OFFSET
0,3
COMMENTS
The a(n)-th composition in standard order lists the leaders of strictly increasing runs in the n-th composition in standard order.
The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences 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.
LINKS
EXAMPLE
The 813th composition in standard order is (1,3,2,1,2,1), with strictly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.
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], Less]], {n, 0, 100}]
CROSSREFS
Ranks of rows of A374683.
The weak version is A375123.
The weak opposite version is A375124.
The opposite version is A375126.
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 transformation is A353847.
Six types of runs:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 02 2024
STATUS
approved