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A350355
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Numbers k such that the k-th composition in standard order is up/down.
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6
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0, 1, 2, 4, 6, 8, 12, 13, 16, 20, 24, 25, 32, 40, 41, 48, 49, 50, 54, 64, 72, 80, 81, 82, 96, 97, 98, 102, 108, 109, 128, 144, 145, 160, 161, 162, 166, 192, 193, 194, 196, 198, 204, 205, 216, 217, 256, 272, 288, 289, 290, 320, 321, 322, 324, 326, 332, 333, 384
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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 up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2).
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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)
6: (1,2)
8: (4)
12: (1,3)
13: (1,2,1)
16: (5)
20: (2,3)
24: (1,4)
25: (1,3,1)
32: (6)
40: (2,4)
41: (2,3,1)
48: (1,5)
49: (1,4,1)
50: (1,3,2)
54: (1,2,1,2)
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MATHEMATICA
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updoQ[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], updoQ[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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