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A335374
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Numbers k such that the k-th composition in standard order (A066099) is not co-unimodal.
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4
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13, 25, 27, 29, 41, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 81, 82, 83, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 145, 153, 155, 157, 161, 162, 163, 165, 166, 167, 169, 173, 177
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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 co-unimodal if it is the concatenation of a weakly decreasing and a weakly increasing sequence, implying that its negation is unimodal.
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:
13: (1,2,1)
25: (1,3,1)
27: (1,2,1,1)
29: (1,1,2,1)
41: (2,3,1)
45: (2,1,2,1)
49: (1,4,1)
50: (1,3,2)
51: (1,3,1,1)
53: (1,2,2,1)
54: (1,2,1,2)
55: (1,2,1,1,1)
57: (1,1,3,1)
59: (1,1,2,1,1)
61: (1,1,1,2,1)
77: (3,1,2,1)
81: (2,4,1)
82: (2,3,2)
83: (2,3,1,1)
89: (2,1,3,1)
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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, 100], !unimodQ[-stc[#]]&]
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CROSSREFS
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This is the dual version of A335373.
The case that is not 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.
Co-unimodal compositions are A332578.
Numbers with non-co-unimodal unsorted prime signature are A332642.
Non-co-unimodal compositions are A332669.
Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.
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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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