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A335373 Numbers k such that the k-th composition in standard order (A066099) is not unimodal.

%I A335373 #8 Jun 05 2020 09:56:42
%S A335373 22,38,44,45,46,54,70,76,77,78,86,88,89,90,91,92,93,94,102,108,109,
%T A335373 110,118,134,140,141,142,148,150,152,153,154,155,156,157,158,166,172,
%U A335373 173,174,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,198
%N A335373 Numbers k such that the k-th composition in standard order (A066099) is not unimodal.
%C A335373 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A335373 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 A335373 The sequence together with the corresponding compositions begins:
%e A335373   22: (2,1,2)
%e A335373   38: (3,1,2)
%e A335373   44: (2,1,3)
%e A335373   45: (2,1,2,1)
%e A335373   46: (2,1,1,2)
%e A335373   54: (1,2,1,2)
%e A335373   70: (4,1,2)
%e A335373   76: (3,1,3)
%e A335373   77: (3,1,2,1)
%e A335373   78: (3,1,1,2)
%e A335373   86: (2,2,1,2)
%e A335373   88: (2,1,4)
%e A335373   89: (2,1,3,1)
%e A335373   90: (2,1,2,2)
%e A335373   91: (2,1,2,1,1)
%e A335373   92: (2,1,1,3)
%e A335373   93: (2,1,1,2,1)
%e A335373   94: (2,1,1,1,2)
%t A335373 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A335373 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A335373 Select[Range[0,200],!unimodQ[stc[#]]&]
%Y A335373 The dual version (non-co-unimodal compositions) is A335374.
%Y A335373 The case that is not co-unimodal either is A335375.
%Y A335373 Unimodal compositions are A001523.
%Y A335373 Unimodal normal sequences are A007052.
%Y A335373 Unimodal permutations are A011782.
%Y A335373 Non-unimodal permutations are A059204.
%Y A335373 Non-unimodal compositions are A115981.
%Y A335373 Non-unimodal normal sequences are A328509.
%Y A335373 Numbers with non-unimodal unsorted prime signature are A332282.
%Y A335373 Partitions with non-unimodal 0-appended first differences are A332284.
%Y A335373 Non-unimodal permutations of the multiset of prime indices of n are A332671.
%Y A335373 Cf. A000120, A029931, A048793, A066099, A070939, A334299.
%Y A335373 Cf. A072704, A332281, A332286, A332287, A332639, A332642, A332669, A332672.
%K A335373 nonn
%O A335373 1,1
%A A335373 _Gus Wiseman_, Jun 03 2020