OFFSET
1,1
COMMENTS
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.
EXAMPLE
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)
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, 200], !unimodQ[stc[#]]&]
CROSSREFS
The dual version (non-co-unimodal compositions) is A335374.
The case that is not co-unimodal either is A335375.
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.
Partitions with non-unimodal 0-appended first differences are A332284.
Non-unimodal permutations of the multiset of prime indices of n are A332671.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 03 2020
STATUS
approved