A335374 Numbers k such that the k-th composition in standard order (A066099) is not co-unimodal.

13, 25, 27, 29, 41, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 81, 82, 83, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 145, 153, 155, 157, 161, 162, 163, 165, 166, 167, 169, 173, 177

Offset

1,1

Comments

A sequence of integers is co-unimodal if it is the concatenation of a weakly decreasing and a weakly increasing sequence, implying that 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.

Links

Table of n, a(n) for n=1..57.

Example

The sequence together with the corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  41: (2,3,1)
  45: (2,1,2,1)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  81: (2,4,1)
  82: (2,3,2)
  83: (2,3,1,1)
  89: (2,1,3,1)

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[#]]&]

Crossrefs

This is the dual version of A335373.

The case that is not 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.

Co-unimodal compositions are A332578.

Numbers with non-co-unimodal unsorted prime signature are A332642.

Non-co-unimodal compositions are A332669.

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

Cf. A112798, A227038, A329398, A332281, A332286, A332287, A332638, A332639, A332643, A332670, A332873, A333146.

Sequence in context: A018991 A018948 A256475 * A335466 A186403 A032478

Adjacent sequences: A335371 A335372 A335373 * A335375 A335376 A335377

Keyword

nonn

Author

Gus Wiseman, Jun 03 2020

Status

approved