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Sum of partial sums of the n-th composition in standard order (A066099).
19

%I #12 Apr 16 2023 06:32:19

%S 0,1,2,3,3,5,4,6,4,7,6,9,5,8,7,10,5,9,8,12,7,11,10,14,6,10,9,13,8,12,

%T 11,15,6,11,10,15,9,14,13,18,8,13,12,17,11,16,15,20,7,12,11,16,10,15,

%U 14,19,9,14,13,18,12,17,16,21,7,13,12,18,11,17,16,22

%N Sum of partial sums of the n-th composition in standard order (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 The 29th composition in standard order is (1,1,2,1), with partial sums (1,2,4,5), with sum 12, so a(29) = 12.

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

%t Table[Total[Accumulate[stc[n]]],{n,0,100}]

%Y See link for sequences related to standard compositions.

%Y Each n appears A000009(n) times.

%Y The reverse version is A029931.

%Y Comps counted by this statistic are A053632, ptns A264034, rev ptns A358194.

%Y This is the sum of partial sums of rows of A066099.

%Y The version for Heinz numbers of partitions is A318283, row sums of A358136.

%Y Row sums of A358134.

%Y A011782 counts compositions.

%Y A065120 gives first part of standard compositions, last A001511.

%Y A242628 lists adjusted partial sums, ranked by A253565, row sums A359043.

%Y A358135 gives last minus first of standard compositions.

%Y Cf. A000120, A029837, A070939, A133494, A253566, A358133, A358137.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 20 2022