OFFSET
0,3
COMMENTS
The a(n)-th composition in standard order lists the leaders of anti-runs of the n-th composition in standard order.
The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
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.
Does this sequence contain all nonnegative integers?
LINKS
EXAMPLE
The 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2). This is the 42nd composition in standard order, so a(346) = 42.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n], UnsameQ]], {n, 0, 100}]
CROSSREFS
Ranks of rows of A374515.
A011782 counts compositions.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Parts are listed by A066099.
- Run-length transform is A333627.
- Run-sum transform is A353847.
Six types of runs:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 02 2024
STATUS
approved