OFFSET
0,3
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 152 in standard order is (3,1,4), and the a(152) = 6 choices are: (1,1,1), (1,1,2), (1,1,4), (3,1,1), (3,1,2), (3,1,4).
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Times@@Length/@Divisors/@stc[n], {n, 0, 100}]
CROSSREFS
Positions of 1's are A000079 (after the first).
An unordered version (using prime indices) is A355731:
- firsts A355732,
- resorted A355733,
- weakly increasing A355735,
- relatively prime A355737,
- strict A355739.
A000005 counts divisors.
A003963 multiplies together the prime indices of n.
A005811 counts runs in binary expansion.
A066099 lists the compositions in standard order.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 23 2022
STATUS
approved