A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 15, 16, 17, 18, 21, 22, 31, 32, 33, 34, 36, 37, 42, 45, 63, 64, 65, 66, 68, 69, 73, 76, 85, 86, 90, 127, 128, 129, 130, 132, 133, 136, 137, 146, 148, 153, 170, 173, 181, 182

Offset

1,3

Comments

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

Links

Table of n, a(n) for n=1..50.

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

Example

The 18789th composition in standard order is (3,3,2,1,3,2,1), with strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so 18789 is in the sequence.
The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  21: (2,2,1)
  22: (2,1,2)
  31: (1,1,1,1,1)
  32: (6)
  33: (5,1)
  34: (4,2)
  36: (3,3)
  37: (3,2,1)

Mathematica

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

Crossrefs

For leaders of anti-runs we have A374519 (counted by A374517).

For leaders of weakly increasing runs we have A374633, counted by A374631.

The opposite version is A374685 (counted by A374686).

The weak version is A374744.

Compositions of this type are counted by A374760.

For distinct instead of identical runs we have A374767 (counted by A374761).

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.

- Run-length transform is A333627, sum A070939.

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

- Ranks of contiguous compositions are A374249, counted by A274174.

- Ranks of non-contiguous compositions are A374253, counted by A335548.

Six types of runs:

- Count: A124766, A124765, A124768, A124769, A333381, A124767.

- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.

- Rank: A375123, A375124, A375125, A375126, A375127, A373948.

Cf. A000961, A065120, A106356, A188920, A189076, A238343, A272919, A333213, A374698, A374706, A374758, A375128.

Sequence in context: A299767 A129268 A271317 * A330220 A039169 A156068

Adjacent sequences: A374756 A374757 A374758 * A374760 A374761 A374762

Keyword

nonn

Author

Gus Wiseman, Jul 29 2024

Status

approved