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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: St000141
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
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
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [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
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: St000306
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: St000662
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [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]
=> [4,2,3,1] => 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]
=> [1,5,3,4,2] => 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
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: St001046
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001046: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001046: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [(1,2)]
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 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),(5,6),(7,8),(9,10)]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 3
Description
The maximal number of arcs nesting a given arc of a perfect matching.
This is also the largest weight of a down step in the histoire d'Hermite corresponding to the perfect matching.
Matching statistic: St000720
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [(1,2)]
=> 1 = 0 + 1
[1,2] => [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3 = 2 + 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3 = 2 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3 = 2 + 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3 = 2 + 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3 = 2 + 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 4 = 3 + 1
Description
The size of the largest partition in the oscillating tableau corresponding to the perfect matching.
Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.
Matching statistic: St000094
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The depth of an ordered tree.
Matching statistic: St001039
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: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
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
[2,4,6,7,1,3,5,8] => [5,6,7,4,1,2,3,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[2,4,5,1,6,7,3,8] => [4,7,3,1,5,6,2,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[2,4,5,6,1,7,3,8] => [5,7,3,4,1,6,2,8] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 5 + 1
[2,4,5,6,7,1,3,8] => [6,7,3,4,5,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[2,5,6,1,7,3,4,8] => [4,7,6,1,5,3,2,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[2,6,7,1,3,4,5,8] => [4,7,6,1,5,3,2,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[3,5,1,2,6,7,4,8] => [4,7,3,1,5,6,2,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[3,5,1,6,2,7,4,8] => [5,7,3,4,1,6,2,8] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 5 + 1
[3,5,1,6,7,2,4,8] => [6,7,3,4,5,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[3,5,6,7,1,2,4,8] => [6,7,5,4,3,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[3,5,7,1,2,4,6,8] => [5,6,7,4,1,2,3,8] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[1,3,5,7,8,2,4,6] => [1,6,7,8,5,2,3,4] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 4 + 1
[1,3,5,6,2,7,8,4] => [1,5,8,4,2,6,7,3] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
[1,3,5,6,7,2,8,4] => [1,6,8,4,5,2,7,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,3,5,6,7,8,2,4] => [1,7,8,4,5,6,2,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[3,6,1,2,4,7,5,8] => [4,7,3,1,5,6,2,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[1,3,6,7,2,4,8,5] => [1,5,8,6,2,4,7,3] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
[1,3,6,7,8,2,4,5] => [1,6,8,7,5,2,4,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,3,7,8,2,4,5,6] => [1,5,8,7,2,6,4,3] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
[3,7,1,2,4,5,6,8] => [4,7,3,1,5,6,2,8] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 + 1
[4,6,1,2,7,3,5,8] => [6,7,3,4,5,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[4,6,1,7,2,3,5,8] => [6,7,3,5,4,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[4,6,7,1,2,3,5,8] => [6,7,5,4,3,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[4,6,1,2,3,7,5,8] => [5,7,3,4,1,6,2,8] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 5 + 1
[1,4,6,2,3,7,8,5] => [1,5,8,4,2,6,7,3] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
[1,4,6,2,7,3,8,5] => [1,6,8,4,5,2,7,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,4,6,7,2,3,8,5] => [1,6,8,5,4,2,7,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,4,6,2,7,8,3,5] => [1,7,8,4,5,6,2,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,4,6,7,2,8,3,5] => [1,7,8,5,4,6,2,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,4,6,7,8,2,3,5] => [1,7,8,6,5,4,2,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,4,6,8,2,3,5,7] => [1,6,7,8,5,2,3,4] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 4 + 1
[1,4,7,2,3,8,5,6] => [1,5,8,4,2,7,6,3] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
[1,4,7,2,8,3,5,6] => [1,6,8,4,7,2,5,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,4,7,8,2,3,5,6] => [1,6,8,7,5,2,4,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,4,7,2,3,5,8,6] => [1,5,8,4,2,6,7,3] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
[4,7,1,2,3,5,6,8] => [5,7,3,4,1,6,2,8] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 5 + 1
[1,4,8,2,3,5,6,7] => [1,5,8,4,2,6,7,3] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
[1,5,7,2,3,8,4,6] => [1,7,8,4,5,6,2,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,5,7,2,8,3,4,6] => [1,7,8,4,6,5,2,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,5,7,8,2,3,4,6] => [1,7,8,6,5,4,2,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,5,7,2,3,4,8,6] => [1,6,8,4,5,2,7,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[5,7,1,2,3,4,6,8] => [6,7,3,4,5,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[1,5,8,2,3,4,6,7] => [1,6,8,4,5,2,7,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,6,8,2,3,4,5,7] => [1,7,8,4,5,6,2,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[4,7,3,5,1,2,6,8] => [6,7,5,4,3,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[5,7,2,4,1,3,6,8] => [6,7,5,4,3,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[5,7,3,4,1,2,6,8] => [6,7,5,4,3,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[4,7,3,6,1,2,5,8] => [6,7,5,4,3,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 5 + 1
[1,3,5,8,2,7,6,4] => [1,5,8,7,2,6,4,3] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000442
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 70%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 70%
Values
[1] => [1] => [1,0]
=> ? = 0
[1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[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] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[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] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[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] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [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]
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [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
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[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
[6,7,5,4,3,2,1,8] => [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
[7,5,6,4,3,2,1,8] => [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
[6,5,7,4,3,2,1,8] => [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
[5,6,7,4,3,2,1,8] => [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
[7,6,4,5,3,2,1,8] => [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
[6,7,4,5,3,2,1,8] => [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
[7,5,4,6,3,2,1,8] => [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
[7,4,5,6,3,2,1,8] => [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
[6,5,4,7,3,2,1,8] => [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
[5,6,4,7,3,2,1,8] => [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
[5,4,6,7,3,2,1,8] => [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
[4,5,6,7,3,2,1,8] => [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
[7,6,5,3,4,2,1,8] => [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
[6,7,5,3,4,2,1,8] => [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
[7,5,6,3,4,2,1,8] => [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
[6,5,7,3,4,2,1,8] => [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
[5,6,7,3,4,2,1,8] => [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
[7,6,4,3,5,2,1,8] => [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
[6,7,4,3,5,2,1,8] => [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
[7,6,3,4,5,2,1,8] => [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
[7,5,4,3,6,2,1,8] => [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
[7,4,5,3,6,2,1,8] => [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
[7,3,4,5,6,2,1,8] => [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
[6,5,4,3,7,2,1,8] => [7,6,4,3,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[5,6,4,3,7,2,1,8] => [7,6,4,3,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[5,4,6,3,7,2,1,8] => [7,6,4,3,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[5,4,3,6,7,2,1,8] => [7,6,3,4,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[3,4,5,6,7,2,1,8] => [7,6,3,4,5,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6
[7,6,5,4,2,3,1,8] => [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
[6,7,5,4,2,3,1,8] => [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
[7,5,6,4,2,3,1,8] => [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
[5,6,7,4,2,3,1,8] => [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
[7,6,4,5,2,3,1,8] => [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
[6,7,4,5,2,3,1,8] => [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
[7,5,4,6,2,3,1,8] => [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
[7,4,5,6,2,3,1,8] => [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
[6,5,4,7,2,3,1,8] => [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
[5,6,4,7,2,3,1,8] => [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
[5,4,6,7,2,3,1,8] => [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
[4,5,6,7,2,3,1,8] => [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
[7,6,5,3,2,4,1,8] => [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
[6,5,7,3,2,4,1,8] => [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
[5,6,7,3,2,4,1,8] => [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
[7,6,5,2,3,4,1,8] => [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
[6,7,5,2,3,4,1,8] => [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
[7,5,6,2,3,4,1,8] => [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
[6,5,7,2,3,4,1,8] => [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
[5,6,7,2,3,4,1,8] => [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
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000503
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 70%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 70%
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
[6,7,4,5,3,2,1,8] => [7,6,5,3,4,1,2,8] => [2,7,4,1,3,5,6,8] => ?
=> ? = 6
[7,5,4,6,3,2,1,8] => [7,6,5,3,2,4,1,8] => [7,4,2,1,3,5,6,8] => ?
=> ? = 6
[5,6,4,7,3,2,1,8] => [7,6,5,3,1,2,4,8] => [2,4,1,7,3,5,6,8] => ?
=> ? = 6
[5,4,6,7,3,2,1,8] => [7,6,5,2,1,3,4,8] => [3,1,4,7,2,5,6,8] => ?
=> ? = 6
[5,6,7,3,4,2,1,8] => [7,6,4,5,1,2,3,8] => [2,3,7,5,1,4,6,8] => ?
=> ? = 6
[6,7,4,3,5,2,1,8] => [7,6,4,3,5,1,2,8] => [2,7,5,3,1,4,6,8] => ?
=> ? = 6
[7,3,4,5,6,2,1,8] => [7,6,2,3,4,5,1,8] => [7,3,4,5,1,2,6,8] => ?
=> ? = 6
[6,5,4,3,7,2,1,8] => [7,6,4,3,2,1,5,8] => [5,1,2,3,7,4,6,8] => ?
=> ? = 6
[5,6,7,4,2,3,1,8] => [7,5,6,4,1,2,3,8] => [2,3,7,1,6,4,5,8] => ?
=> ? = 6
[6,7,4,5,2,3,1,8] => [7,5,6,3,4,1,2,8] => [2,7,4,1,6,3,5,8] => ?
=> ? = 6
[7,5,4,6,2,3,1,8] => [7,5,6,3,2,4,1,8] => [7,4,2,1,6,3,5,8] => ?
=> ? = 6
[6,5,4,7,2,3,1,8] => [7,5,6,3,2,1,4,8] => [4,1,2,7,6,3,5,8] => ?
=> ? = 6
[5,6,4,7,2,3,1,8] => [7,5,6,3,1,2,4,8] => [2,4,1,7,6,3,5,8] => ?
=> ? = 6
[6,5,7,3,2,4,1,8] => [7,5,4,6,2,1,3,8] => [3,1,7,6,4,2,5,8] => ?
=> ? = 6
[5,6,7,3,2,4,1,8] => [7,5,4,6,1,2,3,8] => [2,3,7,6,4,1,5,8] => ?
=> ? = 6
[7,6,5,2,3,4,1,8] => [7,4,5,6,3,2,1,8] => [7,1,2,5,6,3,4,8] => ?
=> ? = 6
[5,6,7,2,3,4,1,8] => [7,4,5,6,1,2,3,8] => [2,3,7,5,6,1,4,8] => ?
=> ? = 6
[6,7,4,3,2,5,1,8] => [7,5,4,3,6,1,2,8] => [2,7,6,3,4,1,5,8] => ?
=> ? = 6
[6,7,3,4,2,5,1,8] => [7,5,3,4,6,1,2,8] => [2,7,4,6,3,1,5,8] => ?
=> ? = 6
[7,6,3,2,4,5,1,8] => [7,4,3,5,6,2,1,8] => [7,1,5,3,6,2,4,8] => ?
=> ? = 6
[6,7,3,2,4,5,1,8] => [7,4,3,5,6,1,2,8] => [2,7,5,3,6,1,4,8] => ?
=> ? = 6
[7,6,2,3,4,5,1,8] => [7,3,4,5,6,2,1,8] => [7,1,4,5,6,2,3,8] => ?
=> ? = 6
[6,7,2,3,4,5,1,8] => [7,3,4,5,6,1,2,8] => [2,7,4,5,6,1,3,8] => ?
=> ? = 6
[7,4,5,3,2,6,1,8] => [7,5,4,2,3,6,1,8] => [7,3,6,2,4,1,5,8] => ?
=> ? = 6
[7,4,3,5,2,6,1,8] => [7,5,3,2,4,6,1,8] => [7,4,2,6,3,1,5,8] => ?
=> ? = 6
[7,4,2,3,5,6,1,8] => [7,3,4,2,5,6,1,8] => ? => ?
=> ? = 6
[7,2,3,4,5,6,1,8] => [7,2,3,4,5,6,1,8] => [7,3,4,5,6,1,2,8] => ?
=> ? = 6
[5,6,4,3,2,7,1,8] => ? => ? => ?
=> ? = 6
[5,4,6,3,2,7,1,8] => [7,5,4,2,1,3,6,8] => ? => ?
=> ? = 6
[4,3,5,6,2,7,1,8] => [7,5,2,1,3,4,6,8] => [3,1,4,6,2,7,5,8] => ?
=> ? = 6
[6,5,3,2,4,7,1,8] => [7,4,3,5,2,1,6,8] => [6,1,5,3,2,7,4,8] => ?
=> ? = 6
[6,4,2,3,5,7,1,8] => [7,3,4,2,5,1,6,8] => ? => ?
=> ? = 6
[6,2,3,4,5,7,1,8] => [7,2,3,4,5,1,6,8] => [6,3,4,5,1,7,2,8] => ?
=> ? = 6
[7,6,5,4,3,1,2,8] => [6,7,5,4,3,2,1,8] => [7,1,2,3,4,6,5,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[6,7,5,4,3,1,2,8] => [6,7,5,4,3,1,2,8] => [2,7,1,3,4,6,5,8] => ?
=> ? = 6
[7,5,6,4,3,1,2,8] => [6,7,5,4,2,3,1,8] => [7,3,1,2,4,6,5,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[7,6,4,5,3,1,2,8] => [6,7,5,3,4,2,1,8] => [7,1,4,2,3,6,5,8] => ?
=> ? = 6
[7,4,5,6,3,1,2,8] => [6,7,5,2,3,4,1,8] => [7,3,4,1,2,6,5,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[5,4,6,7,3,1,2,8] => [6,7,5,2,1,3,4,8] => [3,1,4,7,2,6,5,8] => ?
=> ? = 6
[4,5,6,7,3,1,2,8] => [6,7,5,1,2,3,4,8] => [2,3,4,7,1,6,5,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[7,6,5,3,4,1,2,8] => [6,7,4,5,3,2,1,8] => [7,1,2,5,3,6,4,8] => ?
=> ? = 6
[7,5,6,3,4,1,2,8] => [6,7,4,5,2,3,1,8] => [7,3,1,5,2,6,4,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[6,5,7,3,4,1,2,8] => [6,7,4,5,2,1,3,8] => [3,1,7,5,2,6,4,8] => ?
=> ? = 6
[6,7,3,4,5,1,2,8] => [6,7,3,4,5,1,2,8] => [2,7,4,5,1,6,3,8] => ?
=> ? = 6
[7,5,3,4,6,1,2,8] => [6,7,3,4,2,5,1,8] => [7,5,4,2,1,6,3,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[6,5,4,3,7,1,2,8] => [6,7,4,3,2,1,5,8] => [5,1,2,3,7,6,4,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[6,4,5,3,7,1,2,8] => [6,7,4,2,3,1,5,8] => [5,3,1,2,7,6,4,8] => ?
=> ? = 6
[5,6,3,4,7,1,2,8] => [6,7,3,4,1,2,5,8] => [2,5,4,1,7,6,3,8] => ?
=> ? = 6
[6,4,3,5,7,1,2,8] => [6,7,3,2,4,1,5,8] => [5,4,2,1,7,6,3,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
Description
The maximal difference between two elements in a common block.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000651The maximal size of a rise in a permutation. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000028The number of stack-sorts needed to sort a permutation. St000209Maximum difference of elements in cycles. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000956The maximal displacement of a permutation. 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. 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. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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