A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

13, 22, 25, 27, 29, 38, 41, 44, 45, 46, 49, 50, 51, 53, 54, 55, 57, 59, 61, 70, 76, 77, 78, 81, 82, 83, 86, 88, 89, 90, 91, 92, 93, 94, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 134, 140, 141, 142

Offset: 1

Author

Gus Wiseman, Sep 18 2024

Keywords

• nonn

Comments

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.

Examples

			The terms and corresponding compositions begin:
  13: (1,2,1)
  22: (2,1,2)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  38: (3,1,2)
  41: (2,3,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  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)
		

Links

• Eric Weisstein's World of Mathematics, Unimodal Sequence.

Crossrefs

The version for run-lengths of compositions is A332833.

Compositions of this type are counted by A332834, complement maybe A329398.

A001523 counts unimodal compositions, ranks too dense.

A011782 counts compositions.

A114994 ranks weakly decreasing compositions, complement A335485.

A115981 counts non-unimodal compositions, ranked by A335373.

A225620 ranks weakly increasing compositions, complement A335486.

A238130, A238279, A333755 count compositions by number of runs.

A332835 counts compositions with weakly incr. or weakly decr. run-lengths.

All of the following pertain to compositions in standard order:

- Length is A000120.

- Sum is A029837(n+1).

- Parts are listed by A066099.

- Number of adjacent equal pairs is A124762, unequal A333382.

- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.

- Ranks of strict compositions are A233564.

- Ranks of constant compositions are A272919.

- Anti-runs are ranked by A333489, counted by A003242.

- Run-length transform is A333627, sum A070939.

Cf. A001511, A002051, A065120, A329744, A332745, A332836, A332870, A333218.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!LessEqual@@stc[#]&&!GreaterEqual@@stc[#]&]

Formula

Intersection of A335485 and A335486.