OFFSET
1,1
COMMENTS
Such a composition cannot be strict.
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.
LINKS
EXAMPLE
The terms together with their standard compositions begin:
13: (1,2,1)
22: (2,1,2)
25: (1,3,1)
45: (2,1,2,1)
49: (1,4,1)
54: (1,2,1,2)
76: (3,1,3)
77: (3,1,2,1)
82: (2,3,2)
89: (2,1,3,1)
97: (1,5,1)
101: (1,3,2,1)
102: (1,3,1,2)
105: (1,2,3,1)
108: (1,2,1,3)
109: (1,2,1,2,1)
141: (4,1,2,1)
148: (3,2,3)
150: (3,2,1,2)
153: (3,1,3,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], Length[Split[stc[#]]] == Length[stc[#]]&&!UnsameQ@@First/@Split[stc[#]]&]
CROSSREFS
Compositions of this type are counted by A285981.
Permutations of prime indices of this type are counted by A335460.
A011782 counts compositions.
A066099 lists compositions in standard order.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335456 counts patterns matched by compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
A335465 counts minimal patterns avoided by a standard composition.
A373948 encodes run-compression using compositions in standard order.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 14 2024
STATUS
approved