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A350356
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Numbers k such that the k-th composition in standard order is down/up.
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6
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0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 22, 32, 33, 34, 38, 44, 45, 64, 65, 66, 68, 70, 76, 77, 88, 89, 128, 129, 130, 132, 134, 140, 141, 148, 152, 153, 176, 177, 178, 182, 256, 257, 258, 260, 262, 264, 268, 269, 276, 280, 281, 296, 297, 304, 305, 306, 310, 352, 353
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OFFSET
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1,3
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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.
A composition is down/up if it is alternately strictly increasing and strictly decreasing, starting with a decrease. For example, the partition (3,2,2,2,1) has no down/up permutations, even though it does have the anti-run permutation (2,1,2,3,2).
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LINKS
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FORMULA
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EXAMPLE
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The terms together with the corresponding compositions begin:
0: ()
1: (1)
2: (2)
4: (3)
5: (2,1)
8: (4)
9: (3,1)
16: (5)
17: (4,1)
18: (3,2)
22: (2,1,2)
32: (6)
33: (5,1)
34: (4,2)
38: (3,1,2)
44: (2,1,3)
45: (2,1,2,1)
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MATHEMATICA
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doupQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<y[[m+1]], y[[m]]>y[[m+1]]], {m, 1, Length[y]-1}];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], doupQ[stc[#]]&]
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CROSSREFS
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The case of permutations is counted by A000111.
Counting patterns of this type gives A350354.
A003242 counts anti-run compositions.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Constant compositions are A272919.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.
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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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