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Sorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).
9

%I #5 Dec 18 2023 08:28:46

%S 1,4,20,52,64,68,84,116,308,372,820,884,1088,1092,1108,1140,1396,1908,

%T 2868,2932,3956,5184,5188,5204,5236,5492,6004,8052,13376,13380,13396,

%U 13428,13684,14196,16244,17204,17268,18292,19252,19316,20340,22388,24436,30580

%N Sorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).

%C 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}.

%e The terms together with the corresponding set-systems begin:

%e 1: {{1}}

%e 4: {{1,2}}

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

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

%e 64: {{1,2,3}}

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

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

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

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

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

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

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

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

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

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

%Y For multisets we have A367915, unsorted A367913, firsts A367912.

%Y Sorted positions of first appearances in A368109.

%Y The unsorted version is A368111.

%Y A048793 lists binary indices, length A000120, sum A029931.

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

%Y A070939 gives length of binary expansion.

%Y A096111 gives product of binary indices.

%Y Cf. A072639, A253317, A326031, A326702, A326753, A355741, A367771, A367905, A367906, A367911, A368184.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 17 2023