login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 27 19:29 EDT 2022. Contains 354898 sequences. (Running on oeis4.)