OFFSET
0,4
COMMENTS
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.
EXAMPLE
Composition number 981 in standard order is (1,1,1,2,2,2,1), with partial runs (1), (2), (1,1), (2,2), (1,1,1), (2,2,2), so a(981) = 6.
As a triangle:
1
1 2
1 2 2 3
1 2 2 3 2 2 3 4
1 2 2 3 2 3 2 4 2 2 3 3 3 3 4 5
1 2 2 3 2 3 3 4 2 3 3 4 3 2 3 5 2 2 3 3 3 3 2 4 3 3 4 3 4 4 5 6
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
pre[y_]:=NestWhileList[Most, y, Length[#]>1&];
Table[Length[Union[Join@@pre/@Split[stc[n]]]], {n, 0, 100}]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jun 13 2022
STATUS
approved