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A335375
Numbers k such that the k-th composition in standard order (A066099) is neither unimodal nor co-unimodal.
2
45, 54, 77, 89, 91, 93, 102, 108, 109, 110, 118, 141, 153, 155, 157, 166, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189, 198, 204, 205, 206, 214, 216, 217, 218, 219, 220, 221, 222, 230, 236, 237, 238, 246, 269, 281, 283, 285, 297, 301, 305, 306, 307, 309
OFFSET
1,1
COMMENTS
A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. It is co-unimodal if 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.
EXAMPLE
The sequence together with the corresponding compositions begins:
45: (2,1,2,1)
54: (1,2,1,2)
77: (3,1,2,1)
89: (2,1,3,1)
91: (2,1,2,1,1)
93: (2,1,1,2,1)
102: (1,3,1,2)
108: (1,2,1,3)
109: (1,2,1,2,1)
110: (1,2,1,1,2)
118: (1,1,2,1,2)
141: (4,1,2,1)
153: (3,1,3,1)
155: (3,1,2,1,1)
157: (3,1,1,2,1)
166: (2,3,1,2)
173: (2,2,1,2,1)
177: (2,1,4,1)
178: (2,1,3,2)
179: (2,1,3,1,1)
MATHEMATICA
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[#]]&&!unimodQ[-stc[#]]&]
CROSSREFS
Non-unimodal compositions are ranked by A335373.
Non-co-unimodal compositions are ranked by A335374.
Unimodal compositions are A001523.
Unimodal normal sequences are A007052.
Unimodal permutations are A011782.
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.
Sequence in context: A295802 A242267 A242226 * A039423 A043246 A044026
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 04 2020
STATUS
approved