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A124769
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Number of strictly decreasing runs for compositions in standard order.
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19
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0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 2, 2, 2, 3, 4, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 4, 5, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 3, 3, 2, 2, 3, 4, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 4, 5, 6, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 2, 2, 2, 3, 4, 2, 2, 2, 3, 3, 3, 3, 4, 2, 2, 3, 3, 3, 3, 4, 5, 2, 2, 2, 3, 3, 2, 3, 4, 3
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OFFSET
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0,4
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COMMENTS
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The standard order of compositions is given by A066099.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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. a(n) is the number of maximal strictly decreasing runs in this composition. Alternatively, a(n) is one plus the number of weak ascents in the same composition. For example, the strictly decreasing runs of the 1234567th composition are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so a(1234567) = 7. The 6 weak ascents together with the strict descents are: 3 > 2 > 1 <= 2 <= 2 > 1 <= 2 <= 5 > 1 <= 1 <= 1. - Gus Wiseman, Apr 08 2020
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LINKS
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FORMULA
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a(0) = 0, a(n) = A124764(n) + 1 for n > 0.
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EXAMPLE
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Composition number 11 is 2,1,1; the strictly increasing runs are 2,1; 1; so a(11) = 2.
The table starts:
0
1
1 2
1 1 2 3
1 1 2 2 2 2 3 4
1 1 1 2 2 2 2 3 2 2 3 3 3 3 4 5
1 1 1 2 2 1 2 3 2 2 3 3 2 2 3 4 2 2 2 3 3 3 3 4 3 3 4 4 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;
Table[Length[Split[stc[n], Greater]], {n, 0, 100}] (* Gus Wiseman, Apr 08 2020 *)
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CROSSREFS
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Compositions of n with k weak ascents are A333213.
All of the following pertain to compositions in standard order (A066099):
- Partial sums from the right are A048793 (triangle).
- Weakly decreasing compositions are A114994.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769 (this sequence).
- Reversed initial intervals A164894.
- Weakly increasing compositions are A225620.
- Constant compositions are A272919.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
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KEYWORD
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easy,nonn,tabf
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AUTHOR
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STATUS
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approved
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