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A334299
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Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).
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28
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1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 6, 4, 7, 6, 5, 2, 4, 4, 6, 4, 6, 7, 8, 4, 7, 6, 10, 6, 10, 8, 6, 2, 4, 4, 6, 3, 8, 8, 8, 4, 8, 4, 9, 8, 12, 11, 10, 4, 7, 8, 10, 8, 11, 12, 13, 6, 10, 9, 14, 8, 13, 10, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 12, 7, 14, 12, 10, 4
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1
2
2 3
2 4 4 4
2 4 3 6 4 7 6 5
2 4 4 6 4 6 7 8 4 7 6 10 6 10 8 6
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 subsequences of the composition with STC-number n:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 1 0 2 2 3 0 4 2 5 4 6 6 7
0 1 1 1 1 0 3 1 5 3 3
0 0 0 0 2 0 3 2 1
1 2 1 0
0 1 0
0
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Subsets[stc[n]]]], {n, 0, 100}]
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CROSSREFS
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Looking only at contiguous subsequences gives A124771.
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.
Disallowing empty subsequences gives A334300.
Subsequence-sums are counted by A334968.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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