A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

13, 25, 27, 29, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 82, 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, 153, 155, 157, 162, 165, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189

Offset: 1

Author

Gus Wiseman, Aug 12 2024

Keywords

• nonn

Comments

First differs from the non-dashed version in lacking 166, corresponding to the composition (2,3,1,2).

The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 1-32 or 1-21.

Examples

			The sequence together with corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  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)
  82: (2,3,2)
  89: (2,1,3,1)
  91: (2,1,2,1,1)
  93: (2,1,1,2,1)
		

Links

• Wikipedia, Permutation pattern.
• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

For leaders of identical runs we have A335485.

Positions of non-strictly decreasing rows in A374629 (sums A374630).

For identical leaders we have A374633, counted by A374631.

Matching 1-32 only gives A375137, reverse A375138, both counted by A374636.

Interchanging weak/strict gives A375139, counted by A375135.

Compositions of this type are counted by A375140, complement A188920.

The reverse version is A375296.

A003242 counts anti-runs, ranks A333489.

A011782 counts compositions.

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

A274174 counts contiguous compositions, ranks A374249.

A374637 counts compositions by sum of leaders of weakly increasing runs.

All of the following pertain to compositions in standard order:

- Length is A000120.

- Sum is A029837(n+1).

- Leader is A065120.

- Parts are listed by A066099, reverse A228351.

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

- Strict compositions are A233564.

- Constant compositions are A272919.

- Run-length transform is A333627, sum A070939.

- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.

- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.

Cf. A056823, A106356, A188919, A189076, A238343, A261982, A333213, A335480, A335482, A373948, A374746, A374768, A375123.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!Greater@@First/@Split[stc[#],LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,z_,y_,_}/;x<=y