cp's OEIS Frontend

A358134 Triangle read by rows whose n-th row lists the partial sums of the n-th composition in standard order (row n of A066099).

%I A358134 #9 Jan 02 2023 15:25:50
%S A358134 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 A358134 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 A358134 2,4,5,1,2,3,5,1,2,3,4,5,6,5,6,4,6,4,5
%N A358134 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 A358134 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 A358134 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 A358134 Triangle begins:
%e A358134   1
%e A358134   2
%e A358134   1 2
%e A358134   3
%e A358134   2 3
%e A358134   1 3
%e A358134   1 2 3
%e A358134   4
%e A358134   3 4
%e A358134   2 4
%e A358134   2 3 4
%e A358134   1 4
%e A358134   1 3 4
%e A358134   1 2 4
%e A358134   1 2 3 4
%t A358134 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358134 Join@@Table[Accumulate[stc[n]],{n,100}]
%Y A358134 See link for sequences related to standard compositions.
%Y A358134 First element in each row is A065120.
%Y A358134 Rows are the partial sums of rows of A066099.
%Y A358134 Last element in each row is A070939.
%Y A358134 An adjusted version is A242628, ranked by A253565.
%Y A358134 The first differences instead of partial sums are A358133.
%Y A358134 The version for Heinz numbers of partitions is A358136, ranked by A358137.
%Y A358134 Row sums are A359042.
%Y A358134 A011782 counts compositions.
%Y A358134 A351014 counts distinct runs in standard compositions.
%Y A358134 A358135 gives last minus first of standard compositions.
%Y A358134 Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A253566, A355536, A357135.
%K A358134 nonn,tabf
%O A358134 1,2
%A A358134 _Gus Wiseman_, Oct 31 2022