|
| |
|
|
A334300
|
|
Number of distinct nonempty subsequences (not necessarily contiguous) in the n-th composition in standard order (A066099).
|
|
3
|
|
|
|
0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 5, 3, 6, 5, 4, 1, 3, 3, 5, 3, 5, 6, 7, 3, 6, 5, 9, 5, 9, 7, 5, 1, 3, 3, 5, 2, 7, 7, 7, 3, 7, 3, 8, 7, 11, 10, 9, 3, 6, 7, 9, 7, 10, 11, 12, 5, 9, 8, 13, 7, 12, 9, 6, 1, 3, 3, 5, 3, 7, 7, 7, 3, 5, 5, 11, 6, 13, 11, 9, 3, 7, 6
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Looking only at contiguous subsequences, or restrictions to a subinterval, gives A124770.
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
|
|
|
|
FORMULA
|
|
|
|
EXAMPLE
|
Triangle begins:
1
1 2
1 3 3 3
1 3 2 5 3 6 5 4
1 3 3 5 3 5 6 7 3 6 5 9 5 9 7 5
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The n-th column below lists the STC-numbers of the nonempty subsequences of the composition with STC-number n:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 2 2 3 4 2 5 4 6 6 7
1 1 1 1 3 1 5 3 3
2 3 2 1
1 2 1
1
|
|
|
MATHEMATICA
|
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Rest[Subsets[stc[n]]]]], {n, 0, 100}]
|
|
|
CROSSREFS
|
Looking only at contiguous subsequences gives A124770.
The contiguous case with empty subsequences allowed is A124771.
Allowing empty subsequences gives A334299.
Compositions where every subinterval has a different sum are A333222.
Contiguous positive subsequence-sums are counted by A333224.
Contiguous subsequence-sums are counted by A333257.
Subsequence-sums are counted by A334968.
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
