A337565 Irregular triangle read by rows where row k is the sequence of maximal anti-run lengths in the k-th composition in standard order.

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 3, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 3, 4, 2, 2, 2, 1, 1, 1, 2, 3, 3

Offset

0,6

Comments

An anti-run is a sequence with no adjacent equal parts.

A composition of n is a finite sequence of positive integers summing to n. 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..84.

Example

The first column below lists various selected n; the second column gives the corresponding composition; the third column gives the corresponding row of the triangle, i.e., the anti-run lengths.
    1:           (1) -> (1)
    3:         (1,1) -> (1,1)
    5:         (2,1) -> (2)
    7:       (1,1,1) -> (1,1,1)
   11:       (2,1,1) -> (2,1)
   13:       (1,2,1) -> (3)
   14:       (1,1,2) -> (1,2)
   15:     (1,1,1,1) -> (1,1,1,1)
   23:     (2,1,1,1) -> (2,1,1)
   27:     (1,2,1,1) -> (3,1)
   29:     (1,1,2,1) -> (1,3)
   30:     (1,1,1,2) -> (1,1,2)
   31:   (1,1,1,1,1) -> (1,1,1,1,1)
   43:     (2,2,1,1) -> (1,2,1)
   45:     (2,1,2,1) -> (4)
   46:     (2,1,1,2) -> (2,2)
   47:   (2,1,1,1,1) -> (2,1,1,1)
   55:   (1,2,1,1,1) -> (3,1,1)
   59:   (1,1,2,1,1) -> (1,3,1)
   61:   (1,1,1,2,1) -> (1,1,3)
   62:   (1,1,1,1,2) -> (1,1,1,2)
   63: (1,1,1,1,1,1) -> (1,1,1,1,1,1)
For example, the 119th composition is (1,1,2,1,1,1), with maximal anti-runs ((1),(1,2,1),(1),(1)), so row 119 is (1,3,1,1).

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length/@Split[stc[n], UnsameQ], {n, 0, 50}]

Crossrefs

A000120 gives row sums.

A333381 gives row lengths.

A333769 is the version for runs.

A003242 counts anti-run compositions.

A011782 counts compositions.

A106351 counts anti-run compositions by length.

A329744 is a triangle counting compositions by runs-resistance.

A333755 counts compositions by number of runs.

All of the following pertain to compositions in standard order (A066099):

- Sum is A070939.

- Adjacent equal pairs are counted by A124762.

- Runs are counted by A124767.

- Strict compositions are A233564.

- Constant compositions are A272919.

- Patterns are A333217.

- Heinz number is A333219.

- Anti-runs are counted by A333381.

- Anti-run compositions are A333489.

- Runs-resistance is A333628.

- Combinatory separations are A334030.

Cf. A106356, A113835, A114994, A124767, A181819, A228351, A238279, A318928, A333216, A333627, A333630.

Sequence in context: A184318 A030410 A085301 * A138385 A030614 A328615

Adjacent sequences: A337562 A337563 A337564 * A337566 A337567 A337568

Keyword

nonn,tabf

Author

Gus Wiseman, Sep 07 2020

Status

approved