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A359043 Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.

%I A359043 #9 Dec 21 2022 20:11:44
%S A359043 0,1,2,2,3,4,3,3,4,6,5,6,4,5,4,4,5,8,7,9,6,8,7,8,5,7,6,7,5,6,5,5,6,10,
%T A359043 9,12,8,11,10,12,7,10,9,11,8,10,9,10,6,9,8,10,7,9,8,9,6,8,7,8,6,7,6,6,
%U A359043 7,12,11,15,10,14,13,16,9,13,12,15,11,14,13
%N A359043 Sum of adjusted partial sums of the n-th composition in standard order (A066099). Row sums of A242628.
%C A359043 We define the adjusted partial sums of a composition to be obtained by subtracting one from all parts, taking partial sums, and adding one back to all parts.
%C A359043 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 A359043 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 A359043 The 29th composition in standard order is (1,1,2,1), with adjusted partial sums (1,1,2,2), with sum 6, so a(29) = 6.
%t A359043 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A359043 Table[Total[Accumulate[stc[n]-1]+1],{n,0,100}]
%Y A359043 See link for sequences related to standard compositions.
%Y A359043 The unadjusted reverse version is A029931, row sums of A048793.
%Y A359043 The reverse version is A161511, row sums of A125106.
%Y A359043 Row sums of A242628, ranked by A253565.
%Y A359043 The unadjusted version is A359042, row sums of A358134.
%Y A359043 A011782 counts compositions.
%Y A359043 A066099 lists standard compositions.
%Y A359043 A358135 gives last minus first of standard compositions.
%Y A359043 A358194 counts partitions by sum and weighted sum.
%Y A359043 Cf. A000120, A005940, A019565, A029837, A059893, A253566, A358133, A358170.
%K A359043 nonn
%O A359043 0,3
%A A359043 _Gus Wiseman_, Dec 21 2022