cp's OEIS Frontend

A357187 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.".

%I A357187 #12 Jul 23 2025 16:04:02
%S A357187 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 A357187 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 A357187 -3,1,1,0,0,1,0,0,-1,0,1,0,-1,1,0,0,-1,1,0
%N A357187 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 A357187 Are there any terms > 1?
%C A357187 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 A357187 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 A357187 a(n) = A357186(n + 1) - A357186(n).
%e A357187 We have A357186(5) - A357186(4) = 3 - 2 = 1, so a(4) = 1.
%t A357187 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357187 Differences[Table[stc/@stc[n]/.List->Plus,{n,0,100}]]
%Y A357187 See link for sequences related to standard compositions.
%Y A357187 Positions of first appearances appear to all belong to A052955.
%Y A357187 Differences of A357186 (row-sums of A357135).
%Y A357187 The version for partitions is A357458, differences of A325033.
%Y A357187 Cf. A000120, A029837, A029931, A048896, A058891, A070939, A133494, A333766, A357134, A357137.
%K A357187 sign
%O A357187 0,32
%A A357187 _Gus Wiseman_, Sep 28 2022