A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

26, 50, 53, 58, 90, 98, 100, 101, 106, 107, 114, 117, 122, 154, 164, 178, 181, 186, 194, 196, 197, 201, 202, 203, 210, 212, 213, 214, 215, 218, 226, 228, 229, 234, 235, 242, 245, 250, 282, 306, 309, 314, 324, 329, 346, 354, 356, 357, 362, 363, 370, 373, 378

Offset: 1

Author

Gus Wiseman, Aug 12 2024

Keywords

• nonn

Comments

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

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 together with corresponding compositions begin:
   26: (1,2,2)
   50: (1,3,2)
   53: (1,2,2,1)
   58: (1,1,2,2)
   90: (2,1,2,2)
   98: (1,4,2)
  100: (1,3,3)
  101: (1,3,2,1)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  114: (1,1,3,2)
  117: (1,1,2,2,1)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)
  178: (2,1,3,2)
  181: (2,1,2,2,1)
  186: (2,1,1,2,2)
		

Links

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

Crossrefs

For leaders of identical runs we have A335485.

Ranked by positions of non-weakly decreasing rows in A374683.

For identical leaders we have A374685, counted by A374686.

The complement is counted by A374697.

For distinct leaders we have A374698, counted by A374687.

Compositions of this type are counted by A375135.

Weakly increasing leaders: A375137, counts A374636, complement A189076.

Interchanging weak/strict: A375295, counted by A375140, complement A188920.

A003242 counts anti-run compositions, ranks A333489.

A374700 counts compositions by sum of leaders of strictly 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.

- Strict compositions are A233564.

- Run-length transform is A333627, sum A070939.

- Run-compression transform is A373948, sum A373953, excess A373954.

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

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

- Run-rank: A375123, A375124, A375125, A375126, A375127.

Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.

Programs

Mathematica

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