A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

0, 1, 2, 2, 3, 2, 1, 3, 4, 3, 4, 3, 1, 1, 2, 4, 5, 4, 3, 4, 2, 4, 2, 4, 1, 1, 3, 2, 2, 2, 3, 5, 6, 5, 4, 5, 6, 3, 3, 5, 2, 2, 6, 5, 2, 2, 3, 5, 1, 1, 1, 2, 1, 3, 1, 3, 2, 2, 4, 3, 3, 3, 4, 6, 7, 6, 5, 6, 4, 4, 4, 6, 3, 6, 5, 4, 3, 3, 4, 6, 2, 2, 2, 3, 4, 6, 4

Offset

0,3

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.

Links

Table of n, a(n) for n=0..86.

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

Example

The 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with maximal anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) is 3 + 2 + 1 + 1 = 7.

Mathematica

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Total[First/@Split[stc[n], UnsameQ]], {n, 0, 100}]

Crossrefs

For length instead of sum we have A333381.

Row-sums of A374515.

Other types of runs (instead of anti-):

- For identical runs we have A373953, row-sums of A374251.

- For weakly increasing runs we have A374630, row-sums of A374629.

- For strictly increasing runs we have A374684, row-sums of A374683.

- For weakly decreasing runs we have A374741, row-sums of A374740.

- For strictly decreasing runs we have A374758, row-sums of A374757.

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.

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

Six types of maximal runs:

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

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

Cf. A029931, A228351, A233564, A238343, A272919, A373949, A374517, A374518, A374519, A374638.

Sequence in context: A242266 A239617 A304737 * A369028 A092565 A021452

Adjacent sequences: A374513 A374514 A374515 * A374517 A374518 A374519

Keyword

nonn

Author

Gus Wiseman, Jul 31 2024

Status

approved