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Matching statistic: St000651
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(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1] => [3,2,1] => [2,3,1] => [2,1,3] => 2
[3,1,2] => [3,2,1] => [2,3,1] => [2,1,3] => 2
[3,2,1] => [3,2,1] => [2,3,1] => [2,1,3] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 2
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 2
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,1,3] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 2
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 3
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2
[2,4,3,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 3
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 2
[3,1,4,2] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 3
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,2,4,1] => 2
[3,2,4,1] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 3
[3,4,1,2] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 3
[3,4,2,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 3
[4,1,2,3] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 3
[4,1,3,2] => [4,2,3,1] => [2,3,4,1] => [3,2,1,4] => 3
[4,2,1,3] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 3
[4,2,3,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 3
[4,3,1,2] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 3
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => [5,3,2,1,4] => 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,4,2,5,3] => [5,2,4,1,3] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,5,2] => [5,3,2,1,4] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => [5,3,2,1,4] => 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => [5,2,3,1,4] => 3
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: 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: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●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
[5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[3,4,5,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[5,4,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[4,5,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[5,2,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[7,3,2,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[4,3,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? = 7 + 1
[3,4,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? = 7 + 1
[2,4,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? = 7 + 1
[3,5,4,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[2,5,4,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[2,4,5,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,2,4,3,5,7,6,8] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,3,2,4,6,5,7,8] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[2,1,3,5,4,6,7,8] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[2,5,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[4,2,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? = 7 + 1
[5,2,4,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[2,3,5,1,6,4,7,8] => [4,2,6,1,5,3,7,8] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 1
[2,3,6,1,4,5,7,8] => [4,2,6,1,5,3,7,8] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 1
[3,4,5,1,6,7,2,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[1,3,5,6,2,4,8,7] => [1,5,6,4,2,3,8,7] => ?
=> ? = 3 + 1
[1,3,5,6,2,8,4,7] => [1,5,7,4,2,8,3,6] => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 4 + 1
[1,3,5,2,6,7,4,8] => [1,4,7,2,5,6,3,8] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 4 + 1
[1,3,6,2,4,7,5,8] => [1,4,7,2,5,6,3,8] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 4 + 1
[3,1,6,2,4,5,7,8] => [4,2,6,1,5,3,7,8] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,3,4,6,7,2,5,8] => [1,6,3,7,5,2,4,8] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,3,7,2,4,5,6,8] => [1,4,7,2,5,6,3,8] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 4 + 1
[4,5,1,2,6,7,3,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[1,4,2,6,7,3,5,8] => [1,6,3,7,5,2,4,8] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,4,6,2,3,8,5,7] => [1,5,7,4,2,8,3,6] => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 4 + 1
[1,4,6,2,3,5,8,7] => [1,5,6,4,2,3,8,7] => ?
=> ? = 3 + 1
[1,5,2,7,3,4,6,8] => [1,6,3,7,5,2,4,8] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,2,3,5,7,4,6,8] => [1,2,3,6,7,4,5,8] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[3,2,6,1,4,5,7,8] => [4,2,6,1,5,3,7,8] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 1
[3,1,5,2,7,4,9,6,8] => [4,2,6,1,8,3,9,5,7] => [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[3,2,6,1,5,4,7,8] => [4,2,6,1,5,3,7,8] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 1
[8,2,1,3,4,6,5,7] => [8,3,2,4,5,7,6,1] => ?
=> ? = 7 + 1
[6,3,2,1,5,7,4,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[3,2,5,1,6,4,7,8] => [4,2,6,1,5,3,7,8] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 1
[6,3,2,1,4,7,5,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6 + 1
[7,2,1,3,4,8,6,5] => [8,3,2,4,5,7,6,1] => ?
=> ? = 7 + 1
[1,4,2,7,6,3,5,8] => [1,6,3,7,5,2,4,8] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,3,6,4,2,8,5,7] => [1,5,7,4,2,8,3,6] => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 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: 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: 73% ●values known / values provided: 95%●distinct values known / distinct values provided: 73%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 95%●distinct values known / distinct values provided: 73%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[3,4,5,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[5,4,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[4,5,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[5,2,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[7,3,2,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
[4,3,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 7 + 1
[3,4,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 7 + 1
[2,4,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 7 + 1
[1,2,3,6,5,7,8,4] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,2,3,5,6,7,8,4] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[1,2,4,3,6,5,8,7] => [1,2,4,3,6,5,8,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,2,3,4,6,5,8,7] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[1,2,3,5,4,6,8,7] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 + 1
[3,5,4,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[2,5,4,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[2,4,5,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 1
[1,2,4,3,5,7,6,8] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1 + 1
[2,5,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[4,2,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 7 + 1
[5,2,4,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[3,4,5,1,6,7,2,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[1,3,5,6,2,4,8,7] => [1,5,6,4,2,3,8,7] => ?
=> ?
=> ? = 3 + 1
[1,3,5,6,2,8,4,7] => [1,5,7,4,2,8,3,6] => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 4 + 1
[1,3,5,2,6,7,4,8] => [1,4,7,2,5,6,3,8] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 + 1
[1,3,6,2,4,7,5,8] => [1,4,7,2,5,6,3,8] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 + 1
[1,3,7,2,4,5,6,8] => [1,4,7,2,5,6,3,8] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 + 1
[4,5,1,2,6,7,3,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 1
[1,4,6,2,3,8,5,7] => [1,5,7,4,2,8,3,6] => [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 4 + 1
[1,4,6,2,3,5,8,7] => [1,5,6,4,2,3,8,7] => ?
=> ?
=> ? = 3 + 1
[1,2,3,5,7,4,6,8] => [1,2,3,6,7,4,5,8] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[1,2,3,6,4,7,8,5] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,2,3,7,4,5,8,6] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,2,3,8,4,5,6,7] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[9,1,2,3,4,5,6,7,8] => [9,2,3,4,5,6,7,8,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[8,1,2,3,4,5,6,9,7] => [9,2,3,4,5,6,7,8,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[7,1,2,3,4,5,8,9,6] => [9,2,3,4,5,6,7,8,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
[6,1,2,3,4,7,8,9,5] => [9,2,3,4,5,6,7,8,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
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
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [1,2] => [1,0,1,0]
=> [1,2] => 0
[2,1] => [2,1] => [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[2,3,5,1,7,4,6] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => ? = 3
[2,3,5,6,7,1,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[2,3,6,4,7,1,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[2,3,6,5,7,1,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[2,3,7,4,5,1,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[2,3,7,4,6,1,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,5,2,7,4,6] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => ? = 3
[3,1,5,6,7,2,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,6,4,7,2,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,6,5,7,2,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,7,4,5,2,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,7,4,6,2,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,5,1,7,4,6] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => ? = 3
[3,2,5,6,7,1,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,6,4,7,1,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,6,5,7,1,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,7,4,5,1,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,7,4,6,1,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,6,2,7,3,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,6,3,7,2,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,7,2,5,3,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,7,2,6,3,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,7,3,5,2,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,7,3,6,2,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[5,1,7,2,3,4,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[5,1,7,2,4,3,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[3,4,5,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[5,4,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[4,5,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[5,2,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[7,3,2,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[4,3,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? => ? = 7
[3,4,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? => ? = 7
[2,4,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? => ? = 7
[1,2,3,6,5,7,8,4] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => ? = 4
[1,2,3,5,6,7,8,4] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => ? = 4
[1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ? = 2
[1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ? = 2
[1,2,4,3,6,5,8,7] => [1,2,4,3,6,5,8,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,8,7] => ? = 1
[1,2,3,4,6,5,8,7] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => ? = 1
[1,2,3,5,4,6,8,7] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => ? = 1
[1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => ? = 1
[1,3,2,4,5,6,8,7] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,6,8,7] => ? = 1
[3,5,4,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[2,5,4,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[2,4,5,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[1,2,4,3,5,7,6,8] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5,7,6,8] => ? = 1
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: 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: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 95%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1,0]
=> [1] => 0
[1,2] => [1,2] => [1,0,1,0]
=> [1,2] => 0
[2,1] => [2,1] => [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => ? = 1
[1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 1
[2,3,5,1,7,4,6] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => ? = 3
[2,3,5,6,7,1,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[2,3,6,4,7,1,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[2,3,6,5,7,1,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[2,3,7,4,5,1,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[2,3,7,4,6,1,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,5,2,7,4,6] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => ? = 3
[3,1,5,6,7,2,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,6,4,7,2,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,6,5,7,2,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,7,4,5,2,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,1,7,4,6,2,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,5,1,7,4,6] => [4,2,6,1,7,3,5] => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [4,3,6,5,7,2,1] => ? = 3
[3,2,5,6,7,1,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,6,4,7,1,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,6,5,7,1,4] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,7,4,5,1,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[3,2,7,4,6,1,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,6,2,7,3,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,6,3,7,2,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,7,2,5,3,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,7,2,6,3,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,7,3,5,2,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[4,1,7,3,6,2,5] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[5,1,7,2,3,4,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[5,1,7,2,4,3,6] => [6,2,7,4,5,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[3,4,5,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[5,4,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[4,5,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[5,2,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[7,3,2,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[4,3,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? => ? = 7
[3,4,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? => ? = 7
[2,4,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ? => ? = 7
[1,2,3,6,5,7,8,4] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => ? = 4
[1,2,3,5,6,7,8,4] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => ? = 4
[1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ? = 2
[1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ? = 2
[1,2,4,3,6,5,8,7] => [1,2,4,3,6,5,8,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,8,7] => ? = 1
[1,2,3,4,6,5,8,7] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => ? = 1
[1,2,3,5,4,6,8,7] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => ? = 1
[1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => ? = 1
[1,3,2,4,5,6,8,7] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,6,8,7] => ? = 1
[3,5,4,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
[2,5,4,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? => ? = 6
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: 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
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 72%●distinct values known / distinct values provided: 64%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 72%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1,0]
=> [1,0]
=> ? = 0
[1,2] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [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]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [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]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [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]
=> [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]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [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]
=> [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]
=> [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]
=> [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]
=> [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]
=> [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]
=> [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]
=> [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]
=> [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]
=> [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]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[5,4,3,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,3,5,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[3,4,5,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[6,2,3,4,5,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[5,4,3,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,5,3,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[5,3,4,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,3,5,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[3,4,5,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[5,4,2,3,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,5,2,3,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[5,3,2,4,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[5,2,3,4,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,3,2,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[3,4,2,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,2,3,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[3,2,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[8,2,3,4,1,5,6,7] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[8,4,1,2,3,5,6,7] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[8,3,2,1,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[8,2,3,1,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[8,3,1,2,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[8,2,1,3,4,5,6,7] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[8,1,2,3,4,5,6,7] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6
[3,4,5,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6
[5,4,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6
[4,5,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6
[5,2,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6
[7,3,2,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6
[5,4,3,2,8,7,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[5,4,3,2,7,8,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,5,3,2,8,7,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[3,5,4,2,7,8,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,3,5,2,8,7,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[3,4,5,2,7,8,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,5,4,3,8,7,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,5,4,3,7,8,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,4,5,3,8,7,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,4,5,3,7,8,6,1] => [8,4,3,2,7,6,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[4,3,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 7
[3,4,2,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 7
[2,4,3,5,6,8,7,1] => [8,3,2,4,5,7,6,1] => ?
=> ?
=> ? = 7
[3,2,6,5,7,4,8,1] => [8,2,6,4,5,3,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[3,2,5,6,7,4,8,1] => [8,2,6,4,5,3,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
[2,3,5,6,7,4,8,1] => [8,2,6,4,5,3,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7
Description
The maximal area to the right of an up step of a Dyck path.
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: 64% ●values known / values provided: 70%●distinct values known / distinct values provided: 64%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 64% ●values known / values provided: 70%●distinct values known / distinct values provided: 64%
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
[5,4,3,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[4,3,5,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[3,4,5,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[6,2,3,4,5,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[5,4,3,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[4,5,3,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[5,3,4,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[4,3,5,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[3,4,5,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[5,4,2,3,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[4,5,2,3,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[5,3,2,4,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[5,2,3,4,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[4,3,2,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[3,4,2,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[4,2,3,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[3,2,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[8,2,3,4,1,5,6,7] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[8,4,1,2,3,5,6,7] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[8,3,2,1,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[8,2,3,1,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[8,3,1,2,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[8,2,1,3,4,5,6,7] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[8,1,2,3,4,5,6,7] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 7 + 2
[5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 2
[3,4,5,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 2
[5,4,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 2
[4,5,2,3,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 2
[5,2,3,4,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 2
[7,3,2,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 2
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 2
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 6 + 2
[5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[4,5,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[5,3,4,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[4,3,5,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[3,4,5,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[5,4,2,3,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[4,5,2,3,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[5,3,2,4,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[5,2,3,4,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[3,2,4,5,1,6,7,8] => [5,2,3,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[2,3,4,5,1,6,7,8] => [5,2,3,4,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[5,4,3,1,2,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[5,3,4,1,2,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[4,3,5,1,2,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[3,4,5,1,2,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[5,4,2,1,3,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
[4,5,2,1,3,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 4 + 2
Description
The depth of an ordered tree.
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: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 73%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 73%
Values
[1] => [1] => [1] => {{1}}
=> ? = 0
[1,2] => [1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => [3,1,2] => [2,3,1] => {{1,2,3}}
=> 2
[3,1,2] => [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,2,1] => [3,2,1] => [3,1,2] => {{1,2,3}}
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 1
[2,3,1,4] => [3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => [4,1,2,3] => [2,3,4,1] => {{1,2,3,4}}
=> 3
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => {{1,2,3,4}}
=> 3
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,1,4,2] => [2,4,1,3] => [3,2,4,1] => {{1,3,4},{2}}
=> 3
[3,2,1,4] => [3,2,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 2
[3,2,4,1] => [4,2,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 3
[3,4,1,2] => [3,4,1,2] => [2,4,3,1] => {{1,2,4},{3}}
=> 3
[3,4,2,1] => [4,3,1,2] => [2,4,1,3] => {{1,2,3,4}}
=> 3
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[4,1,3,2] => [2,4,3,1] => [4,2,1,3] => {{1,3,4},{2}}
=> 3
[4,2,1,3] => [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[4,2,3,1] => [4,2,3,1] => [4,3,1,2] => {{1,2,3,4}}
=> 3
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => {{1,2,4},{3}}
=> 3
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => {{1,2,3,4}}
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3
[1,3,5,2,4] => [1,4,2,5,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => {{1},{2,3,4,5}}
=> 3
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 2
[1,4,2,5,3] => [1,3,5,2,4] => [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2
[1,4,3,5,2] => [1,5,3,2,4] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 3
[1,4,5,2,3] => [1,4,5,2,3] => [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3
[1,4,5,3,2] => [1,5,4,2,3] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 3
[8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => [8,3,4,5,2,6,7,1] => {{1,8},{2,3,4,5},{6},{7}}
=> ? = 7
[8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => [8,5,3,4,2,6,7,1] => {{1,8},{2,5},{3},{4},{6},{7}}
=> ? = 7
[8,3,2,1,4,5,6,7] => [4,3,2,5,6,7,8,1] => [8,4,2,3,5,6,7,1] => {{1,8},{2,3,4},{5},{6},{7}}
=> ? = 7
[8,2,3,1,4,5,6,7] => [4,2,3,5,6,7,8,1] => [8,3,4,2,5,6,7,1] => {{1,8},{2,3,4},{5},{6},{7}}
=> ? = 7
[8,3,1,2,4,5,6,7] => [3,4,2,5,6,7,8,1] => [8,4,3,2,5,6,7,1] => {{1,8},{2,4},{3},{5},{6},{7}}
=> ? = 7
[8,2,1,3,4,5,6,7] => [3,2,4,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 7
[8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 7
[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
[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
[7,2,3,4,5,6,1,8] => [7,2,3,4,5,6,1,8] => [7,3,4,5,6,1,2,8] => ?
=> ? = 6
[3,4,5,2,6,7,1,8] => [7,4,1,2,3,5,6,8] => [2,3,5,1,6,7,4,8] => ?
=> ? = 6
[5,4,2,3,6,7,1,8] => [7,3,4,2,1,5,6,8] => [5,1,4,2,6,7,3,8] => ?
=> ? = 6
[5,2,3,4,6,7,1,8] => [7,2,3,4,1,5,6,8] => [5,3,4,1,6,7,2,8] => ?
=> ? = 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
[4,3,5,6,7,1,2,8] => [6,7,2,1,3,4,5,8] => [3,1,4,5,7,6,2,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
[3,4,5,6,7,1,2,8] => [6,7,1,2,3,4,5,8] => [2,3,4,5,7,6,1,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 6
Description
The maximal difference between two elements in a common block.
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: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
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
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,4,6,5,7,1] => [7,2,3,5,4,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,5,6,4,7,1] => [7,2,5,4,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,5,6,7,4,1] => [7,2,6,4,5,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,5,7,6,4,1] => [7,2,6,5,4,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,6,4,5,7,1] => [7,2,5,4,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,6,4,7,5,1] => [7,2,6,4,5,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,6,5,4,7,1] => [7,2,5,4,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,6,5,7,4,1] => [7,2,6,4,5,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,6,7,4,5,1] => [7,2,6,5,4,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,6,7,5,4,1] => [7,2,6,5,4,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,7,4,5,6,1] => [7,2,6,4,5,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,7,4,6,5,1] => [7,2,6,4,5,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,7,5,4,6,1] => [7,2,6,5,4,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,7,5,6,4,1] => [7,2,6,5,4,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,7,6,4,5,1] => [7,2,6,5,4,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,3,7,6,5,4,1] => [7,2,6,5,4,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,4,3,5,6,7,1] => [7,3,2,4,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,4,5,3,6,7,1] => [7,4,3,2,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,4,5,6,3,7,1] => [7,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,4,5,6,7,3,1] => [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,4,6,5,3,7,1] => [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,5,3,4,6,7,1] => [7,4,3,2,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,5,3,6,4,7,1] => [7,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,5,3,6,7,4,1] => [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,5,4,3,6,7,1] => [7,4,3,2,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,5,4,6,3,7,1] => [7,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,5,4,6,7,3,1] => [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,5,6,3,4,7,1] => [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,5,6,4,3,7,1] => [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,6,3,4,5,7,1] => [7,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,6,3,4,7,5,1] => [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,6,3,5,4,7,1] => [7,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,6,3,5,7,4,1] => [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,6,4,3,5,7,1] => [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,6,4,5,3,7,1] => [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,6,5,3,4,7,1] => [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,6,5,4,3,7,1] => [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,7,3,4,5,6,1] => [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,7,3,4,6,5,1] => [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,4,5,6,7,2] => [7,2,3,4,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,4,6,5,7,2] => [7,2,3,5,4,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,5,6,4,7,2] => [7,2,5,4,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,5,6,7,4,2] => [7,2,6,4,5,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,5,7,6,4,2] => [7,2,6,5,4,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,6,4,5,7,2] => [7,2,5,4,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,6,4,7,5,2] => [7,2,6,4,5,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,6,5,4,7,2] => [7,2,5,4,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[3,1,6,5,7,4,2] => [7,2,6,4,5,3,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
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: St001203
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
St001203: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1 = 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]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 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]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 3 = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 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]
=> [1,1,0,0,1,0,1,0]
=> 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]
=> [1,0,1,1,0,1,0,0]
=> 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]
=> [1,0,1,1,0,1,0,0]
=> 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [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]
=> [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]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[5,4,3,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[4,3,5,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[3,4,5,6,2,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[6,2,3,4,5,7,8,1] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[5,4,3,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[4,5,3,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[5,3,4,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[4,3,5,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[3,4,5,2,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[5,4,2,3,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[4,5,2,3,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[5,3,2,4,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[5,2,3,4,6,7,8,1] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[4,3,2,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[3,4,2,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[4,2,3,5,6,7,8,1] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[3,2,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[2,3,4,5,6,7,8,1] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[8,2,3,4,1,5,6,7] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[8,4,1,2,3,5,6,7] => [8,5,3,4,2,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[8,3,2,1,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[8,2,3,1,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[8,3,1,2,4,5,6,7] => [8,4,3,2,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[8,2,1,3,4,5,6,7] => [8,3,2,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[8,1,2,3,4,5,6,7] => [8,2,3,4,5,6,7,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 + 1
Description
We 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:
In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra.
Example:
[5,6,6,6,6] goes into [2,2,2,2,1].
Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra.
The statistic return the global dimension of the CNakayama algebra divided by 2.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000730The maximal arc length of a set partition. St000028The number of stack-sorts needed to sort a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001330The hat guessing number of a graph. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001589The nesting number of a perfect matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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