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A335373
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Numbers k such that the k-th composition in standard order (A066099) is not unimodal.
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33
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22, 38, 44, 45, 46, 54, 70, 76, 77, 78, 86, 88, 89, 90, 91, 92, 93, 94, 102, 108, 109, 110, 118, 134, 140, 141, 142, 148, 150, 152, 153, 154, 155, 156, 157, 158, 166, 172, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 198
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OFFSET
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1,1
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COMMENTS
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A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
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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The sequence together with the corresponding compositions begins:
22: (2,1,2)
38: (3,1,2)
44: (2,1,3)
45: (2,1,2,1)
46: (2,1,1,2)
54: (1,2,1,2)
70: (4,1,2)
76: (3,1,3)
77: (3,1,2,1)
78: (3,1,1,2)
86: (2,2,1,2)
88: (2,1,4)
89: (2,1,3,1)
90: (2,1,2,2)
91: (2,1,2,1,1)
92: (2,1,1,3)
93: (2,1,1,2,1)
94: (2,1,1,1,2)
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MATHEMATICA
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unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 200], !unimodQ[stc[#]]&]
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CROSSREFS
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The dual version (non-co-unimodal compositions) is A335374.
The case that is not co-unimodal either is A335375.
Unimodal normal sequences are A007052.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers with non-unimodal unsorted prime signature are A332282.
Partitions with non-unimodal 0-appended first differences are A332284.
Non-unimodal permutations of the multiset of prime indices of n are A332671.
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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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