Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000141
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 2
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 3
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => 3
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [2,1,4,3] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => 2
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 3
[3,4,2,1] => [1,4,3,2] => [4,3,1,2] => 3
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 3
[4,2,1,3] => [4,2,1,3] => [2,1,4,3] => 1
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 3
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[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] => [5,1,2,3,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 3
[1,2,4,5,3] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => 3
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 4
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => 3
[1,3,4,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => 3
[1,3,4,5,2] => [1,2,3,5,4] => [5,1,2,3,4] => 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => 4
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,4,2,5,3] => [1,3,2,5,4] => [3,5,1,2,4] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => 3
[1,4,3,5,2] => [1,3,2,5,4] => [3,5,1,2,4] => 3
[1,4,5,2,3] => [1,3,5,2,4] => [5,3,1,2,4] => 4
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
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => [1,1,0,0]
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[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] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,1,3,4] => [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] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[3,1,4,2] => [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[3,4,2,1] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,2,1,3] => [4,2,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4,3,2,1] => [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] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[1,2,4,5,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,4,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[1,3,4,5,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,2,5,3] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[1,4,3,5,2] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,5,2,3] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[5,4,6,3,7,2,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,6,5,7,2,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,2,6,5,7,4,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,5,2,7,6,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,2,5,4,7,6,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,1,6,5,7,4,8,3] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,1,5,4,7,6,8,3] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,2,4,1,7,6,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,1,4,3,7,6,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,5,2,6,1,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,2,5,4,6,1,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,1,5,4,6,3,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,2,4,1,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[1,3,4,2,6,7,5,8] => [1,2,4,3,5,7,6,8] => [4,7,1,2,3,5,6,8] => ?
 => ? = 5 + 1
[2,1,4,3,7,5,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,1,5,3,6,4,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,1,5,3,7,4,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[2,1,6,3,7,4,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,1,4,2,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,1,4,2,7,5,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,1,5,2,6,4,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,1,5,2,7,4,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,1,6,2,7,4,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,1,5,2,6,3,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,1,5,2,7,3,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,1,6,2,7,3,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,1,2,3,8,5,6,7] => [4,1,2,3,8,5,6,7] => [1,4,2,8,3,5,6,7] => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[5,1,6,2,7,3,8,4] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[1,5,2,7,3,4,6,8] => [1,5,2,7,3,4,6,8] => [1,5,2,7,3,4,6,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[6,1,8,2,3,4,5,7] => [6,1,8,2,3,4,5,7] => [1,2,3,6,4,8,5,7] => [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[5,4,1,2,8,3,6,7] => [5,4,1,2,8,3,6,7] => [1,5,4,2,8,3,6,7] => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 3 + 1
[7,4,1,8,2,3,5,6] => [6,4,1,8,2,3,5,7] => [1,4,2,6,3,8,5,7] => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[4,1,5,3,7,2,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[6,4,1,8,2,3,5,7] => [6,4,1,8,2,3,5,7] => [1,4,2,6,3,8,5,7] => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[7,4,1,6,2,3,5,8] => [7,4,1,6,2,3,5,8] => [1,4,2,7,3,6,5,8] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[4,8,1,6,2,3,5,7] => [4,8,1,6,2,3,5,7] => [1,4,2,8,3,6,5,7] => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,2,5,1,6,4,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,2,5,4,7,1,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,2,5,3,7,6,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,2,5,3,7,1,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,2,5,3,6,1,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,2,5,1,6,3,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[3,1,5,4,6,2,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,1,5,3,6,2,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,5,1,6,2,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,5,2,7,1,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,6,2,7,5,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,6,2,7,1,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 1
[4,3,6,1,7,2,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4 + 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: St000442
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 88%
Values
[1] => [1] => [1] => [1,0]
 => ? = 0
[1,2] => [1,2] => [1,2] => [1,0,1,0]
 => 0
[2,1] => [2,1] => [2,1] => [1,1,0,0]
 => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,3,2] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 2
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[2,3,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 2
[3,1,2] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 3
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 3
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 3
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[3,1,4,2] => [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 3
[3,4,2,1] => [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 3
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 3
[4,2,1,3] => [4,2,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 3
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,2,4,5,3] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 3
[1,3,4,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,3,4,5,2] => [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,5,3] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 3
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,4,3,5,2] => [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 3
[1,4,5,2,3] => [1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,4,5,3,2] => [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 4
[8,7,6,5,4,3,2,1] => [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
[5,6,7,8,4,3,2,1] => [1,2,3,8,7,6,5,4] => [8,7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[5,4,6,3,7,2,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,8,7] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[7,8,6,5,4,3,1,2] => [2,8,7,6,5,4,1,3] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[6,7,8,5,4,1,2,3] => [2,4,8,7,6,1,3,5] => [8,7,6,4,2,1,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[6,7,5,8,3,1,2,4] => [2,6,5,8,4,1,3,7] => [6,8,5,4,2,1,3,7] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 6
[7,5,6,8,1,2,3,4] => [6,3,5,8,1,2,4,7] => [6,1,8,3,2,5,4,7] => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 5
[5,6,7,8,1,2,3,4] => [2,4,6,8,1,3,5,7] => [8,6,4,2,1,3,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[8,7,6,4,3,2,1,5] => [8,7,6,4,3,2,1,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[8,6,5,4,3,2,1,7] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 5
[8,6,4,3,2,1,5,7] => [8,6,4,3,2,1,5,7] => [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[3,5,8,7,6,4,2,1] => [1,2,8,7,6,5,4,3] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[3,5,6,7,8,4,2,1] => [1,2,3,4,8,7,6,5] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[2,6,8,7,5,4,3,1] => [1,2,8,7,6,5,4,3] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[2,5,6,7,8,4,3,1] => [1,2,3,4,8,7,6,5] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[2,3,5,6,8,7,4,1] => [1,2,3,4,8,7,6,5] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[2,3,5,6,7,8,4,1] => [1,2,3,4,5,8,7,6] => [8,7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,3,6,5,7,2,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[3,2,6,5,7,4,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[4,3,5,2,7,6,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[3,2,5,4,7,6,8,1] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[1,7,8,6,5,4,3,2] => [1,2,8,7,6,5,4,3] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[1,5,6,7,8,4,3,2] => [1,2,3,4,8,7,6,5] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[1,3,5,7,8,6,4,2] => [1,2,3,4,8,7,6,5] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[1,3,5,6,7,8,4,2] => [1,2,3,4,5,8,7,6] => [8,7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[2,1,6,5,7,4,8,3] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,1,5,4,7,6,8,3] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,1,4,5,6,7,8,3] => [2,1,3,4,5,6,8,7] => [2,8,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[1,2,6,7,8,5,4,3] => [1,2,3,4,8,7,6,5] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[1,2,5,6,7,8,4,3] => [1,2,3,4,5,8,7,6] => [8,7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[1,2,3,5,6,7,8,4] => [1,2,3,4,5,6,8,7] => [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[3,2,4,1,7,6,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,1,4,3,7,6,8,5] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[1,2,3,4,8,7,6,5] => [1,2,3,4,8,7,6,5] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[4,3,5,2,6,1,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[3,2,5,4,6,1,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,1,5,4,6,3,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[3,2,4,1,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,8,7] => [2,8,1,3,4,5,6,7] => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 6
[1,3,4,2,6,7,5,8] => [1,2,4,3,5,7,6,8] => [4,7,1,2,3,5,6,8] => ?
 => ? = 5
[2,4,6,8,1,3,5,7] => [2,4,6,8,1,3,5,7] => [8,6,4,2,1,3,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 7
[2,1,4,3,7,5,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,1,5,3,6,4,8,7] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
[2,1,5,3,7,4,8,6] => [2,1,4,3,6,5,8,7] => [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 4
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 88%
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]
 => [1,1,0,1,0,0]
 => 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1
[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]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,3,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 3
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 3
[2,4,3,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 3
[3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 1
[3,1,4,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 2
[3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? = 2
[3,4,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 3
[3,4,2,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 3
[4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 1
[4,2,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 2
[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]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,2,4,5,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 4
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,3,4,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[1,3,4,5,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,3,5,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 4
[1,3,5,4,2] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2
[1,4,2,5,3] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 3
[1,4,3,5,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[1,4,5,2,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 4
[1,4,5,3,2] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 4
[1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,5,3,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,5,3,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? = 2
[2,1,4,5,3] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 3
[2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 4
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? = 4
[2,3,1,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[2,3,1,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[2,3,4,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3
[2,3,4,5,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[2,3,5,1,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 4
[2,3,5,4,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 4
[2,4,1,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 3
[2,4,1,5,3] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[2,4,3,1,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 3
[2,4,3,5,1] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? = 3
[2,4,5,1,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 4
[2,4,5,3,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? = 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,2,3,5,6,4] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,2,4,5,3,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,2,4,5,6,3] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,3,4,2,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,3,4,5,2,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[1,3,4,5,6,2] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[2,3,1,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[2,3,4,1,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[2,3,4,5,1,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4
[2,3,4,5,6,1] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5
[1,2,3,4,6,7,5] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
[1,2,3,5,6,4,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5
[1,2,3,5,6,7,4] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3
[1,2,4,5,3,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001207
(load all 3 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 3
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
[1,3,4,2] => [1,2,4,3] => [4,1,2,3] => 3
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 3
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => 2
[2,3,1,4] => [1,3,2,4] => [3,1,2,4] => 2
[2,3,4,1] => [1,2,4,3] => [4,1,2,3] => 3
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 3
[2,4,3,1] => [1,4,3,2] => [4,3,1,2] => 3
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [2,1,4,3] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => [2,4,1,3] => 2
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 3
[3,4,2,1] => [1,4,3,2] => [4,3,1,2] => 3
[4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 3
[4,2,1,3] => [4,2,1,3] => [2,1,4,3] => 1
[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 3
[4,3,1,2] => [4,3,1,2] => [1,4,3,2] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[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] => [5,1,2,3,4] => ? = 4
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ? = 3
[1,2,4,5,3] => [1,2,3,5,4] => [5,1,2,3,4] => ? = 4
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ? = 3
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ? = 4
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ? = 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ? = 3
[1,3,4,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ? = 3
[1,3,4,5,2] => [1,2,3,5,4] => [5,1,2,3,4] => ? = 4
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => ? = 4
[1,3,5,4,2] => [1,2,5,4,3] => [5,4,1,2,3] => ? = 4
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 2
[1,4,2,5,3] => [1,3,2,5,4] => [3,5,1,2,4] => ? = 3
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ? = 3
[1,4,3,5,2] => [1,3,2,5,4] => [3,5,1,2,4] => ? = 3
[1,4,5,2,3] => [1,3,5,2,4] => [5,3,1,2,4] => ? = 4
[1,4,5,3,2] => [1,2,5,4,3] => [5,4,1,2,3] => ? = 4
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ? = 2
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ? = 4
[1,5,3,2,4] => [1,5,3,2,4] => [5,1,3,2,4] => ? = 4
[1,5,3,4,2] => [1,5,2,4,3] => [5,1,4,2,3] => ? = 4
[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,2,3] => ? = 3
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ? = 4
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => ? = 3
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => ? = 2
[2,1,4,5,3] => [2,1,3,5,4] => [2,5,1,3,4] => ? = 3
[2,1,5,3,4] => [2,1,5,3,4] => [5,2,1,3,4] => ? = 4
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => ? = 4
[2,3,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ? = 2
[2,3,1,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ? = 3
[2,3,4,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ? = 3
[2,3,4,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => ? = 4
[2,3,5,1,4] => [1,3,5,2,4] => [5,3,1,2,4] => ? = 4
[2,3,5,4,1] => [1,2,5,4,3] => [5,4,1,2,3] => ? = 4
[2,4,1,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => ? = 3
[2,4,1,5,3] => [1,3,2,5,4] => [3,5,1,2,4] => ? = 3
[2,4,3,1,5] => [1,4,3,2,5] => [4,3,1,2,5] => ? = 3
[2,4,3,5,1] => [1,3,2,5,4] => [3,5,1,2,4] => ? = 3
[2,4,5,1,3] => [1,3,5,2,4] => [5,3,1,2,4] => ? = 4
[2,4,5,3,1] => [1,2,5,4,3] => [5,4,1,2,3] => ? = 4
[2,5,1,3,4] => [2,5,1,3,4] => [2,1,5,3,4] => ? = 2
[2,5,1,4,3] => [2,5,1,4,3] => [2,5,4,1,3] => ? = 3
[2,5,3,1,4] => [1,5,3,2,4] => [5,1,3,2,4] => ? = 4
[2,5,3,4,1] => [1,5,2,4,3] => [5,1,4,2,3] => ? = 4
[2,5,4,1,3] => [2,5,4,1,3] => [5,4,2,1,3] => ? = 4
[2,5,4,3,1] => [1,5,4,3,2] => [5,4,3,1,2] => ? = 4
[3,1,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => ? = 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.