A357135 - OEIS

%I #7 Sep 27 2022 09:00:12

%S 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 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 1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1

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

%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 Row n is the A357134(n)-th composition in standard order.

%e Triangle begins:

%e    0:

%e    1: 1

%e    2: 2

%e    3: 1 1

%e    4: 1 1

%e    5: 2 1

%e    6: 1 2

%e    7: 1 1 1

%e    8: 3

%e    9: 1 1 1

%e   10: 2 2

%e   11: 2 1 1

%e   12: 1 1 1

%e   13: 1 2 1

%e   14: 1 1 2

%e   15: 1 1 1 1

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

%t Join@@Table[Join@@stc/@stc[n],{n,0,30}]

%Y See link for sequences related to standard compositions.

%Y Row n is the A357134(n)-th composition in standard order.

%Y The version for Heinz numbers of partitions is A357139, cf. A003963.

%Y Row sums are A357186, differences A357187.

%Y Cf. A000120, A001511, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.

%K nonn

%O 0,2

%A _Gus Wiseman_, Sep 26 2022