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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 204 compositions to match this statistic)
(load all 204 compositions to match this statistic)
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 3
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 2
[2,4,1,3] => 2
[2,4,3,1] => 3
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 3
[4,1,2,3] => 2
[4,1,3,2] => 3
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 3
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern ([1],(1,1)), i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [2,1] => [2,1] => [2]
=> 1
[2,1] => [1,2] => [1,2] => [1,1]
=> 2
[1,2,3] => [3,2,1] => [3,1,2] => [3]
=> 1
[1,3,2] => [2,3,1] => [3,2,1] => [2,1]
=> 2
[2,1,3] => [3,1,2] => [2,3,1] => [3]
=> 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[3,1,2] => [2,1,3] => [2,1,3] => [2,1]
=> 2
[3,2,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,2,3,4] => [4,3,2,1] => [4,1,2,3] => [4]
=> 1
[1,2,4,3] => [3,4,2,1] => [4,1,3,2] => [3,1]
=> 2
[1,3,2,4] => [4,2,3,1] => [4,3,1,2] => [4]
=> 1
[1,3,4,2] => [2,4,3,1] => [4,2,1,3] => [3,1]
=> 2
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [2,2]
=> 2
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 3
[2,1,3,4] => [4,3,1,2] => [2,4,1,3] => [4]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,4,3,1] => [3,1]
=> 2
[2,3,1,4] => [4,1,3,2] => [3,4,2,1] => [4]
=> 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,1]
=> 2
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 3
[3,1,2,4] => [4,2,1,3] => [3,1,4,2] => [4]
=> 1
[3,1,4,2] => [2,4,1,3] => [3,2,4,1] => [3,1]
=> 2
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [4]
=> 1
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [3,1]
=> 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[4,1,2,3] => [3,2,1,4] => [3,1,2,4] => [3,1]
=> 2
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 3
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [3,1]
=> 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => [5]
=> 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => [4,1]
=> 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => [5]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [5,1,3,2,4] => [4,1]
=> 2
[1,2,5,3,4] => [4,3,5,2,1] => [5,1,4,3,2] => [3,2]
=> 2
[1,2,5,4,3] => [3,4,5,2,1] => [5,1,3,4,2] => [3,1,1]
=> 3
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,1,2,4] => [5]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,1,4,2] => [4,1]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,1,3,2] => [5]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [5,2,1,3,4] => [4,1]
=> 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,1,2,3] => [3,2]
=> 2
[1,3,5,4,2] => [2,4,5,3,1] => [5,2,1,4,3] => [3,1,1]
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,2,1,3] => [5]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,1,2] => [4,1]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [5,3,4,1,2] => [5]
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [5,2,4,1,3] => [4,1]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [5,3,2,1,4] => [3,2]
=> 2
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => [1,5,8,3,4,2,6,7] => ?
=> ? = 2
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => [3,1,2,5,4,6,7,8] => ?
=> ? = 5
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [3,1,2,6,4,5,7,8] => ?
=> ? = 4
[6,7,4,5,8,1,2,3] => [3,2,1,8,5,4,7,6] => [3,1,2,7,4,8,6,5] => ?
=> ? = 2
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => [5,1,2,3,4,7,6,8] => ?
=> ? = 3
[8,5,6,3,4,1,2,7] => [7,2,1,4,3,6,5,8] => [4,1,6,3,7,5,2,8] => ?
=> ? = 2
[8,6,4,3,2,1,5,7] => [7,5,1,2,3,4,6,8] => [2,3,4,6,1,7,5,8] => ?
=> ? = 2
[8,2,3,4,1,5,6,7] => [7,6,5,1,4,3,2,8] => [4,7,2,3,1,5,6,8] => ?
=> ? = 2
[8,2,3,1,4,5,6,7] => [7,6,5,4,1,3,2,8] => [3,7,2,1,4,5,6,8] => ?
=> ? = 2
[5,6,7,4,3,2,1,8] => [8,1,2,3,4,7,6,5] => [2,3,4,7,8,5,6,1] => ?
=> ? = 1
[7,6,4,3,5,2,1,8] => [8,1,2,5,3,4,6,7] => [2,5,4,6,3,7,8,1] => ?
=> ? = 1
[7,6,3,4,5,2,1,8] => [8,1,2,5,4,3,6,7] => [2,5,6,3,4,7,8,1] => ?
=> ? = 1
[2,3,4,5,6,1,7,8] => [8,7,1,6,5,4,3,2] => [6,8,2,3,4,5,1,7] => ?
=> ? = 1
[4,5,6,1,2,3,7,8] => [8,7,3,2,1,6,5,4] => [6,1,2,8,4,5,3,7] => ?
=> ? = 1
[2,6,8,7,5,4,3,1] => [1,3,4,5,7,8,6,2] => [1,8,3,4,5,2,7,6] => ?
=> ? = 6
[2,3,5,6,8,7,4,1] => [1,4,7,8,6,5,3,2] => [1,8,2,4,3,5,7,6] => ?
=> ? = 4
[3,2,6,5,4,8,7,1] => [1,7,8,4,5,6,2,3] => [1,3,8,5,6,2,7,4] => ?
=> ? = 3
[1,6,5,4,3,8,7,2] => [2,7,8,3,4,5,6,1] => [8,2,4,5,6,1,7,3] => ?
=> ? = 3
[2,1,5,4,7,6,8,3] => [3,8,6,7,4,5,1,2] => [2,8,3,5,1,7,4,6] => ?
=> ? = 2
[3,2,1,6,5,8,7,4] => [4,7,8,5,6,1,2,3] => [2,3,8,4,6,1,7,5] => ?
=> ? = 3
[1,2,3,5,6,7,8,4] => [4,8,7,6,5,3,2,1] => [8,1,2,4,3,5,6,7] => ?
=> ? = 2
[2,1,4,3,7,6,8,5] => [5,8,6,7,3,4,1,2] => [2,8,4,1,5,7,3,6] => ?
=> ? = 2
[3,2,5,4,6,1,8,7] => [7,8,1,6,4,5,2,3] => [6,3,8,5,2,4,7,1] => ?
=> ? = 2
[2,3,4,5,6,1,8,7] => [7,8,1,6,5,4,3,2] => [6,8,2,3,4,5,7,1] => ?
=> ? = 2
[2,1,4,6,5,3,8,7] => [7,8,3,5,6,4,1,2] => [2,8,5,1,6,4,7,3] => ?
=> ? = 2
[2,1,5,4,6,3,8,7] => [7,8,3,6,4,5,1,2] => [2,8,6,5,1,4,7,3] => ?
=> ? = 2
[2,4,3,1,6,5,8,7] => [7,8,5,6,1,3,4,2] => [3,8,4,2,6,1,7,5] => ?
=> ? = 2
[1,8,7,4,5,6,3,2] => [2,3,6,5,4,7,8,1] => [8,2,3,6,4,5,7,1] => ?
=> ? = 5
[1,8,7,4,6,5,3,2] => [2,3,5,6,4,7,8,1] => [8,2,3,6,5,4,7,1] => ?
=> ? = 6
[1,8,7,5,4,6,3,2] => [2,3,6,4,5,7,8,1] => [8,2,3,5,6,4,7,1] => ?
=> ? = 5
[1,8,7,5,6,4,3,2] => [2,3,4,6,5,7,8,1] => [8,2,3,4,6,5,7,1] => ?
=> ? = 6
[2,1,4,5,8,6,7,3] => [3,7,6,8,5,4,1,2] => [2,8,3,1,4,7,6,5] => ?
=> ? = 3
[2,3,4,8,5,6,7,1] => [1,7,6,5,8,4,3,2] => [1,8,2,3,7,5,6,4] => ?
=> ? = 3
[8,2,7,4,6,5,3,1] => [1,3,5,6,4,7,2,8] => [1,7,3,6,5,4,2,8] => ?
=> ? = 6
[4,3,2,7,5,6,8,1] => [1,8,6,5,7,2,3,4] => [1,3,4,8,7,5,2,6] => ?
=> ? = 2
[2,4,7,8,1,3,5,6] => [6,5,3,1,8,7,4,2] => [6,8,1,2,3,5,4,7] => ?
=> ? = 2
[2,1,5,3,6,4,8,7] => [7,8,4,6,3,5,1,2] => [2,8,5,6,1,3,7,4] => ?
=> ? = 2
[2,5,7,8,1,3,4,6] => [6,4,3,1,8,7,5,2] => [6,8,1,3,2,4,5,7] => ?
=> ? = 2
[2,7,8,1,3,4,5,6] => [6,5,4,3,1,8,7,2] => [6,8,1,3,4,5,2,7] => ?
=> ? = 2
[2,1,3,8,4,5,6,7] => [7,6,5,4,8,3,1,2] => [2,8,1,7,4,5,6,3] => ?
=> ? = 2
[3,4,6,8,1,2,5,7] => [7,5,2,1,8,6,4,3] => [7,1,8,3,2,4,5,6] => ?
=> ? = 2
[3,5,6,8,1,2,4,7] => [7,4,2,1,8,6,5,3] => [7,1,8,2,3,5,4,6] => ?
=> ? = 2
[3,1,6,2,7,4,8,5] => [5,8,4,7,2,6,1,3] => [3,6,8,7,5,1,2,4] => ?
=> ? = 2
[1,3,2,8,4,5,6,7] => [7,6,5,4,8,2,3,1] => [8,3,1,7,4,5,6,2] => ?
=> ? = 2
[4,1,5,2,6,3,8,7] => [7,8,3,6,2,5,1,4] => [4,5,6,8,1,2,7,3] => ?
=> ? = 2
[4,1,5,2,7,3,8,6] => [6,8,3,7,2,5,1,4] => [4,5,7,8,1,6,2,3] => ?
=> ? = 2
[4,1,6,2,7,3,8,5] => [5,8,3,7,2,6,1,4] => [4,6,7,8,5,1,2,3] => ?
=> ? = 2
[1,4,2,7,3,8,5,6] => [6,5,8,3,7,2,4,1] => [8,4,7,1,6,5,2,3] => ?
=> ? = 2
[1,4,2,7,3,5,6,8] => [8,6,5,3,7,2,4,1] => [8,4,7,1,3,5,2,6] => ?
=> ? = 1
[1,4,2,8,3,5,6,7] => [7,6,5,3,8,2,4,1] => [8,4,7,1,3,5,6,2] => ?
=> ? = 2
Description
The length of the partition.
Matching statistic: St000340
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 6 - 1
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 4 - 1
[7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[6,5,4,3,7,2,8,1] => [1,8,2,7,3,4,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[5,2,3,4,6,7,8,1] => [1,8,7,6,4,3,2,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 1
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 6 - 1
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 - 1
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2 - 1
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 4 - 1
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 5 - 1
[8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 4 - 1
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[6,7,4,5,8,1,2,3] => [3,2,1,8,5,4,7,6] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 2 - 1
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[7,5,6,8,2,3,1,4] => [4,1,3,2,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[6,5,7,8,2,1,3,4] => [4,3,1,2,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns 110 and 001.
Matching statistic: St000011
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 83%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 1
[1,2] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,4,3,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,3,4] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,1,4] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3,4,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,4,3,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,1,2,4] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,2,4,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,1,2,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,1,3,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,2,1,3] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,3,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,2,4,3,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,5,3,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,2,5,4,3] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2,4,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 6
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => [.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[6,5,4,3,7,2,8,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 4
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => [[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5
[8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[6,7,4,5,8,1,2,3] => [3,2,1,8,5,4,7,6] => [[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2
[7,5,6,8,2,3,1,4] => [4,1,3,2,8,6,5,7] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 4
[8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3
[7,8,5,4,2,1,3,6] => [6,3,1,2,4,5,8,7] => [[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[7,8,5,4,1,2,3,6] => [6,3,2,1,4,5,8,7] => [[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[8,5,6,3,4,1,2,7] => [7,2,1,4,3,6,5,8] => [[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 2
[8,6,4,3,2,1,5,7] => [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[6,5,7,3,2,4,1,8] => [8,1,4,2,3,7,5,6] => [[.,[[.,[.,.]],[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 1
[3,5,8,7,6,4,2,1] => [1,2,4,6,7,8,5,3] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[4,3,6,5,8,7,2,1] => [1,2,7,8,5,6,3,4] => [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[2,6,8,7,5,4,3,1] => [1,3,4,5,7,8,6,2] => [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[2,3,5,6,8,7,4,1] => [1,4,7,8,6,5,3,2] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[1,6,5,4,3,8,7,2] => [2,7,8,3,4,5,6,1] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[1,4,3,6,5,8,7,2] => [2,7,8,5,6,3,4,1] => [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[1,2,6,7,8,5,4,3] => [3,4,5,8,7,6,2,1] => ?
=> ?
=> ? = 4
[3,4,2,1,7,8,6,5] => [5,6,8,7,1,2,4,3] => [[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[2,4,3,1,7,6,8,5] => [5,8,6,7,1,3,4,2] => [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[3,2,4,1,6,8,7,5] => [5,7,8,6,1,4,2,3] => [[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[2,3,4,1,6,7,8,5] => [5,8,7,6,1,4,3,2] => [[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[2,1,4,3,8,7,6,5] => [5,6,7,8,3,4,1,2] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[2,1,4,3,6,8,7,5] => [5,7,8,6,3,4,1,2] => [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
[3,2,5,4,1,8,7,6] => [6,7,8,1,4,5,2,3] => [[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3
[3,2,1,4,5,8,7,6] => [6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3
[2,1,4,6,5,3,8,7] => [7,8,3,5,6,4,1,2] => [[[.,[.,.]],[[.,.],[.,.]]],[.,.]]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[4,3,2,1,6,5,8,7] => [7,8,5,6,1,2,3,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000297
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%
Mp00130: Permutations —descent tops⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => => => ? = 1 - 1
[1,2] => [1,2] => 0 => 0 => 0 = 1 - 1
[2,1] => [2,1] => 1 => 1 => 1 = 2 - 1
[1,2,3] => [1,2,3] => 00 => 00 => 0 = 1 - 1
[1,3,2] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[2,1,3] => [2,1,3] => 10 => 01 => 0 = 1 - 1
[2,3,1] => [1,3,2] => 01 => 10 => 1 = 2 - 1
[3,1,2] => [3,1,2] => 01 => 10 => 1 = 2 - 1
[3,2,1] => [3,2,1] => 11 => 11 => 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,3,2,4] => [1,3,2,4] => 010 => 010 => 0 = 1 - 1
[1,3,4,2] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[1,4,2,3] => [1,4,2,3] => 001 => 100 => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[2,1,3,4] => [2,1,3,4] => 100 => 001 => 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[2,3,1,4] => [1,3,2,4] => 010 => 010 => 0 = 1 - 1
[2,3,4,1] => [1,2,4,3] => 001 => 100 => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => 001 => 100 => 1 = 2 - 1
[2,4,3,1] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[3,1,2,4] => [3,1,2,4] => 010 => 010 => 0 = 1 - 1
[3,1,4,2] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => 110 => 011 => 0 = 1 - 1
[3,2,4,1] => [2,1,4,3] => 101 => 101 => 1 = 2 - 1
[3,4,1,2] => [2,4,1,3] => 001 => 100 => 1 = 2 - 1
[3,4,2,1] => [1,4,3,2] => 011 => 110 => 2 = 3 - 1
[4,1,2,3] => [4,1,2,3] => 001 => 100 => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => 011 => 110 => 2 = 3 - 1
[4,2,1,3] => [4,2,1,3] => 101 => 101 => 1 = 2 - 1
[4,2,3,1] => [4,1,3,2] => 011 => 110 => 2 = 3 - 1
[4,3,1,2] => [4,3,1,2] => 011 => 110 => 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => 111 => 111 => 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,5,3,4] => [1,2,5,3,4] => 0001 => 1000 => 1 = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0010 => 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,3,4,2,5] => [1,2,4,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,5,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,3,5,4,2] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[1,4,2,3,5] => [1,4,2,3,5] => 0010 => 0100 => 0 = 1 - 1
[1,4,2,5,3] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => 0110 => 0110 => 0 = 1 - 1
[1,4,3,5,2] => [1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,4,5,2,3] => [1,3,5,2,4] => 0001 => 1000 => 1 = 2 - 1
[1,4,5,3,2] => [1,2,5,4,3] => 0011 => 1100 => 2 = 3 - 1
[7,6,8,5,4,3,2,1] => ? => ? => ? => ? = 6 - 1
[7,8,5,6,4,3,2,1] => ? => ? => ? => ? = 6 - 1
[7,8,6,4,5,3,2,1] => ? => ? => ? => ? = 6 - 1
[8,6,7,4,5,3,2,1] => ? => ? => ? => ? = 6 - 1
[7,6,5,4,8,3,2,1] => ? => ? => ? => ? = 4 - 1
[6,5,7,4,8,3,2,1] => ? => ? => ? => ? = 4 - 1
[7,8,6,5,3,4,2,1] => ? => ? => ? => ? = 6 - 1
[8,6,7,5,3,4,2,1] => ? => ? => ? => ? = 6 - 1
[6,7,8,3,4,5,2,1] => ? => ? => ? => ? = 4 - 1
[7,8,6,5,4,2,3,1] => ? => ? => ? => ? = 6 - 1
[8,7,5,6,4,2,3,1] => ? => ? => ? => ? = 6 - 1
[8,7,6,4,5,2,3,1] => ? => ? => ? => ? = 6 - 1
[6,7,8,5,2,3,4,1] => ? => ? => ? => ? = 4 - 1
[7,8,3,2,4,5,6,1] => [5,8,2,1,3,4,7,6] => ? => ? => ? = 3 - 1
[7,6,5,4,3,2,8,1] => ? => ? => ? => ? = 2 - 1
[6,5,7,4,3,2,8,1] => ? => ? => ? => ? = 2 - 1
[6,5,4,3,7,2,8,1] => ? => ? => ? => ? = 2 - 1
[3,4,5,2,6,7,8,1] => ? => ? => ? => ? = 2 - 1
[5,2,3,4,6,7,8,1] => ? => ? => ? => ? = 2 - 1
[4,3,2,5,6,7,8,1] => ? => ? => ? => ? = 2 - 1
[3,4,2,5,6,7,8,1] => ? => ? => ? => ? = 2 - 1
[4,2,3,5,6,7,8,1] => ? => ? => ? => ? = 2 - 1
[3,2,4,5,6,7,8,1] => ? => ? => ? => ? = 2 - 1
[8,7,6,5,4,3,1,2] => ? => ? => ? => ? = 7 - 1
[8,6,7,5,4,3,1,2] => ? => ? => ? => ? = 6 - 1
[8,7,5,6,4,3,1,2] => ? => ? => ? => ? = 6 - 1
[8,7,6,4,5,3,1,2] => ? => ? => ? => ? = 6 - 1
[8,7,6,5,3,4,1,2] => ? => ? => ? => ? = 6 - 1
[8,7,5,6,3,4,1,2] => ? => ? => ? => ? = 5 - 1
[8,7,6,5,4,1,2,3] => ? => ? => ? => ? = 6 - 1
[8,5,6,7,4,1,2,3] => ? => ? => ? => ? = 4 - 1
[8,7,6,4,5,1,2,3] => ? => ? => ? => ? = 5 - 1
[8,6,7,4,5,1,2,3] => ? => ? => ? => ? = 4 - 1
[8,7,4,5,6,1,2,3] => ? => ? => ? => ? = 4 - 1
[7,8,4,5,6,1,2,3] => ? => ? => ? => ? = 3 - 1
[6,7,4,5,8,1,2,3] => ? => ? => ? => ? = 2 - 1
[7,8,5,4,1,2,3,6] => [6,8,5,4,1,2,3,7] => ? => ? => ? = 2 - 1
[8,7,1,2,3,4,5,6] => ? => ? => ? => ? = 3 - 1
[8,2,3,4,1,5,6,7] => ? => ? => ? => ? = 2 - 1
[8,4,1,2,3,5,6,7] => ? => ? => ? => ? = 2 - 1
[8,3,2,1,4,5,6,7] => ? => ? => ? => ? = 2 - 1
[8,2,3,1,4,5,6,7] => ? => ? => ? => ? = 2 - 1
[8,3,1,2,4,5,6,7] => ? => ? => ? => ? = 2 - 1
[8,2,1,3,4,5,6,7] => ? => ? => ? => ? = 2 - 1
[6,7,5,4,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[6,5,7,4,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[5,6,7,4,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[7,6,4,5,3,2,1,8] => ? => ? => ? => ? = 1 - 1
[6,5,4,7,3,2,1,8] => ? => ? => ? => ? = 1 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000745
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [[1]]
=> [[1]]
=> 1
[1,2] => [1,2] => [[1,2]]
=> [[1,2]]
=> 1
[2,1] => [2,1] => [[1],[2]]
=> [[1],[2]]
=> 2
[1,2,3] => [1,2,3] => [[1,2,3]]
=> [[1,2,3]]
=> 1
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[3,1,2] => [3,1,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> [[1],[2],[3]]
=> 3
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> [[1,2,3,4]]
=> 1
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 2
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 1
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 2
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 2
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 3
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 2
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 1
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 2
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 2
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 3
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 1
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 1
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 2
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 2
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 3
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 2
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 3
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> 2
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 3
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 3
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 1
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 2
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 1
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 2
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 1
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> 1
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 2
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 2
[7,6,8,5,4,3,2,1] => ? => ?
=> ?
=> ? = 6
[7,8,5,6,4,3,2,1] => ? => ?
=> ?
=> ? = 6
[7,8,6,4,5,3,2,1] => ? => ?
=> ?
=> ? = 6
[8,6,7,4,5,3,2,1] => ? => ?
=> ?
=> ? = 6
[7,6,5,4,8,3,2,1] => ? => ?
=> ?
=> ? = 4
[6,5,7,4,8,3,2,1] => ? => ?
=> ?
=> ? = 4
[7,8,6,5,3,4,2,1] => ? => ?
=> ?
=> ? = 6
[8,6,7,5,3,4,2,1] => ? => ?
=> ?
=> ? = 6
[6,7,8,3,4,5,2,1] => ? => ?
=> ?
=> ? = 4
[7,8,6,5,4,2,3,1] => ? => ?
=> ?
=> ? = 6
[8,7,5,6,4,2,3,1] => ? => ?
=> ?
=> ? = 6
[8,7,6,4,5,2,3,1] => ? => ?
=> ?
=> ? = 6
[8,6,5,7,3,2,4,1] => [8,4,3,7,2,1,6,5] => ?
=> ?
=> ? = 4
[6,7,8,5,2,3,4,1] => ? => ?
=> ?
=> ? = 4
[8,5,6,7,2,3,4,1] => [8,2,4,7,1,3,6,5] => ?
=> ?
=> ? = 4
[7,8,3,2,4,5,6,1] => [5,8,2,1,3,4,7,6] => ?
=> ?
=> ? = 3
[7,6,5,4,3,2,8,1] => ? => ?
=> ?
=> ? = 2
[6,5,7,4,3,2,8,1] => ? => ?
=> ?
=> ? = 2
[6,5,4,3,7,2,8,1] => ? => ?
=> ?
=> ? = 2
[3,4,5,2,6,7,8,1] => ? => ?
=> ?
=> ? = 2
[5,2,3,4,6,7,8,1] => ? => ?
=> ?
=> ? = 2
[4,3,2,5,6,7,8,1] => ? => ?
=> ?
=> ? = 2
[3,4,2,5,6,7,8,1] => ? => ?
=> ?
=> ? = 2
[4,2,3,5,6,7,8,1] => ? => ?
=> ?
=> ? = 2
[3,2,4,5,6,7,8,1] => ? => ?
=> ?
=> ? = 2
[8,7,6,5,4,3,1,2] => ? => ?
=> ?
=> ? = 7
[8,6,7,5,4,3,1,2] => ? => ?
=> ?
=> ? = 6
[8,7,5,6,4,3,1,2] => ? => ?
=> ?
=> ? = 6
[8,7,6,4,5,3,1,2] => ? => ?
=> ?
=> ? = 6
[8,7,6,5,3,4,1,2] => ? => ?
=> ?
=> ? = 6
[8,7,5,6,3,4,1,2] => ? => ?
=> ?
=> ? = 5
[7,8,3,4,5,6,1,2] => [5,8,1,2,4,7,3,6] => ?
=> ?
=> ? = 3
[8,7,6,5,4,1,2,3] => ? => ?
=> ?
=> ? = 6
[8,5,6,7,4,1,2,3] => ? => ?
=> ?
=> ? = 4
[8,7,6,4,5,1,2,3] => ? => ?
=> ?
=> ? = 5
[8,6,7,4,5,1,2,3] => ? => ?
=> ?
=> ? = 4
[8,7,4,5,6,1,2,3] => ? => ?
=> ?
=> ? = 4
[7,8,4,5,6,1,2,3] => ? => ?
=> ?
=> ? = 3
[6,7,4,5,8,1,2,3] => ? => ?
=> ?
=> ? = 2
[7,5,6,8,2,3,1,4] => [6,2,5,8,1,4,3,7] => ?
=> ?
=> ? = 2
[6,5,7,8,2,1,3,4] => [4,3,6,8,2,1,5,7] => ?
=> ?
=> ? = 2
[7,8,5,6,1,2,3,4] => [5,8,4,7,1,2,3,6] => ?
=> ?
=> ? = 3
[7,8,5,4,1,2,3,6] => [6,8,5,4,1,2,3,7] => ?
=> ?
=> ? = 2
[8,7,1,2,3,4,5,6] => ? => ?
=> ?
=> ? = 3
[8,5,6,3,4,1,2,7] => [8,3,6,2,5,1,4,7] => ?
=> ?
=> ? = 2
[8,2,3,4,1,5,6,7] => ? => ?
=> ?
=> ? = 2
[8,4,1,2,3,5,6,7] => ? => ?
=> ?
=> ? = 2
[8,3,2,1,4,5,6,7] => ? => ?
=> ?
=> ? = 2
[8,2,3,1,4,5,6,7] => ? => ?
=> ?
=> ? = 2
[8,3,1,2,4,5,6,7] => ? => ?
=> ?
=> ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000925
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 58%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 58%
Values
[1] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,2] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[2,1] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 3
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 2
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3
[8,7,6,5,4,3,2,1] => [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},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 6
[7,8,5,6,4,3,2,1] => [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},{5,6},{7,8}}
=> ? = 6
[7,8,6,4,5,3,2,1] => [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},{4,5},{6},{7,8}}
=> ? = 6
[8,6,7,4,5,3,2,1] => [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},{2},{3},{4,5},{6,7},{8}}
=> ? = 6
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 4
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2},{3},{4,5,6,7,8}}
=> ? = 4
[7,8,6,5,3,4,2,1] => [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},{3,4},{5},{6},{7,8}}
=> ? = 6
[8,6,7,5,3,4,2,1] => [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},{3,4},{5},{6,7},{8}}
=> ? = 6
[8,7,5,6,3,4,2,1] => [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},{2},{3,4},{5,6},{7},{8}}
=> ? = 6
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5},{6,7,8}}
=> ? = 4
[7,8,6,5,4,2,3,1] => [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},{2,3},{4},{5},{6},{7,8}}
=> ? = 6
[8,6,7,5,4,2,3,1] => [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},{2,3},{4},{5},{6,7},{8}}
=> ? = 6
[8,7,5,6,4,2,3,1] => [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},{2,3},{4},{5,6},{7},{8}}
=> ? = 6
[8,7,6,4,5,2,3,1] => [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},{2,3},{4,5},{6},{7},{8}}
=> ? = 6
[8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 4
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> {{1},{2,3,4},{5},{6,7,8}}
=> ? = 4
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 4
[7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> {{1},{2,3,4,5,6},{7,8}}
=> ? = 3
[8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 7
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 6
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4},{5},{6,7},{8}}
=> ? = 6
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4},{5,6},{7},{8}}
=> ? = 6
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4,5},{6},{7},{8}}
=> ? = 6
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 6
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 5
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 4
[5,6,7,8,3,4,1,2] => [2,1,4,3,8,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4},{5,6,7,8}}
=> ? = 3
[7,8,3,4,5,6,1,2] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2},{3,4,5,6},{7,8}}
=> ? = 3
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 6
[7,6,8,5,4,2,1,3] => [3,1,2,4,5,8,6,7] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 4
[8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 6
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 4
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> {{1,2,3},{4},{5,6,7},{8}}
=> ? = 4
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 5
[8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 4
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 4
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 3
[6,7,4,5,8,1,2,3] => [3,2,1,8,5,4,7,6] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 2
[7,6,5,8,3,2,1,4] => [4,1,2,3,8,5,6,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[7,5,6,8,2,3,1,4] => [4,1,3,2,8,6,5,7] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[6,7,5,8,3,1,2,4] => [4,2,1,3,8,5,7,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[6,5,7,8,2,1,3,4] => [4,3,1,2,8,7,5,6] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 5
[8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 4
[7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 3
[8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 3
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
Description
The number of topologically connected components of a set partition.
For example, the set partition {{1,5},{2,3},{4,6}} has the two connected components {1,4,5,6} and {2,3}.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000546
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 83%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [.,.]
=> [1] => 0 = 1 - 1
[1,2] => [2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [.,[.,.]]
=> [2,1] => 1 = 2 - 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[1,3,2] => [2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 1 = 2 - 1
[2,1,3] => [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 0 = 1 - 1
[2,3,1] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1 = 2 - 1
[3,1,2] => [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 1 = 2 - 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 2 - 1
[1,4,3,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 3 - 1
[2,1,3,4] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 2 - 1
[2,4,3,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 3 - 1
[3,1,2,4] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1 = 2 - 1
[3,2,1,4] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2 = 3 - 1
[4,1,2,3] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 3 - 1
[4,2,1,3] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2 = 3 - 1
[4,3,1,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1 = 2 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0 = 1 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1 = 2 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2 = 3 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 0 = 1 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1 = 2 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2 = 3 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0 = 1 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1 = 2 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1 = 2 - 1
[1,4,6,7,2,3,5] => [5,3,2,7,6,4,1] => [[[[.,.],.],[.,.]],[[.,.],.]]
=> [6,7,4,1,2,3,5] => ? = 2 - 1
[1,4,6,7,5,3,2] => [2,3,5,7,6,4,1] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 4 - 1
[1,4,7,6,5,3,2] => [2,3,5,6,7,4,1] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 5 - 1
[1,5,4,7,6,3,2] => [2,3,6,7,4,5,1] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 4 - 1
[1,5,6,4,7,3,2] => [2,3,7,4,6,5,1] => [[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => ? = 3 - 1
[1,5,6,7,4,3,2] => [2,3,4,7,6,5,1] => [[.,.],[.,[.,[[[.,.],.],.]]]]
=> [5,6,7,4,3,1,2] => ? = 4 - 1
[1,5,7,6,4,3,2] => [2,3,4,6,7,5,1] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 5 - 1
[1,6,5,4,3,7,2] => [2,7,3,4,5,6,1] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [6,5,4,3,7,1,2] => ? = 2 - 1
[1,6,5,4,7,3,2] => [2,3,7,4,5,6,1] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,3,1,2] => ? = 3 - 1
[1,6,5,7,4,3,2] => [2,3,4,7,5,6,1] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,3,1,2] => ? = 4 - 1
[1,6,7,5,4,3,2] => [2,3,4,5,7,6,1] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 5 - 1
[1,7,4,3,6,5,2] => [2,5,6,3,4,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 4 - 1
[1,7,4,5,3,6,2] => [2,6,3,5,4,7,1] => [[.,.],[[.,[[.,.],.]],[.,.]]]
=> [7,4,5,3,6,1,2] => ? = 3 - 1
[1,7,4,5,6,3,2] => [2,3,6,5,4,7,1] => [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [7,4,5,6,3,1,2] => ? = 4 - 1
[1,7,4,6,5,3,2] => [2,3,5,6,4,7,1] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 5 - 1
[1,7,5,4,3,6,2] => [2,6,3,4,5,7,1] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 3 - 1
[1,7,5,4,6,3,2] => [2,3,6,4,5,7,1] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 4 - 1
[1,7,6,4,5,3,2] => [2,3,5,4,6,7,1] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 5 - 1
[2,1,6,3,4,5,7] => [7,5,4,3,6,1,2] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 5 - 1
[2,3,4,6,1,7,5] => [5,7,1,6,4,3,2] => [[.,[[[.,.],.],.]],[[.,.],.]]
=> [6,7,2,3,4,1,5] => ? = 2 - 1
[2,3,4,6,7,1,5] => [5,1,7,6,4,3,2] => [[.,[[[.,.],.],.]],[[.,.],.]]
=> [6,7,2,3,4,1,5] => ? = 2 - 1
[2,3,5,6,7,1,4] => [4,1,7,6,5,3,2] => [[.,[[.,.],.]],[[[.,.],.],.]]
=> [5,6,7,2,3,1,4] => ? = 2 - 1
[2,4,1,3,5,6,7] => [7,6,5,3,1,4,2] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[2,5,1,3,4,6,7] => [7,6,4,3,1,5,2] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[2,5,7,1,3,4,6] => [6,4,3,1,7,5,2] => [[[[.,[.,.]],.],[.,.]],[.,.]]
=> [7,5,2,1,3,4,6] => ? = 2 - 1
[2,6,1,3,4,5,7] => [7,5,4,3,1,6,2] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[2,6,1,7,3,4,5] => [5,4,3,7,1,6,2] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [6,7,2,1,3,4,5] => ? = 2 - 1
[2,6,5,4,3,1,7] => [7,1,3,4,5,6,2] => [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => ? = 1 - 1
[2,6,7,1,3,4,5] => [5,4,3,1,7,6,2] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [6,7,2,1,3,4,5] => ? = 2 - 1
[3,5,6,1,2,4,7] => [7,4,2,1,6,5,3] => [[[[.,.],[.,.]],[[.,.],.]],.]
=> [5,6,3,1,2,4,7] => ? = 1 - 1
[3,5,6,4,2,1,7] => [7,1,2,4,6,5,3] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [5,6,3,4,2,1,7] => ? = 1 - 1
[3,6,1,2,4,5,7] => [7,5,4,2,1,6,3] => [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => ? = 1 - 1
[4,1,6,2,3,5,7] => [7,5,3,2,6,1,4] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 1 - 1
[4,2,5,1,3,6,7] => [7,6,3,1,5,2,4] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [4,5,2,1,3,6,7] => ? = 1 - 1
[4,6,1,2,3,5,7] => [7,5,3,2,1,6,4] => [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => ? = 1 - 1
[4,6,1,2,3,7,5] => [5,7,3,2,1,6,4] => [[[[.,.],.],[.,.]],[[.,.],.]]
=> [6,7,4,1,2,3,5] => ? = 2 - 1
[4,6,5,3,2,1,7] => [7,1,2,3,5,6,4] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [6,4,5,3,2,1,7] => ? = 1 - 1
[5,2,6,7,1,3,4] => [4,3,1,7,6,2,5] => [[[.,[.,.]],.],[[[.,.],.],.]]
=> [5,6,7,2,1,3,4] => ? = 2 - 1
[5,3,2,6,1,4,7] => [7,4,1,6,2,3,5] => [[[.,[.,[.,.]]],[[.,.],.]],.]
=> [5,6,3,2,1,4,7] => ? = 1 - 1
[5,3,6,7,1,2,4] => [4,2,1,7,6,3,5] => [[[.,.],[.,.]],[[[.,.],.],.]]
=> [5,6,7,3,1,2,4] => ? = 2 - 1
[5,4,3,2,1,7,6] => [6,7,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [7,5,4,3,2,1,6] => ? = 2 - 1
[6,2,4,5,3,1,7] => [7,1,3,5,4,2,6] => [[.,[[.,.],[[.,.],[.,.]]]],.]
=> [6,4,5,2,3,1,7] => ? = 1 - 1
[6,2,5,4,3,1,7] => [7,1,3,4,5,2,6] => [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => ? = 1 - 1
[6,3,2,5,4,1,7] => [7,1,4,5,2,3,6] => [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [6,5,3,2,4,1,7] => ? = 1 - 1
[6,4,3,2,7,1,5] => [5,1,7,2,3,4,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [6,7,4,3,2,1,5] => ? = 2 - 1
[6,4,7,2,5,1,3] => [3,1,5,2,7,4,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => ? = 3 - 1
[6,4,7,3,5,1,2] => [2,1,5,3,7,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ? = 3 - 1
[6,5,8,7,4,3,2,1] => [1,2,3,4,7,8,5,6] => [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [8,6,5,7,4,3,2,1] => ? = 6 - 1
[4,3,6,5,8,7,2,1] => [1,2,7,8,5,6,3,4] => [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [8,6,4,3,5,7,2,1] => ? = 4 - 1
Description
The number of global descents of a permutation.
The global descents are the integers in the set
C(π)={i∈[n−1]:∀1≤j≤i<k≤n:π(j)>π(k)}.
In particular, if i∈C(π) then i is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St000069
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 92%
St000069: Posets ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 92%
Values
[1] => ([],1)
=> 1
[1,2] => ([(0,1)],2)
=> 1
[2,1] => ([],2)
=> 2
[1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,1,3] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => ([(1,2)],3)
=> 2
[3,1,2] => ([(1,2)],3)
=> 2
[3,2,1] => ([],3)
=> 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> 3
[4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> 3
[4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> 3
[4,3,1,2] => ([(2,3)],4)
=> 3
[4,3,2,1] => ([],4)
=> 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,2,6,3,4,7,5] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 2
[1,3,7,2,4,5,6] => ([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
=> ? = 2
[1,4,6,7,2,3,5] => ([(0,4),(0,5),(2,6),(3,1),(4,2),(5,3),(5,6)],7)
=> ? = 2
[1,4,7,2,3,5,6] => ([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
=> ? = 2
[1,5,2,3,4,7,6] => ([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 2
[1,5,2,7,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(2,6),(3,1),(4,3),(4,6)],7)
=> ? = 2
[1,5,6,2,7,3,4] => ([(0,4),(0,5),(2,6),(3,1),(4,2),(5,3),(5,6)],7)
=> ? = 2
[1,5,7,2,3,4,6] => ([(0,4),(0,5),(2,6),(3,2),(4,3),(5,1),(5,6)],7)
=> ? = 2
[1,6,2,3,4,7,5] => ([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
=> ? = 2
[1,6,2,3,7,4,5] => ([(0,2),(0,5),(2,6),(3,1),(4,3),(4,6),(5,4)],7)
=> ? = 2
[2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 2
[2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 2
[2,3,4,6,7,1,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7)
=> ? = 2
[2,3,5,6,7,1,4] => ([(0,6),(1,5),(3,4),(4,2),(5,3),(5,6)],7)
=> ? = 2
[2,5,1,3,4,6,7] => ([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> ? = 1
[2,5,7,1,3,4,6] => ([(0,6),(1,4),(1,6),(3,5),(4,2),(4,5),(6,3)],7)
=> ? = 2
[2,6,1,3,4,5,7] => ([(0,6),(1,3),(1,6),(2,5),(3,5),(4,2),(6,4)],7)
=> ? = 1
[2,6,1,7,3,4,5] => ([(0,5),(0,6),(1,3),(1,6),(3,5),(4,2),(6,4)],7)
=> ? = 2
[3,1,2,4,5,7,6] => ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ? = 2
[3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[3,5,6,1,2,4,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,2),(6,5)],7)
=> ? = 1
[3,6,1,2,4,5,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,5),(6,2)],7)
=> ? = 1
[4,1,2,3,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 2
[4,1,6,2,3,5,7] => ([(0,2),(0,6),(1,5),(1,6),(2,3),(3,5),(5,4),(6,4)],7)
=> ? = 1
[4,5,1,6,2,3,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,2),(6,5)],7)
=> ? = 1
[4,6,1,2,3,7,5] => ([(0,3),(0,6),(1,4),(2,5),(2,6),(3,5),(4,2)],7)
=> ? = 2
[5,1,2,3,4,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 2
[5,1,2,3,6,4,7] => ([(0,6),(1,4),(2,5),(3,2),(3,6),(4,3),(6,5)],7)
=> ? = 1
[5,1,2,6,3,4,7] => ([(0,6),(1,4),(2,5),(3,2),(4,3),(4,6),(6,5)],7)
=> ? = 1
[5,1,6,2,3,4,7] => ([(0,6),(1,4),(1,6),(2,5),(3,2),(4,3),(6,5)],7)
=> ? = 1
[5,3,6,7,1,2,4] => ([(0,5),(1,5),(1,6),(2,3),(3,6),(5,4)],7)
=> ? = 2
[5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[6,1,2,3,7,4,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7)
=> ? = 2
[6,1,2,7,3,4,5] => ([(0,6),(1,5),(3,4),(4,2),(5,3),(5,6)],7)
=> ? = 2
[6,3,4,7,1,2,5] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 2
[6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 2
[6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[7,6,8,5,4,3,2,1] => ([(5,7),(6,7)],8)
=> ? = 6
[7,6,5,4,8,3,2,1] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[6,5,7,4,8,3,2,1] => ([(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 4
[7,8,3,2,4,5,6,1] => ([(1,7),(2,7),(3,4),(5,6),(7,5)],8)
=> ? = 3
[7,6,5,4,3,2,8,1] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[6,5,7,4,3,2,8,1] => ([(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 2
[6,5,4,3,7,2,8,1] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> ? = 2
[5,4,6,3,7,2,8,1] => ([(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 2
[3,4,5,2,6,7,8,1] => ([(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
=> ? = 2
[5,2,3,4,6,7,8,1] => ([(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
=> ? = 2
Description
The number of maximal elements of a poset.
Matching statistic: St001124
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 42% ●values known / values provided: 51%●distinct values known / distinct values provided: 42%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 42% ●values known / values provided: 51%●distinct values known / distinct values provided: 42%
Values
[1] => [1] => [1,0]
=> []
=> ? = 1 - 2
[1,2] => [2,1] => [1,1,0,0]
=> []
=> ? = 1 - 2
[2,1] => [1,2] => [1,0,1,0]
=> [1]
=> ? = 2 - 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? = 1 - 2
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> ? = 2 - 2
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 1 - 2
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0 = 2 - 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0 = 2 - 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1 = 3 - 2
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 2
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? = 2 - 2
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 2
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0 = 2 - 2
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0 = 2 - 2
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1 = 3 - 2
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 2
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> ? = 2 - 2
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 2
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0 = 2 - 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0 = 2 - 2
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1 = 3 - 2
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0 = 2 - 2
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? = 1 - 2
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0 = 2 - 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0 = 2 - 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1 = 3 - 2
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0 = 2 - 2
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1 = 3 - 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0 = 2 - 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1 = 3 - 2
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1 = 3 - 2
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2 = 4 - 2
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0 = 2 - 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0 = 2 - 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1 = 3 - 2
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0 = 2 - 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0 = 2 - 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1 = 3 - 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0 = 2 - 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0 = 2 - 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0 = 2 - 2
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1 = 3 - 2
[1,5,2,3,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0 = 2 - 2
[1,5,2,4,3] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1 = 3 - 2
[1,5,3,2,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0 = 2 - 2
[1,5,3,4,2] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1 = 3 - 2
[1,5,4,2,3] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1 = 3 - 2
[1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2 = 4 - 2
[2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 2
[2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0 = 2 - 2
[2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0 = 2 - 2
[2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1 = 3 - 2
[2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 2
[2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0 = 2 - 2
[2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0 = 2 - 2
[2,3,5,4,1] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1 = 3 - 2
[2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0 = 2 - 2
[2,4,3,1,5] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0 = 2 - 2
[2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0 = 2 - 2
[2,4,5,3,1] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 1 = 3 - 2
[2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0 = 2 - 2
[2,5,1,4,3] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1 = 3 - 2
[2,5,3,1,4] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0 = 2 - 2
[2,5,3,4,1] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1 = 3 - 2
[2,5,4,1,3] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1 = 3 - 2
[3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[3,1,2,5,4] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 2
[3,1,4,2,5] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[3,2,1,4,5] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? = 2 - 2
[3,2,4,1,5] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[3,4,2,1,5] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[4,1,2,3,5] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[4,1,3,2,5] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[4,2,1,3,5] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[4,2,3,1,5] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[4,3,1,2,5] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[4,3,2,1,5] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,2,3,4,6,5] => [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? = 2 - 2
[1,2,3,5,4,6] => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 1 - 2
[1,2,4,3,6,5] => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? = 2 - 2
[1,2,4,5,3,6] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 1 - 2
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity gλμ,ν of the Specht module Sλ in Sμ⊗Sν:
Sμ⊗Sν=⨁λgλμ,νSλ
This statistic records the Kronecker coefficient g(n−1)1λ,λ, for λ⊢n>1. For n≤1 the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000912The number of maximal antichains in a poset. St000167The number of leaves of an ordered tree. St000068The number of minimal elements in a poset. St000031The number of cycles in the cycle decomposition of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000105The number of blocks in the set partition. St000678The number of up steps after the last double rise of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000054The first entry of the permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000740The last entry of a permutation. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St000990The first ascent of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000015The number of peaks of a Dyck path. St000133The "bounce" of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000239The number of small weak excedances. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000061The number of nodes on the left branch of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000702The number of weak deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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