- FindStat

Your data matches 66 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000834
(load all 46 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [2,1] => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1,2] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1

Description

The number of right outer peaks of a permutation. A right outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$n$ if

$w_n > w_{n-1}$ . In other words, it is a peak in the word

$[w_1,..., w_n,0]$ .

Matching statistic: St001280
(load all 4 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [1]
 => 0
[1,2] => [2,1] => [1,1] => [1,1]
 => 0
[2,1] => [1,2] => [2] => [2]
 => 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,1,1]
 => 0
[1,3,2] => [2,3,1] => [2,1] => [2,1]
 => 1
[2,1,3] => [3,1,2] => [1,2] => [2,1]
 => 1
[2,3,1] => [1,3,2] => [2,1] => [2,1]
 => 1
[3,1,2] => [2,1,3] => [1,2] => [2,1]
 => 1
[3,2,1] => [1,2,3] => [3] => [3]
 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,1,1,1]
 => 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [2,1,1]
 => 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [2,1,1]
 => 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [2,1,1]
 => 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [2,1,1]
 => 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [3,1]
 => 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [2,1,1]
 => 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [2,2]
 => 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [2,1,1]
 => 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [2,1,1]
 => 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [2,1,1]
 => 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [3,1]
 => 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [2,1,1]
 => 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [2,2]
 => 2
[3,2,1,4] => [4,1,2,3] => [1,3] => [3,1]
 => 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [2,2]
 => 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [2,1,1]
 => 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [3,1]
 => 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [2,1,1]
 => 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [2,2]
 => 2
[4,2,1,3] => [3,1,2,4] => [1,3] => [3,1]
 => 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [2,2]
 => 2
[4,3,1,2] => [2,1,3,4] => [1,3] => [3,1]
 => 1
[4,3,2,1] => [1,2,3,4] => [4] => [4]
 => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,1,1,1,1]
 => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [2,1,1,1]
 => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [2,1,1,1]
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [2,1,1,1]
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [2,1,1,1]
 => 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [3,1,1]
 => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [2,1,1,1]
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [2,2,1]
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [2,1,1,1]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [2,1,1,1]
 => 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [2,1,1,1]
 => 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [3,1,1]
 => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [2,1,1,1]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [2,2,1]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [3,1,1]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [2,2,1]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [2,1,1,1]
 => 1
[8,3,4,5,2,6,7,1] => [1,7,6,2,5,4,3,8] => ? => ?
 => ? = 3
[7,4,5,6,2,3,8,1] => [1,8,3,2,6,5,4,7] => ? => ?
 => ? = 3
[3,4,5,1,2,6,7,8] => [8,7,6,2,1,5,4,3] => ? => ?
 => ? = 1
[2,3,7,8,6,5,4,1] => [1,4,5,6,8,7,3,2] => ? => ?
 => ? = 1
[3,2,5,6,4,7,8,1] => [1,8,7,4,6,5,2,3] => ? => ?
 => ? = 3
[1,4,6,7,8,5,3,2] => [2,3,5,8,7,6,4,1] => ? => ?
 => ? = 1
[1,3,6,7,8,5,4,2] => [2,4,5,8,7,6,3,1] => ? => ?
 => ? = 1
[2,1,4,5,3,6,8,7] => [7,8,6,3,5,4,1,2] => ? => ?
 => ? = 3
[2,1,3,8,5,6,7,4] => [4,7,6,5,8,3,1,2] => ? => ?
 => ? = 3
[2,7,3,5,4,6,1,8] => ? => ? => ?
 => ? = 3
[6,2,4,3,5,1,8,7] => [7,8,1,5,3,4,2,6] => ? => ?
 => ? = 4
[8,2,5,4,7,6,3,1] => [1,3,6,7,4,5,2,8] => ? => ?
 => ? = 3
[8,3,4,2,6,5,7,1] => [1,7,5,6,2,4,3,8] => ? => ?
 => ? = 4
[6,4,3,2,5,1,8,7] => [7,8,1,5,2,3,4,6] => ? => ?
 => ? = 3
[2,4,5,8,1,3,6,7] => [7,6,3,1,8,5,4,2] => ? => ?
 => ? = 1
[2,5,6,1,3,4,7,8] => [8,7,4,3,1,6,5,2] => ? => ?
 => ? = 1
[2,6,1,3,7,8,4,5] => [5,4,8,7,3,1,6,2] => ? => ?
 => ? = 2
[2,1,3,6,4,5,8,7] => [7,8,5,4,6,3,1,2] => ? => ?
 => ? = 3
[3,5,8,1,2,4,6,7] => [7,6,4,2,1,8,5,3] => ? => ?
 => ? = 1
[1,3,2,4,6,8,5,7] => [7,5,8,6,4,2,3,1] => ? => ?
 => ? = 2
[1,3,4,6,7,2,5,8] => [8,5,2,7,6,4,3,1] => ? => ?
 => ? = 1
[3,1,2,7,4,5,8,6] => [6,8,5,4,7,2,1,3] => ? => ?
 => ? = 3
[4,1,5,6,2,3,8,7] => [7,8,3,2,6,5,1,4] => ? => ?
 => ? = 3
[1,4,5,7,2,8,3,6] => [6,3,8,2,7,5,4,1] => ? => ?
 => ? = 2
[1,2,4,6,7,3,5,8] => [8,5,3,7,6,4,2,1] => ? => ?
 => ? = 1
[1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ?
 => ? = 1
[1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ?
 => ? = 1
[5,6,1,2,3,7,4,8] => [8,4,7,3,2,1,6,5] => ? => ?
 => ? = 2
[1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ?
 => ? = 1
[1,2,5,6,8,3,4,7] => [7,4,3,8,6,5,2,1] => ? => ?
 => ? = 1
[1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ?
 => ? = 1
[1,5,2,3,6,7,4,8] => [8,4,7,6,3,2,5,1] => ? => ?
 => ? = 2
[1,2,6,7,3,8,4,5] => [5,4,8,3,7,6,2,1] => ? => ?
 => ? = 2
[3,7,1,8,9,10,2,4,5,6] => [6,5,4,2,10,9,8,1,7,3] => ? => ?
 => ? = 2
[5,6,7,9,1,2,3,10,4,8] => [8,4,10,3,2,1,9,7,6,5] => ? => ?
 => ? = 2
[4,7,2,6,1,3,5,8] => [8,5,3,1,6,2,7,4] => ? => ?
 => ? = 2
[4,8,2,5,1,3,6,7] => [7,6,3,1,5,2,8,4] => ? => ?
 => ? = 2
[4,8,3,5,1,2,6,7] => [7,6,2,1,5,3,8,4] => ? => ?
 => ? = 2
[7,1,2,5,3,4,8,6] => [6,8,4,3,5,2,1,7] => ? => ?
 => ? = 3
[7,1,4,5,2,3,8,6] => [6,8,3,2,5,4,1,7] => ? => ?
 => ? = 3
[4,2,7,1,6,3,8,5] => ? => ? => ?
 => ? = 4
[3,6,5,8,1,2,4,7] => [7,4,2,1,8,5,6,3] => ? => ?
 => ? = 2
[3,7,1,8,4,6,2,5] => [5,2,6,4,8,1,7,3] => ? => ?
 => ? = 3
[2,6,1,8,5,7,3,4] => [4,3,7,5,8,1,6,2] => ? => ?
 => ? = 3
[5,4,8,3,6,1,7,2] => [2,7,1,6,3,8,4,5] => ? => ?
 => ? = 4
[4,5,1,7,3,6,2,8] => ? => ? => ?
 => ? = 3
[4,7,2,5,1,8,3,6] => ? => ? => ?
 => ? = 3
[4,2,6,3,8,1,7,5] => [5,7,1,8,3,6,2,4] => ? => ?
 => ? = 4
[8,4,6,3,2,1,7,5] => [5,7,1,2,3,6,4,8] => ? => ?
 => ? = 3
[2,3,8,6,4,1,5,7] => [7,5,1,4,6,8,3,2] => ? => ?
 => ? = 1

Description

The number of parts of an integer partition that are at least two.

Matching statistic: St000291
(load all 3 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 87%●distinct values known / distinct values provided: 83%

Values

[1] => [1] => [1] => 1 => 0
[1,2] => [2,1] => [1,1] => 11 => 0
[2,1] => [1,2] => [2] => 10 => 1
[1,2,3] => [3,2,1] => [1,1,1] => 111 => 0
[1,3,2] => [2,3,1] => [2,1] => 101 => 1
[2,1,3] => [3,1,2] => [1,2] => 110 => 1
[2,3,1] => [1,3,2] => [2,1] => 101 => 1
[3,1,2] => [2,1,3] => [1,2] => 110 => 1
[3,2,1] => [1,2,3] => [3] => 100 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1111 => 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => 1011 => 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1101 => 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => 1011 => 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => 1101 => 1
[1,4,3,2] => [2,3,4,1] => [3,1] => 1001 => 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => 1110 => 1
[2,1,4,3] => [3,4,1,2] => [2,2] => 1010 => 2
[2,3,1,4] => [4,1,3,2] => [1,2,1] => 1101 => 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => 1011 => 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1101 => 1
[2,4,3,1] => [1,3,4,2] => [3,1] => 1001 => 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => 1110 => 1
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 2
[3,2,1,4] => [4,1,2,3] => [1,3] => 1100 => 1
[3,2,4,1] => [1,4,2,3] => [2,2] => 1010 => 2
[3,4,1,2] => [2,1,4,3] => [1,2,1] => 1101 => 1
[3,4,2,1] => [1,2,4,3] => [3,1] => 1001 => 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => 1110 => 1
[4,1,3,2] => [2,3,1,4] => [2,2] => 1010 => 2
[4,2,1,3] => [3,1,2,4] => [1,3] => 1100 => 1
[4,2,3,1] => [1,3,2,4] => [2,2] => 1010 => 2
[4,3,1,2] => [2,1,3,4] => [1,3] => 1100 => 1
[4,3,2,1] => [1,2,3,4] => [4] => 1000 => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 11111 => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => 10111 => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => 11011 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => 10111 => 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => 11011 => 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => 10011 => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => 11101 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => 10101 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => 11011 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => 10111 => 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => 11011 => 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => 10011 => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => 11101 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => 10101 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 11001 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => 10101 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => 11011 => 1
[8,3,4,5,2,6,7,1] => [1,7,6,2,5,4,3,8] => ? => ? => ? = 3
[7,4,5,6,2,3,8,1] => [1,8,3,2,6,5,4,7] => ? => ? => ? = 3
[3,4,5,1,2,6,7,8] => [8,7,6,2,1,5,4,3] => ? => ? => ? = 1
[2,3,7,8,6,5,4,1] => [1,4,5,6,8,7,3,2] => ? => ? => ? = 1
[3,2,5,6,4,7,8,1] => [1,8,7,4,6,5,2,3] => ? => ? => ? = 3
[1,4,6,7,8,5,3,2] => [2,3,5,8,7,6,4,1] => ? => ? => ? = 1
[1,3,6,7,8,5,4,2] => [2,4,5,8,7,6,3,1] => ? => ? => ? = 1
[2,1,4,5,3,6,8,7] => [7,8,6,3,5,4,1,2] => ? => ? => ? = 3
[2,1,3,8,5,6,7,4] => [4,7,6,5,8,3,1,2] => ? => ? => ? = 3
[2,7,3,5,4,6,1,8] => ? => ? => ? => ? = 3
[6,2,4,3,5,1,8,7] => [7,8,1,5,3,4,2,6] => ? => ? => ? = 4
[8,2,5,4,7,6,3,1] => [1,3,6,7,4,5,2,8] => ? => ? => ? = 3
[8,3,4,2,6,5,7,1] => [1,7,5,6,2,4,3,8] => ? => ? => ? = 4
[6,4,3,2,5,1,8,7] => [7,8,1,5,2,3,4,6] => ? => ? => ? = 3
[2,4,5,8,1,3,6,7] => [7,6,3,1,8,5,4,2] => ? => ? => ? = 1
[2,5,6,1,3,4,7,8] => [8,7,4,3,1,6,5,2] => ? => ? => ? = 1
[2,6,1,3,7,8,4,5] => [5,4,8,7,3,1,6,2] => ? => ? => ? = 2
[2,1,3,6,4,5,8,7] => [7,8,5,4,6,3,1,2] => ? => ? => ? = 3
[3,5,8,1,2,4,6,7] => [7,6,4,2,1,8,5,3] => ? => ? => ? = 1
[1,3,2,4,6,8,5,7] => [7,5,8,6,4,2,3,1] => ? => ? => ? = 2
[1,3,4,6,7,2,5,8] => [8,5,2,7,6,4,3,1] => ? => ? => ? = 1
[3,1,2,7,4,5,8,6] => [6,8,5,4,7,2,1,3] => ? => ? => ? = 3
[4,1,5,6,2,3,8,7] => [7,8,3,2,6,5,1,4] => ? => ? => ? = 3
[1,4,5,7,2,8,3,6] => [6,3,8,2,7,5,4,1] => ? => ? => ? = 2
[1,2,4,6,7,3,5,8] => [8,5,3,7,6,4,2,1] => ? => ? => ? = 1
[1,2,4,7,8,3,5,6] => [6,5,3,8,7,4,2,1] => ? => ? => ? = 1
[1,4,7,2,3,5,6,8] => [8,6,5,3,2,7,4,1] => ? => ? => ? = 1
[5,6,1,2,3,7,4,8] => [8,4,7,3,2,1,6,5] => ? => ? => ? = 2
[1,5,6,8,2,3,4,7] => [7,4,3,2,8,6,5,1] => ? => ? => ? = 1
[1,2,5,6,8,3,4,7] => [7,4,3,8,6,5,2,1] => ? => ? => ? = 1
[1,5,6,2,3,4,7,8] => [8,7,4,3,2,6,5,1] => ? => ? => ? = 1
[1,5,2,3,6,7,4,8] => [8,4,7,6,3,2,5,1] => ? => ? => ? = 2
[1,2,6,7,3,8,4,5] => [5,4,8,3,7,6,2,1] => ? => ? => ? = 2
[3,7,1,8,9,10,2,4,5,6] => [6,5,4,2,10,9,8,1,7,3] => ? => ? => ? = 2
[5,7,8,9,1,10,2,3,4,6] => [6,4,3,2,10,1,9,8,7,5] => [1,1,1,2,2,1,1,1] => 1111010111 => ? = 2
[3,5,1,7,2,9,4,10,6,8] => [8,6,10,4,9,2,7,1,5,3] => [1,2,2,2,2,1] => 1101010101 => ? = 4
[5,6,7,9,1,2,3,10,4,8] => [8,4,10,3,2,1,9,7,6,5] => ? => ? => ? = 2
[4,7,2,6,1,3,5,8] => [8,5,3,1,6,2,7,4] => ? => ? => ? = 2
[4,8,2,5,1,3,6,7] => [7,6,3,1,5,2,8,4] => ? => ? => ? = 2
[4,8,3,5,1,2,6,7] => [7,6,2,1,5,3,8,4] => ? => ? => ? = 2
[7,1,2,5,3,4,8,6] => [6,8,4,3,5,2,1,7] => ? => ? => ? = 3
[7,1,4,5,2,3,8,6] => [6,8,3,2,5,4,1,7] => ? => ? => ? = 3
[6,7,8,9,10,5,4,3,2,1] => [1,2,3,4,5,10,9,8,7,6] => [6,1,1,1,1] => 1000001111 => ? = 1
[4,2,7,1,6,3,8,5] => ? => ? => ? => ? = 4
[3,6,5,8,1,2,4,7] => [7,4,2,1,8,5,6,3] => ? => ? => ? = 2
[2,3,4,5,6,7,8,9,10,1] => [1,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[2,3,4,5,6,7,8,9,1,10] => [10,1,9,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 1
[2,3,4,5,6,7,8,10,1,9] => [9,1,10,8,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 1
[2,3,4,5,6,7,9,10,1,8] => [8,1,10,9,7,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 1
[2,3,4,5,6,8,9,10,1,7] => [7,1,10,9,8,6,5,4,3,2] => [1,2,1,1,1,1,1,1,1] => 1101111111 => ? = 1

Description

The number of descents of a binary word.

Matching statistic: St000390
(load all 13 compositions to match this statistic)

Mapping

Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%

Values

[1] =>  => ? = 0
[1,2] => 0 => 0
[2,1] => 1 => 1
[1,2,3] => 00 => 0
[1,3,2] => 01 => 1
[2,1,3] => 10 => 1
[2,3,1] => 01 => 1
[3,1,2] => 10 => 1
[3,2,1] => 11 => 1
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 1
[1,3,2,4] => 010 => 1
[1,3,4,2] => 001 => 1
[1,4,2,3] => 010 => 1
[1,4,3,2] => 011 => 1
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 2
[2,3,1,4] => 010 => 1
[2,3,4,1] => 001 => 1
[2,4,1,3] => 010 => 1
[2,4,3,1] => 011 => 1
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 2
[3,2,1,4] => 110 => 1
[3,2,4,1] => 101 => 2
[3,4,1,2] => 010 => 1
[3,4,2,1] => 011 => 1
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 2
[4,2,1,3] => 110 => 1
[4,2,3,1] => 101 => 2
[4,3,1,2] => 110 => 1
[4,3,2,1] => 111 => 1
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => 0011 => 1
[1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => 0101 => 2
[1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => 0011 => 1
[1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => 0101 => 2
[1,4,3,2,5] => 0110 => 1
[1,4,3,5,2] => 0101 => 2
[1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => 0011 => 1
[8,6,5,7,3,4,2,1] => ? => ? = 3
[8,5,4,6,3,7,2,1] => ? => ? = 3
[7,6,4,5,3,8,2,1] => ? => ? = 3
[7,3,4,5,2,6,8,1] => ? => ? = 3
[6,7,3,4,5,8,1,2] => ? => ? = 2
[6,7,8,5,3,1,2,4] => ? => ? = 1
[5,6,2,3,4,7,1,8] => ? => ? = 2
[5,6,7,4,3,1,2,8] => ? => ? = 1
[5,6,7,4,2,1,3,8] => ? => ? = 1
[6,7,4,5,1,2,3,8] => ? => ? = 2
[5,6,4,7,1,2,3,8] => ? => ? = 2
[5,6,7,3,2,1,4,8] => ? => ? = 1
[4,5,3,6,1,2,7,8] => ? => ? = 2
[5,6,2,3,1,4,7,8] => ? => ? = 2
[5,6,1,2,3,4,7,8] => ? => ? = 1
[5,4,8,7,6,3,2,1] => ? => ? = 2
[3,6,7,8,5,4,2,1] => ? => ? = 1
[4,3,7,6,8,5,2,1] => ? => ? = 3
[3,4,7,8,6,5,2,1] => ? => ? = 1
[3,4,6,8,7,5,2,1] => ? => ? = 1
[5,4,6,3,8,7,2,1] => ? => ? = 3
[5,4,3,7,6,8,2,1] => ? => ? = 3
[2,5,7,8,6,4,3,1] => ? => ? = 1
[2,5,6,8,7,4,3,1] => ? => ? = 1
[3,2,7,6,8,5,4,1] => ? => ? = 3
[2,3,7,8,6,5,4,1] => ? => ? = 1
[2,3,6,8,7,5,4,1] => ? => ? = 1
[5,4,7,6,3,2,8,1] => ? => ? = 3
[1,5,7,8,6,4,3,2] => ? => ? = 1
[1,5,6,8,7,4,3,2] => ? => ? = 1
[1,3,4,7,8,6,5,2] => ? => ? = 1
[1,3,4,6,8,7,5,2] => ? => ? = 1
[2,1,4,6,5,7,8,3] => ? => ? = 3
[1,2,6,8,7,5,4,3] => ? => ? = 1
[1,2,5,7,8,6,4,3] => ? => ? = 1
[2,1,3,6,5,7,8,4] => ? => ? = 3
[3,2,4,5,1,7,8,6] => ? => ? = 3
[2,1,4,5,3,7,8,6] => ? => ? = 3
[3,2,1,5,4,8,7,6] => ? => ? = 3
[3,2,4,5,1,6,8,7] => ? => ? = 3
[3,7,6,5,4,2,1,8] => ? => ? = 1
[2,4,5,6,3,7,1,8] => ? => ? = 2
[2,4,3,5,6,7,1,8] => ? => ? = 2
[1,2,4,6,5,7,3,8] => ? => ? = 2
[3,5,6,4,2,1,7,8] => ? => ? = 1
[1,3,6,5,4,2,7,8] => ? => ? = 1
[1,2,4,5,3,6,7,8] => ? => ? = 1
[1,3,2,7,5,6,4,8] => ? => ? = 3
[1,4,3,7,5,6,2,8] => ? => ? = 3

Description

The number of runs of ones in a binary word.

Matching statistic: St000658

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 83%

Values

[1] => [1] => [1,0]
 => [1,0]
 => ? = 0
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,6,5,7,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[7,6,5,8,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,7,4,5,3,6,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,6,4,5,3,7,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,5,4,6,3,7,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[7,6,4,5,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[7,5,4,6,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[6,5,4,7,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,5,4,6,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,6,5,4,7,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,6,5,4,8,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,3,4,2,5,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,5,3,4,2,6,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,4,3,5,2,6,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,6,5,4,3,2,7,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,6,5,3,4,2,7,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,6,4,3,5,2,7,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,5,4,3,6,2,7,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[7,6,5,3,4,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,6,4,3,5,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,5,4,3,6,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[6,5,4,3,8,7,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[5,4,3,7,6,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[5,4,3,2,8,7,6,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[5,4,3,6,2,8,7,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[4,3,2,6,5,8,7,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[5,4,3,2,7,6,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[3,2,1,7,6,8,5,4] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[3,2,1,6,5,8,7,4] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3

Description

The number of rises of length 2 of a Dyck path. This is also the number of

$(1,1)$ steps of the associated Łukasiewicz path, see [1]. A related statistic is the number of double rises in a Dyck path, [[St000024]].

Matching statistic: St000659
(load all 6 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 83%

Values

[1] => [1] => [1,0]
 => [1,0]
 => ? = 0
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,6,5,7,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[7,6,5,8,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,7,4,5,3,6,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,6,4,5,3,7,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,5,4,6,3,7,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[7,6,4,5,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[7,5,4,6,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[6,5,4,7,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,5,4,6,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,6,5,4,7,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,6,5,4,8,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,6,3,4,2,5,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,7,5,3,4,2,6,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,4,3,5,2,6,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,6,5,4,3,2,7,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[8,6,5,3,4,2,7,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,6,4,3,5,2,7,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,5,4,3,6,2,7,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,6,5,4,3,2,8,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => ? = 2
[7,6,5,3,4,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,6,4,3,5,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,5,4,3,6,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[6,5,4,3,7,2,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1
[6,5,4,3,8,7,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[5,4,3,7,6,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[5,4,3,2,8,7,6,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[5,4,3,6,2,8,7,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[4,3,2,6,5,8,7,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[5,4,3,2,7,6,8,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[3,2,1,7,6,8,5,4] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[3,2,1,6,5,8,7,4] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 3

Description

The number of rises of length at least 2 of a Dyck path.

Matching statistic: St000035
(load all 94 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 83%

Values

[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [3,2,1] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[2,1,3,5,4,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => ? = 2
[2,1,3,5,4,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,3,6,4,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => ? = 2
[2,1,3,6,4,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,3,6,5,4,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => ? = 2
[2,1,3,6,5,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,3,7,4,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => ? = 2
[2,1,3,7,4,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,3,7,5,4,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => ? = 2
[2,1,3,7,5,6,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,3,7,6,4,5] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => ? = 2
[2,1,3,7,6,5,4] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,4,1] => ? = 2
[2,1,4,3,5,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => ? = 2
[2,1,4,3,5,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => ? = 3
[2,1,4,3,6,5,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => ? = 3
[2,1,4,3,6,7,5] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => ? = 3
[2,1,4,3,7,5,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => ? = 3
[2,1,4,3,7,6,5] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => ? = 3
[2,1,4,5,3,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => ? = 2
[2,1,4,5,3,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,4,6,3,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => ? = 2
[2,1,4,6,3,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,4,6,5,3,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => ? = 2
[2,1,4,6,5,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,4,7,3,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => ? = 2
[2,1,4,7,3,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,4,7,5,3,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => ? = 2
[2,1,4,7,5,6,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,4,7,6,3,5] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => ? = 2
[2,1,4,7,6,5,3] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [7,6,5,2,3,4,1] => ? = 2
[2,1,5,3,4,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => ? = 2
[2,1,5,3,4,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => ? = 3
[2,1,5,3,6,4,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => ? = 3
[2,1,5,3,6,7,4] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => ? = 3
[2,1,5,3,7,4,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => ? = 3
[2,1,5,3,7,6,4] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => ? = 3
[2,1,5,4,3,6,7] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => ? = 2
[2,1,5,4,3,7,6] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [7,5,6,4,2,3,1] => ? = 3
[2,1,5,4,6,3,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => ? = 3
[2,1,5,4,6,7,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [7,4,5,6,2,3,1] => ? = 3
[2,1,5,4,7,3,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => ? = 3
[2,1,5,4,7,6,3] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [7,6,4,5,2,3,1] => ? = 3
[2,1,5,6,3,4,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => ? = 2
[2,1,5,6,3,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,5,6,4,3,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => ? = 2
[2,1,5,6,4,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,5,7,3,4,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [5,6,7,2,3,4,1] => ? = 2
[2,1,5,7,3,6,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3
[2,1,5,7,4,3,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [6,7,5,2,3,4,1] => ? = 2
[2,1,5,7,4,6,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [7,5,6,2,3,4,1] => ? = 3

Description

The number of left outer peaks of a permutation. A left outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ . In other words, it is a peak in the word

$[0,w_1,..., w_n]$ . This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

Matching statistic: St000340

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 67%

Values

[1] => [1] => [1,0]
 => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[7,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 1
[8,6,7,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 2
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 2
[6,7,8,5,4,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 1
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 1
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[7,6,8,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,5,6,4,7,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[7,5,6,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[4,5,6,7,8,3,2,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 1
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[8,6,5,7,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[7,6,5,8,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[8,7,4,5,3,6,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[8,6,4,5,3,7,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[8,5,4,6,3,7,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[7,6,4,5,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[7,5,4,6,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[6,5,4,7,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 3
[3,4,5,6,7,8,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,6,7,5,4,2,3,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[7,6,8,5,4,2,3,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
[8,6,7,4,5,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[7,6,8,4,5,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[8,7,5,4,6,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
[8,6,5,4,7,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
[8,5,6,4,7,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[7,6,5,4,8,2,3,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
[7,5,6,4,8,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[6,5,7,4,8,2,3,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,6,7,5,3,2,4,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[7,6,8,5,3,2,4,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[8,5,6,7,2,3,4,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3
[6,5,7,8,2,3,4,1] => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3
[8,7,6,4,3,2,5,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[8,6,7,4,3,2,5,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[7,6,8,4,3,2,5,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3
[8,7,6,3,4,2,5,1] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
[8,6,7,3,4,2,5,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[7,6,8,3,4,2,5,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

Matching statistic: St000374

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 83%

Values

[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[2,1,3,5,4,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,3,5,4,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,6,4,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,3,6,4,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,6,5,4,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,3,6,5,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,7,4,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,3,7,4,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,7,5,4,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,3,7,5,6,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,7,6,4,5] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,3,7,6,5,4] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,1,3,5,6,7,4] => ? = 2
[2,1,4,3,5,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 2
[2,1,4,3,5,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,4,3,6,5,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,4,3,6,7,5] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,4,3,7,5,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,4,3,7,6,5] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => ? = 3
[2,1,4,5,3,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,4,5,3,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,6,3,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,4,6,3,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,6,5,3,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,4,6,5,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,7,3,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,4,7,3,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,7,5,3,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,4,7,5,6,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,7,6,3,5] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,4,7,6,5,3] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,1,3,5,6,7,4] => ? = 2
[2,1,5,3,4,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 2
[2,1,5,3,4,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,5,3,6,4,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,5,3,6,7,4] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,5,3,7,4,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,5,3,7,6,4] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => ? = 3
[2,1,5,4,3,6,7] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,4,5,3,6,7] => ? = 2
[2,1,5,4,3,7,6] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3,7,6] => ? = 3
[2,1,5,4,6,3,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,5,4,6,7,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,5,4,7,3,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,5,4,7,6,3] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => ? = 3
[2,1,5,6,3,4,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,5,6,3,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,5,6,4,3,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,5,6,4,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,5,7,3,4,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,5,7,3,6,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,5,7,4,3,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,5,7,4,6,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000703
(load all 4 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 83%

Values

[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [1,2] => 0
[2,1] => [1,1] => [1,0,1,0]
 => [2,1] => 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 2
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 1
[2,1,3,5,4,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,3,5,4,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,6,4,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,3,6,4,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,6,5,4,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,3,6,5,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,7,4,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,3,7,4,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,7,5,4,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,3,7,5,6,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,3,7,6,4,5] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,3,7,6,5,4] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,1,3,5,6,7,4] => ? = 2
[2,1,4,3,5,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 2
[2,1,4,3,5,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,4,3,6,5,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,4,3,6,7,5] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,4,3,7,5,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,4,3,7,6,5] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => ? = 3
[2,1,4,5,3,6,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,4,5,3,7,6] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,6,3,5,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,4,6,3,7,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,6,5,3,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,4,6,5,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,7,3,5,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,4,7,3,6,5] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,7,5,3,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,4,7,5,6,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,4,7,6,3,5] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,4,7,6,5,3] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,1,3,5,6,7,4] => ? = 2
[2,1,5,3,4,6,7] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => ? = 2
[2,1,5,3,4,7,6] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,5,3,6,4,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,5,3,6,7,4] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,5,3,7,4,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,5,3,7,6,4] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => ? = 3
[2,1,5,4,3,6,7] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,4,5,3,6,7] => ? = 2
[2,1,5,4,3,7,6] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3,7,6] => ? = 3
[2,1,5,4,6,3,7] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,5,4,6,7,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => ? = 3
[2,1,5,4,7,3,6] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => ? = 3
[2,1,5,4,7,6,3] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => ? = 3
[2,1,5,6,3,4,7] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,5,6,3,7,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,5,6,4,3,7] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,5,6,4,7,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,5,7,3,4,6] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ? = 2
[2,1,5,7,3,6,4] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3
[2,1,5,7,4,3,6] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => ? = 2
[2,1,5,7,4,6,3] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => ? = 3

Description

The number of deficiencies of a permutation. This is defined as

$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$ The number of exceedances is [[St000155]].

The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000251The number of nonsingleton blocks of a set partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001512The minimum rank of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001354The number of series nodes in the modular decomposition of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000021The number of descents of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St001874Lusztig's a-function for the symmetric group. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000092The number of outer peaks of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice.