cp's OEIS Frontend

A357135 Take the k-th composition in standard order for each part k of the n-th composition in standard order; then concatenate.

%I A357135 #7 Sep 27 2022 09:00:12
%S A357135 1,2,1,1,1,1,2,1,1,2,1,1,1,3,1,1,1,2,2,2,1,1,1,1,1,1,2,1,1,1,2,1,1,1,
%T A357135 1,2,1,3,1,1,1,2,1,1,1,1,2,1,1,2,2,1,2,1,2,2,1,1,1,1,3,1,1,1,1,1,2,2,
%U A357135 1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1
%N A357135 Take the k-th composition in standard order for each part k of the n-th composition in standard order; then concatenate.
%C A357135 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 A357135 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 A357135 Row n is the A357134(n)-th composition in standard order.
%e A357135 Triangle begins:
%e A357135    0:
%e A357135    1: 1
%e A357135    2: 2
%e A357135    3: 1 1
%e A357135    4: 1 1
%e A357135    5: 2 1
%e A357135    6: 1 2
%e A357135    7: 1 1 1
%e A357135    8: 3
%e A357135    9: 1 1 1
%e A357135   10: 2 2
%e A357135   11: 2 1 1
%e A357135   12: 1 1 1
%e A357135   13: 1 2 1
%e A357135   14: 1 1 2
%e A357135   15: 1 1 1 1
%t A357135 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A357135 Join@@Table[Join@@stc/@stc[n],{n,0,30}]
%Y A357135 See link for sequences related to standard compositions.
%Y A357135 Row n is the A357134(n)-th composition in standard order.
%Y A357135 The version for Heinz numbers of partitions is A357139, cf. A003963.
%Y A357135 Row sums are A357186, differences A357187.
%Y A357135 Cf. A000120, A001511, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.
%K A357135 nonn
%O A357135 0,2
%A A357135 _Gus Wiseman_, Sep 26 2022