A374519 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are identical.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 76, 77, 80, 81, 82, 84, 85

Offset: 1

Author

Gus Wiseman, Aug 01 2024

Keywords

• nonn

Comments

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) 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 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2), so 346 is in the sequence.
		

Links

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

Crossrefs

Positions of constant rows in A374515.

Compositions of this type are counted by A374517.

The complement is A374520.

For distinct instead of identical leaders we have A374638, counted by A374518.

Other types of runs (instead of anti-):

- For identical runs we have A272919, counted by A000005.

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

- For strictly increasing runs we have A374685, counted by A374686.

- For weakly decreasing runs we have A374744, counted by A374742.

- For strictly decreasing runs we have A374759, counted by A374760.

A065120 gives leaders of standard compositions.

A106356 counts compositions by number of maximal anti-runs.

A238279 counts compositions by number of maximal runs.

A238424 counts partitions whose first differences are an anti-run.

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.

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

- 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.

Six types of maximal runs:

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

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

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

Cf. A029931, A114994, A228351, A233564, A238343, A373949.

Programs

Mathematica

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