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By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.
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%I #7 Nov 11 2022 08:08:22

%S 0,1,1,2,2,2,2,3,1,3,2,3,3,3,3,4,2,2,3,4,3,3,3,4,2,4,3,4,4,4,4,5,2,3,

%T 2,3,4,4,4,5,2,4,3,4,4,4,4,5,3,3,4,5,4,4,4,5,3,5,4,5,5,5,5,6,3,3,3,4,

%U 3,3,3,4,3,5,4,5,5,5,5,6,3,3,4,5,4,4,4

%N By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.

%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>

%F Sum of A000120 over row n of A066099.

%e Composition 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 5.

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

%t Length/@Table[Join@@stc/@stc[n],{n,0,100}]

%Y See link for sequences related to standard compositions (A066099).

%Y Dominates A000120.

%Y Row-lengths of A357135, which is ranked by A357134.

%Y A related sequence is A358330.

%Y Cf. A001511, A029931, A048896, A058891, A070939, A096111, A333766, A357137, A357139, A357186, A357187.

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 10 2022