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A358133 Triangle read by rows whose n-th row lists the first differences of the n-th composition in standard order (row n of A066099).

%I A358133 #9 Oct 31 2022 15:23:45
%S A358133 0,-1,1,0,0,-2,0,-1,0,2,1,-1,0,1,0,0,0,-3,-1,-2,0,1,0,-1,-1,1,-1,0,0,
%T A358133 3,2,-2,1,0,1,-1,0,0,2,0,1,-1,0,0,1,0,0,0,0,-4,-2,-3,0,0,-1,-1,-2,1,
%U A358133 -2,0,0,2,1,-2,0,0,0,-1,0,-1,2,-1,1,-1,-1,0,1,-1
%N A358133 Triangle read by rows whose n-th row lists the first differences of the n-th composition in standard order (row n of A066099).
%C A358133 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%H A358133 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A358133 Triangle begins (dots indicate empty rows):
%e A358133    1:   .
%e A358133    2:   .
%e A358133    3:   0
%e A358133    4:   .
%e A358133    5:  -1
%e A358133    6:   1
%e A358133    7:   0  0
%e A358133    8:   .
%e A358133    9:  -2
%e A358133   10:   0
%e A358133   11:  -1  0
%e A358133   12:   2
%e A358133   13:   1 -1
%e A358133   14:   0  1
%e A358133   15:   0  0  0
%t A358133 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358133 Table[Differences[stc[n]],{n,100}]
%Y A358133 See link for sequences related to standard compositions.
%Y A358133 First differences of rows of A066099.
%Y A358133 The version for Heinz numbers of partitions is A355536, ranked by A253566.
%Y A358133 The partial sums instead of first differences are A358134.
%Y A358133 Row sums are A358135.
%Y A358133 A011782 counts compositions.
%Y A358133 A351014 counts distinct runs in standard compositions.
%Y A358133 Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A242628, A357135, A357187.
%K A358133 sign,tabf
%O A358133 3,6
%A A358133 _Gus Wiseman_, Oct 31 2022