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

22, 38, 44, 45, 46, 54, 70, 76, 77, 78, 86, 88, 89, 90, 91, 92, 93, 94, 102, 108, 109, 110, 118, 134, 140, 141, 142, 148, 150, 152, 153, 154, 155, 156, 157, 158, 166, 172, 173, 174, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 198

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.

Links

Table of n, a(n) for n=1..56.

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.

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

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

Sequence in context: A084141 A259736 A082261 * A337460 A063252 A078540

Adjacent sequences: A335370 A335371 A335372 * A335374 A335375 A335376

Keyword

nonn

Author

Gus Wiseman, Jun 03 2020

Status

approved