A124767 Number of level runs for compositions in standard order.

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 1, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 3, 4, 3, 2, 2, 3, 3, 3, 3, 3, 4, 3, 2, 3, 2, 3, 2, 3, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 4, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 2, 2, 3, 3, 3, 2, 4, 4, 3, 3

Offset

0,6

Comments

The standard order of compositions is given by A066099.

For n > 0, a(n) is one more than the number of adjacent unequal terms in the n-th composition in standard order. Also the number of runs in the same composition. - Gus Wiseman, Apr 08 2020

Links

Table of n, a(n) for n=0..104.

Formula

a(0) = 0, a(n) = 1 + Sum_{1<=i=1<k, b(i)!=b(i+1)} 1 for n > 0.

For n > 0, a(n) = A333382(n) + 1. - Gus Wiseman, Apr 08 2020

Example

Composition number 11 is 2,1,1; the level runs are 2; 1,1; so a(11) = 2.
The table starts:
  0
  1
  1 1
  1 2 2 1
  1 2 1 2 2 3 2 1
  1 2 2 2 2 2 3 2 2 3 2 3 2 3 2 1
  1 2 2 2 1 3 3 2 2 3 1 2 3 4 3 2 2 3 3 3 3 3 4 3 2 3 2 3 2 3 2 1
The 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1) with runs ((3),(2),(1),(2,2),(1),(2),(5),(1,1,1)), so a(1234567) = 8. - Gus Wiseman, Apr 08 2020

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Split[stc[n]]], {n, 0, 100}] (* Gus Wiseman, Apr 17 2020 *)

Crossrefs

Row-lengths are A011782.

Compositions counted by number of runs are A238279 or A333755.

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

- Length is A000120.

- Sum is A070939.

- Weakly decreasing compositions are A114994.

- Adjacent equal pairs are counted by A124762.

- Weakly decreasing runs are counted by A124765.

- Weakly increasing runs are counted by A124766.

- Equal runs are counted by A124767 (this sequence).

- Weakly increasing compositions are A225620.

- Strict compositions A233564.

- Constant compositions are A272919.

- Anti-runs are counted by A333381.

- Adjacent unequal pairs are counted by A333382.

- Anti-run compositions are A333489.

- Runs-resistance is A333628.

- Run-lengths are A333769 (triangle).

Cf. A029931, A048793, A066099, A106356, A228351, A318928, A329744, A333217, A333219, A333627.

Sequence in context: A116514 A334028 A351014 * A319443 A130633 A266499

Adjacent sequences: A124764 A124765 A124766 * A124768 A124769 A124770

Keyword

easy,nonn,tabf

Author

Franklin T. Adams-Watters, Nov 06 2006

Status

approved