cp's OEIS Frontend

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 A357186 #6 Sep 29 2022 07:42:03
%S A357186 0,1,2,2,2,3,3,3,3,3,4,4,3,4,4,4,3,4,4,4,4,5,5,5,4,4,5,5,4,5,5,5,3,4,
%T A357186 5,5,4,5,5,5,5,5,6,6,5,6,6,6,4,5,5,5,5,6,6,6,5,5,6,6,5,6,6,6,3,4,5,5,
%U A357186 5,6,6,6,5,5,6,6,5,6,6,6,5,6,6,6,6,7,7
%N 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 A357186 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 A357186 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 A357186 a(n) = A029837(A357134(n)).
%e A357186 Composition 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 2 + 1 + 1 + 1 + 1 = 6.
%t A357186 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357186 Table[stc/@stc[n]/.List->Plus,{n,0,100}]
%Y A357186 See link for sequences related to standard compositions.
%Y A357186 This is the sum of A029837 over the n-th composition in standard order.
%Y A357186 Vertex degrees are A133494.
%Y A357186 The version for Heinz numbers of partitions is A325033.
%Y A357186 Row sums of A357135.
%Y A357186 First differences are A357187.
%Y A357186 Cf. A000120, A001511, A003963, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.
%K A357186 nonn
%O A357186 0,3
%A A357186 _Gus Wiseman_, Sep 28 2022