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A354582
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Number of distinct contiguous constant subsequences (or partial runs) in the k-th composition in standard order.
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3
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0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 2, 3, 4, 1, 2, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 3, 3, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 2, 3, 5, 2, 2, 3, 3, 3, 3, 2, 4, 3, 3, 4, 3, 4, 4, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 2, 3, 4, 5, 2, 3, 2, 4, 3, 4, 3
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OFFSET
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0,4
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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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EXAMPLE
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Composition number 981 in standard order is (1,1,1,2,2,2,1), with partial runs (1), (2), (1,1), (2,2), (1,1,1), (2,2,2), so a(981) = 6.
As a triangle:
1
1 2
1 2 2 3
1 2 2 3 2 2 3 4
1 2 2 3 2 3 2 4 2 2 3 3 3 3 4 5
1 2 2 3 2 3 3 4 2 3 3 4 3 2 3 5 2 2 3 3 3 3 2 4 3 3 4 3 4 4 5 6
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
pre[y_]:=NestWhileList[Most, y, Length[#]>1&];
Table[Length[Union[Join@@pre/@Split[stc[n]]]], {n, 0, 100}]
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CROSSREFS
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If we allow any constant subsequence we get A063787.
If we allow any contiguous subsequence we get A124771.
Positions of first appearances are A126646.
If we allow any subsequence we get A334299.
A066099 lists all compositions in standard order.
A124767 counts runs in standard compositions.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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