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Numbers k such that the k-th composition in standard order (A066099) is not unimodal.
34

%I #8 Jun 05 2020 09:56:42

%S 22,38,44,45,46,54,70,76,77,78,86,88,89,90,91,92,93,94,102,108,109,

%T 110,118,134,140,141,142,148,150,152,153,154,155,156,157,158,166,172,

%U 173,174,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,198

%N Numbers k such that the k-th composition in standard order (A066099) is not unimodal.

%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%C 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.

%e The sequence together with the corresponding compositions begins:

%e 22: (2,1,2)

%e 38: (3,1,2)

%e 44: (2,1,3)

%e 45: (2,1,2,1)

%e 46: (2,1,1,2)

%e 54: (1,2,1,2)

%e 70: (4,1,2)

%e 76: (3,1,3)

%e 77: (3,1,2,1)

%e 78: (3,1,1,2)

%e 86: (2,2,1,2)

%e 88: (2,1,4)

%e 89: (2,1,3,1)

%e 90: (2,1,2,2)

%e 91: (2,1,2,1,1)

%e 92: (2,1,1,3)

%e 93: (2,1,1,2,1)

%e 94: (2,1,1,1,2)

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,200],!unimodQ[stc[#]]&]

%Y The dual version (non-co-unimodal compositions) is A335374.

%Y The case that is not co-unimodal either is A335375.

%Y Unimodal compositions are A001523.

%Y Unimodal normal sequences are A007052.

%Y Unimodal permutations are A011782.

%Y Non-unimodal permutations are A059204.

%Y Non-unimodal compositions are A115981.

%Y Non-unimodal normal sequences are A328509.

%Y Numbers with non-unimodal unsorted prime signature are A332282.

%Y Partitions with non-unimodal 0-appended first differences are A332284.

%Y Non-unimodal permutations of the multiset of prime indices of n are A332671.

%Y Cf. A000120, A029931, A048793, A066099, A070939, A334299.

%Y Cf. A072704, A332281, A332286, A332287, A332639, A332642, A332669, A332672.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 03 2020