A358133 - OEIS

%I #9 Oct 31 2022 15:23:45

%S 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 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 -2,0,0,2,1,-2,0,0,0,-1,0,-1,2,-1,1,-1,-1,0,1,-1

%N 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 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 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 Triangle begins (dots indicate empty rows):

%e    1:   .

%e    2:   .

%e    3:   0

%e    4:   .

%e    5:  -1

%e    6:   1

%e    7:   0  0

%e    8:   .

%e    9:  -2

%e   10:   0

%e   11:  -1  0

%e   12:   2

%e   13:   1 -1

%e   14:   0  1

%e   15:   0  0  0

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[Differences[stc[n]],{n,100}]

%Y See link for sequences related to standard compositions.

%Y First differences of rows of A066099.

%Y The version for Heinz numbers of partitions is A355536, ranked by A253566.

%Y The partial sums instead of first differences are A358134.

%Y Row sums are A358135.

%Y A011782 counts compositions.

%Y A351014 counts distinct runs in standard compositions.

%Y Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A242628, A357135, A357187.

%K sign,tabf

%O 3,6

%A _Gus Wiseman_, Oct 31 2022