367911, A368184.

allocated

nonn

Gus Wiseman, Dec 17 2023

approved

editing

allocated for Gus Wiseman

allocated

approved