A333769 Irregular triangle read by rows where row k is the sequence of run-lengths of the k-th composition in standard order.

1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1

Offset

0,3

Comments

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..86.

Example

The standard compositions and their run-lengths:
   0:        () -> ()
   1:       (1) -> (1)
   2:       (2) -> (1)
   3:     (1,1) -> (2)
   4:       (3) -> (1)
   5:     (2,1) -> (1,1)
   6:     (1,2) -> (1,1)
   7:   (1,1,1) -> (3)
   8:       (4) -> (1)
   9:     (3,1) -> (1,1)
  10:     (2,2) -> (2)
  11:   (2,1,1) -> (1,2)
  12:     (1,3) -> (1,1)
  13:   (1,2,1) -> (1,1,1)
  14:   (1,1,2) -> (2,1)
  15: (1,1,1,1) -> (4)
  16:       (5) -> (1)
  17:     (4,1) -> (1,1)
  18:     (3,2) -> (1,1)
  19:   (3,1,1) -> (1,2)
For example, the 119th composition is (1,1,2,1,1,1), so row 119 is (2,1,3).

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length/@Split[stc[n]], {n, 0, 30}]

Crossrefs

Row sums are A000120.

Row lengths are A124767.

Row k is the A333627(k)-th standard composition.

A triangle counting compositions by runs-resistance is A329744.

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

- Partial sums from the right are A048793.

- Sum is A070939.

- Adjacent equal pairs are counted by A124762.

- Strict compositions are A233564.

- Partial sums from the left are A272020.

- Constant compositions are A272919.

- Normal compositions are A333217.

- Heinz number is A333219.

- Runs-resistance is A333628.

- First appearances of run-resistances are A333629.

- Combinatory separations are A334030.

Cf. A029931, A098504, A114994, A181819, A182850, A225620, A228351, A238279, A242882, A318928, A329747, A333489, A333630.

Sequence in context: A323719 A220122 A101446 * A259396 A328672 A368465

Adjacent sequences: A333766 A333767 A333768 * A333770 A333771 A333772

Keyword

nonn,tabf

Author

Gus Wiseman, Apr 10 2020

Status

approved