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A385886 Irregular triangle read by rows listing the lengths of maximal anti-runs (sequences of distinct consecutive elements increasing by more than 1) of binary indices, duplicate rows removed.

%I A385886 #10 Jul 19 2025 10:13:11
%S A385886 1,1,1,2,1,1,1,1,2,2,1,1,1,1,1,3,1,1,2,1,2,1,2,1,1,1,1,1,1,1,1,3,2,2,
%T A385886 1,1,1,2,3,1,1,1,2,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,4,1,1,3,1,2,2,2,1,2,
%U A385886 1,1,1,1,2,1,3,1,2,2,1,1,1,1,2,1
%N A385886 Irregular triangle read by rows listing the lengths of maximal anti-runs (sequences of distinct consecutive elements increasing by more than 1) of binary indices, duplicate rows removed.
%C A385886 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.
%C A385886 This is the triangle A384877, except all duplicates after the first instance of each composition are removed. It lists all compositions in order of their first appearance as a row of A384877.
%e A385886 The binary indices of 27 are {1,2,4,5}, with maximal anti-runs ((1),(2,4),(5)), with lengths (1,2,1). After removing duplicates, this is our row 10.
%e A385886 The binary indices of 53 are {1,3,5,6}, with maximal anti-runs ((1,3,5),(6)), with lengths (3,1). After removing duplicates, this is our row 16.
%e A385886 Triangle begins:
%e A385886    0: .
%e A385886    1: 1
%e A385886    2: 1 1
%e A385886    3: 2
%e A385886    4: 1 1 1
%e A385886    5: 1 2
%e A385886    6: 2 1
%e A385886    7: 1 1 1 1
%e A385886    8: 3
%e A385886    9: 1 1 2
%e A385886   10: 1 2 1
%e A385886   11: 2 1 1
%e A385886   12: 1 1 1 1 1
%e A385886   13: 1 3
%e A385886   14: 2 2
%e A385886   15: 1 1 1 2
%e A385886   16: 3 1
%e A385886   17: 1 1 2 1
%e A385886   18: 1 2 1 1
%e A385886   19: 2 1 1 1
%e A385886   20: 1 1 1 1 1 1
%t A385886 DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2!=#1+1&],{n,0,100}]]
%Y A385886 In the following references, "before" is short for "before removing duplicate rows".
%Y A385886 Positions of singleton rows appear to be A001906 = A055588 - 1.
%Y A385886 Positions of rows of the form (1,1,...) appear to be A001911-2, before A023758.
%Y A385886 Row sums appear to be A200648, before A000120.
%Y A385886 Row lengths appear to be A200649, before A384890.
%Y A385886 Standard composition numbers of each row appear to be A348366.
%Y A385886 Before we had A384877, ranks A385816, firsts A052499.
%Y A385886 For runs instead of anti-runs we have A385817, see A245563, A245562, A246029.
%Y A385886 Cf. A044813, A048793, A069010, A164707, A242882, A328592, A384175, A384177, A384878, A384879, A384893.
%K A385886 nonn,tabf
%O A385886 0,4
%A A385886 _Gus Wiseman_, Jul 14 2025