A358134 - OEIS

%I #9 Jan 02 2023 15:25:50

%S 1,2,1,2,3,2,3,1,3,1,2,3,4,3,4,2,4,2,3,4,1,4,1,3,4,1,2,4,1,2,3,4,5,4,

%T 5,3,5,3,4,5,2,5,2,4,5,2,3,5,2,3,4,5,1,5,1,4,5,1,3,5,1,3,4,5,1,2,5,1,

%U 2,4,5,1,2,3,5,1,2,3,4,5,6,5,6,4,6,4,5

%N Triangle read by rows whose n-th row lists the partial sums 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:

%e   1

%e   2

%e   1 2

%e   3

%e   2 3

%e   1 3

%e   1 2 3

%e   4

%e   3 4

%e   2 4

%e   2 3 4

%e   1 4

%e   1 3 4

%e   1 2 4

%e   1 2 3 4

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

%t Join@@Table[Accumulate[stc[n]],{n,100}]

%Y See link for sequences related to standard compositions.

%Y First element in each row is A065120.

%Y Rows are the partial sums of rows of A066099.

%Y Last element in each row is A070939.

%Y An adjusted version is A242628, ranked by A253565.

%Y The first differences instead of partial sums are A358133.

%Y The version for Heinz numbers of partitions is A358136, ranked by A358137.

%Y Row sums are A359042.

%Y A011782 counts compositions.

%Y A351014 counts distinct runs in standard compositions.

%Y A358135 gives last minus first of standard compositions.

%Y Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A253566, A355536, A357135.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Oct 31 2022