OFFSET
0,3
COMMENTS
First differs from A353432 (the consecutive case) in having 0 and 53.
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
The initial terms, their binary expansions, and the corresponding standard compositions:
0: 0 ()
1: 1 (1)
10: 1010 (2,2)
21: 10101 (2,2,1)
26: 11010 (1,2,2)
43: 101011 (2,2,1,1)
53: 110101 (1,2,2,1)
58: 111010 (1,1,2,2)
107: 1101011 (1,2,2,1,1)
117: 1110101 (1,1,2,2,1)
174: 10101110 (2,2,1,1,2)
186: 10111010 (2,1,1,2,2)
292: 100100100 (3,3,3)
314: 100111010 (3,1,1,2,2)
346: 101011010 (2,2,1,2,2)
348: 101011100 (2,2,1,1,3)
349: 101011101 (2,2,1,1,2,1)
373: 101110101 (2,1,1,2,2,1)
430: 110101110 (1,2,2,1,1,2)
442: 110111010 (1,2,1,1,2,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y], Length/@Split[y]];
Select[Range[0, 100], rosQ[stc[#]]&]
CROSSREFS
These compositions are counted by A353390.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A333769 lists run-lengths of compositions in standard order.
Statistics of standard compositions:
- Runs-resistance is A333628.
Classes of standard compositions:
- Runs are A272919.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2022
STATUS
approved