A357187 - OEIS

%I #10 Sep 29 2022 22:05:42

%S 1,1,0,0,1,0,0,0,0,1,0,-1,1,0,0,-1,1,0,0,0,1,0,0,-1,0,1,0,-1,1,0,0,-2,

%T 1,1,0,-1,1,0,0,0,0,1,0,-1,1,0,0,-2,1,0,0,0,1,0,0,-1,0,1,0,-1,1,0,0,

%U -3,1,1,0,0,1,0,0,-1,0,1,0,-1,1,0,0,-1,1,0

%N First differences A357186 = "Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything."

%C Are there any terms > 1?

%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 a(n) = A357186(n + 1) - A357186(n).

%e We have A357186(5) - A357186(4) = 3 - 2 = 1, so a(4) = 1.

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

%t Differences[Table[stc/@stc[n]/.List->Plus,{n,0,100}]]

%Y See link for sequences related to standard compositions.

%Y Positions of first appearances appear to all belong to A052955.

%Y Differences of A357186 (row-sums of A357135).

%Y The version for partitions is A357458, differences of A325033.

%Y Cf. A000120, A029837, A029931, A048896, A058891, A070939, A133494, A333766, A357134, A357137.

%K sign

%O 0,32

%A _Gus Wiseman_, Sep 28 2022