- FindStat

Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000141
(load all 10 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St000013
(load all 4 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 92%●distinct values known / distinct values provided: 91%

Values

[1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 1 + 1
[1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1 + 1
[2,4,6,1,3,5,7,8] => [4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 3 + 1
[2,4,5,6,1,3,7,8] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[2,3,5,7,1,4,6,8] => [5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 4 + 1
[3,5,1,6,2,4,7,8] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[3,1,5,7,2,4,6,8] => [5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 4 + 1
[1,3,4,6,2,7,5,8] => [1,5,3,7,2,6,4,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[1,3,4,7,2,5,6,8] => [1,5,3,7,2,6,4,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[1,4,2,6,3,7,5,8] => [1,5,3,7,2,6,4,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[4,6,1,2,3,5,7,8] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[1,2,4,6,3,5,7,8] => [1,2,5,6,3,4,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,4,2,7,3,5,6,8] => [1,5,3,7,2,6,4,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[3,4,1,2,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[4,5,6,1,2,3,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[2,1,5,6,3,4,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[2,1,6,5,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[2,1,6,7,8,3,4,5,10,9] => [2,1,8,7,6,5,4,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
[3,4,1,2,7,8,5,6,10,9] => [4,3,2,1,8,7,6,5,10,9] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[2,1,4,3,7,8,5,6,10,9] => [2,1,4,3,8,7,6,5,10,9] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[2,1,8,7,6,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
[2,1,4,3,8,9,10,5,6,7] => [2,1,4,3,10,9,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 + 1
[2,1,5,6,3,4,9,10,7,8] => [2,1,6,5,4,3,10,9,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[3,4,1,2,6,5,9,10,7,8] => [4,3,2,1,6,5,10,9,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[2,1,4,3,6,5,9,10,7,8] => [2,1,4,3,6,5,10,9,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[2,1,4,3,6,5,10,9,8,7] => [2,1,4,3,6,5,10,9,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[4,3,2,1,6,5,10,9,8,7] => [4,3,2,1,6,5,10,9,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[2,1,6,5,4,3,10,9,8,7] => [2,1,6,5,4,3,10,9,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[2,1,4,3,10,9,8,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 + 1
[3,2,5,7,1,4,6,8] => [5,2,6,7,1,3,4,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? = 4 + 1
[3,6,2,5,1,4,7,8] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 4 + 1
[3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 + 1
[4,5,6,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 5 + 1
[2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[2,1,6,7,8,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5 + 1
[2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5 + 1
[1,4,2,7,3,6,5,8] => [1,5,3,7,2,6,4,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1 + 1
[2,1,4,3,6,5,8,7,12,11,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3 + 1

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000662

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 91%

Values

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

Description

The staircase size of the code of a permutation. The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ . The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ . This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

Matching statistic: St001039

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 75%●distinct values known / distinct values provided: 73%

Values

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

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

Matching statistic: St000503
(load all 4 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 73%

Values

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

Description

The maximal difference between two elements in a common block.

Matching statistic: St001330
(load all 5 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 73%

Values

[1] => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => ([],2)
 => 1 = 0 + 1
[2,1] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => ([],3)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,5,2,3,4] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 2 = 1 + 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,3,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,4,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[2,4,5,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,5,1,3,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,5,3,1,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,5,4,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,1,4,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,1,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,1,5,4,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[3,2,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,2,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,2,5,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[3,4,1,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[3,4,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[3,5,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,5,2,1,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 1
[4,1,2,3,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[4,1,2,5,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[4,1,3,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[4,1,3,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[4,1,5,2,3] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[4,1,5,3,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[4,2,1,5,3] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[4,2,3,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[4,3,1,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[4,3,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,2,4,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,3,2,4] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,3,4,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,4,2,3] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,1,4,3,2] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[5,2,1,3,4] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1

Description

The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of

$q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number

$HG(G)$ of a graph

$G$ is the largest integer

$q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of

$q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.

Matching statistic: St000651
(load all 5 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 73%

Values

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

Description

The maximal size of a rise in a permutation. This is

$\max_i \sigma_{i+1}-\sigma_i$ , except for the permutations without rises, where it is

$0$ .

Matching statistic: St000028

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 73%

Values

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

Description

The number of stack-sorts needed to sort a permutation. A permutation is (West)

$t$ -stack sortable if it is sortable using

$t$ stacks in series. Let

$W_t(n,k)$ be the number of permutations of size

$n$ with

$k$ descents which are

$t$ -stack sortable. Then the polynomials

$W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have

$W_{n,1}(x) = A_n(x)$ , the Eulerian polynomials. One can show that

$W_{n,1}(x)$ and

$W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern

$231$ have statistic at most

$1$ , see [3]. These are counted by

$\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern

$2341$ and the barred pattern

$3\bar 5241$ have statistic at most

$2$ , see [4]. These are counted by

$\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).

Matching statistic: St000306

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 91%

Values

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

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000094

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 91%

Values

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

Description

The depth of an ordered tree.

The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000442The maximal area to the right of an up step of a Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St000730The maximal arc length of a set partition. St000454The largest eigenvalue of a graph if it is integral. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001589The nesting number of a perfect matching. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .