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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

Table of n, a(n) for n=0..82.

FORMULA

a(n) = A334299(n) - 1.

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

Row lengths are A011782.

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.

Knapsack compositions are A333223.

Contiguous positive subsequence-sums are counted by A333224.

Contiguous subsequence-sums are counted by A333257.

Subsequence-sums are counted by A334968.

Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325676, A334967.

Sequence in context: A110783 A333224 A124770 * A308093 A264154 A302641

Adjacent sequences:  A334297 A334298 A334299 * A334301 A334302 A334303

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Jun 01 2020

STATUS

approved

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Last modified July 31 10:51 EDT 2021. Contains 346373 sequences. (Running on oeis4.)