Processing math: 100%

Your data matches 84 different statistics following compositions of up to 3 maps.
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Mp00069: Permutations complementPermutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [2,1] => 2
[2,1] => [1,2] => 1
[1,2,3] => [3,2,1] => 3
[1,3,2] => [3,1,2] => 2
[2,1,3] => [2,3,1] => 2
[2,3,1] => [2,1,3] => 1
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,2,3] => 1
[1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3] => [4,3,1,2] => 3
[1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2] => [4,2,1,3] => 2
[1,4,2,3] => [4,1,3,2] => 3
[1,4,3,2] => [4,1,2,3] => 2
[2,1,3,4] => [3,4,2,1] => 3
[2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [3,2,4,1] => 2
[2,3,4,1] => [3,2,1,4] => 1
[2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [2,4,3,1] => 3
[3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [2,3,4,1] => 2
[3,2,4,1] => [2,3,1,4] => 1
[3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [1,4,3,2] => 3
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,3,4,2] => 2
[4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => 2
[4,3,2,1] => [1,2,3,4] => 1
[1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [5,4,3,1,2] => 4
[1,2,4,3,5] => [5,4,2,3,1] => 4
[1,2,4,5,3] => [5,4,2,1,3] => 3
[1,2,5,3,4] => [5,4,1,3,2] => 4
[1,2,5,4,3] => [5,4,1,2,3] => 3
[1,3,2,4,5] => [5,3,4,2,1] => 4
[1,3,2,5,4] => [5,3,4,1,2] => 3
[1,3,4,2,5] => [5,3,2,4,1] => 3
[1,3,4,5,2] => [5,3,2,1,4] => 2
[1,3,5,2,4] => [5,3,1,4,2] => 3
[1,3,5,4,2] => [5,3,1,2,4] => 2
[1,4,2,3,5] => [5,2,4,3,1] => 4
[1,4,2,5,3] => [5,2,4,1,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => 3
[1,4,3,5,2] => [5,2,3,1,4] => 2
[1,4,5,2,3] => [5,2,1,4,3] => 3
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].
Mp00066: Permutations inversePermutations
Mp00237: Permutations descent views to invisible inversion bottomsPermutations
Mp00108: Permutations cycle typeInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [1,2] => [1,1]
=> 2
[2,1] => [2,1] => [2,1] => [2]
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,3,2] => [1,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1,3] => [2,1,3] => [2,1]
=> 2
[2,3,1] => [3,1,2] => [3,1,2] => [3]
=> 1
[3,1,2] => [2,3,1] => [3,2,1] => [2,1]
=> 2
[3,2,1] => [3,2,1] => [2,3,1] => [3]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 3
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 3
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [3,1]
=> 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [3,1]
=> 2
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [4]
=> 1
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[2,4,3,1] => [4,1,3,2] => [4,3,1,2] => [4]
=> 1
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 3
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => [3,1]
=> 2
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,1]
=> 2
[3,2,4,1] => [4,2,1,3] => [2,4,1,3] => [4]
=> 1
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [3,1]
=> 2
[3,4,2,1] => [4,3,1,2] => [3,1,4,2] => [4]
=> 1
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 3
[4,1,3,2] => [2,4,3,1] => [3,2,4,1] => [3,1]
=> 2
[4,2,1,3] => [3,2,4,1] => [4,3,2,1] => [2,2]
=> 2
[4,2,3,1] => [4,2,3,1] => [3,4,2,1] => [4]
=> 1
[4,3,1,2] => [3,4,2,1] => [2,4,3,1] => [3,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [4]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [3,1,1]
=> 3
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [3,1,1]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [3,1,1]
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,4,5,2,3] => [2,2,1]
=> 3
[1,3,5,4,2] => [1,5,2,4,3] => [1,5,4,2,3] => [4,1]
=> 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 4
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,3,2,4] => [3,1,1]
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [3,1,1]
=> 3
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,2,4] => [4,1]
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => [3,1,1]
=> 3
[7,5,6,8,3,4,1,2] => [7,8,5,6,2,3,1,4] => [3,6,2,1,8,5,7,4] => ?
=> ? = 2
[7,8,5,3,4,6,1,2] => [7,8,4,5,3,6,1,2] => [6,1,5,8,4,3,7,2] => ?
=> ? = 2
[7,8,5,6,3,1,2,4] => [6,7,5,8,3,4,1,2] => [4,1,8,3,7,6,5,2] => ?
=> ? = 3
[8,7,3,2,1,4,5,6] => [5,4,3,6,7,8,2,1] => [2,8,4,5,3,6,7,1] => ?
=> ? = 4
[8,3,4,5,6,1,2,7] => [6,7,2,3,4,5,8,1] => [8,7,2,3,4,6,5,1] => ?
=> ? = 3
[8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => [8,5,2,3,4,6,7,1] => ?
=> ? = 4
[6,5,4,7,3,2,1,8] => [7,6,5,3,2,1,4,8] => [2,3,5,1,6,7,4,8] => ?
=> ? = 2
[6,7,4,5,2,3,1,8] => [7,5,6,3,4,1,2,8] => [4,1,6,3,7,5,2,8] => ?
=> ? = 2
[5,6,4,7,2,3,1,8] => [7,5,6,3,1,2,4,8] => [3,1,6,2,7,5,4,8] => ?
=> ? = 2
[6,7,3,4,2,5,1,8] => [7,5,3,4,6,1,2,8] => [6,1,5,3,7,4,2,8] => ?
=> ? = 2
[5,6,3,4,2,7,1,8] => [7,5,3,4,1,2,6,8] => [4,1,5,3,7,2,6,8] => ?
=> ? = 2
[3,4,5,6,2,7,1,8] => [7,5,1,2,3,4,6,8] => [5,1,2,3,7,4,6,8] => ?
=> ? = 2
[6,7,3,4,5,1,2,8] => [6,7,3,4,5,1,2,8] => [5,1,7,3,4,6,2,8] => ?
=> ? = 3
[7,6,4,3,1,2,5,8] => [5,6,4,3,7,2,1,8] => [2,7,4,6,5,3,1,8] => ?
=> ? = 4
[7,3,1,2,4,5,6,8] => [3,4,2,5,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 6
[3,4,2,5,1,6,7,8] => [5,3,1,2,4,6,7,8] => [3,1,5,2,4,6,7,8] => ?
=> ? = 4
[3,4,2,6,7,5,8,1] => [8,3,1,2,6,4,5,7] => [3,1,8,6,4,2,5,7] => ?
=> ? = 1
[2,4,5,6,3,7,8,1] => [8,1,5,2,3,4,6,7] => [8,5,2,3,1,4,6,7] => ?
=> ? = 1
[1,6,8,7,5,4,3,2] => [1,8,7,6,5,2,4,3] => [1,5,4,2,6,7,8,3] => ?
=> ? = 2
[1,7,6,8,5,4,3,2] => [1,8,7,6,5,3,2,4] => [1,3,5,2,6,7,8,4] => ?
=> ? = 2
[1,5,8,7,6,4,3,2] => [1,8,7,6,2,5,4,3] => [1,6,4,5,2,7,8,3] => ?
=> ? = 2
[1,7,6,5,8,4,3,2] => [1,8,7,6,4,3,2,5] => [1,3,4,6,2,7,8,5] => ?
=> ? = 2
[2,1,4,5,7,8,6,3] => [2,1,8,3,4,7,5,6] => [2,1,8,3,7,5,4,6] => ?
=> ? = 2
[2,1,5,4,3,8,7,6] => [2,1,5,4,3,8,7,6] => ? => ?
=> ? = 3
[5,7,6,4,3,2,1,8] => [7,6,5,4,1,3,2,8] => [4,3,1,5,6,7,2,8] => ?
=> ? = 2
[4,7,6,5,3,2,1,8] => [7,6,5,1,4,3,2,8] => [5,3,4,1,6,7,2,8] => ?
=> ? = 2
[1,3,8,6,5,7,4,2] => [1,8,2,7,5,4,6,3] => [1,8,6,5,7,4,2,3] => ?
=> ? = 2
[2,3,6,4,5,7,8,1] => [8,1,2,4,5,3,6,7] => [8,1,5,2,4,3,6,7] => ?
=> ? = 1
[7,2,3,4,6,5,1,8] => [7,2,3,4,6,5,1,8] => [5,6,2,3,7,1,4,8] => ?
=> ? = 2
[8,2,7,5,6,4,3,1] => [8,2,7,6,4,5,3,1] => [3,5,8,6,4,7,1,2] => ?
=> ? = 1
[6,3,5,4,2,7,1,8] => [7,5,2,4,3,1,6,8] => [3,4,5,1,7,2,6,8] => ?
=> ? = 2
[2,4,6,1,7,3,8,5] => [4,1,6,2,8,3,5,7] => [4,6,8,1,3,2,5,7] => ?
=> ? = 3
[3,4,1,5,7,2,6,8] => [3,6,1,2,4,7,5,8] => [6,1,3,2,7,4,5,8] => ?
=> ? = 4
[1,2,4,7,3,5,6,8] => [1,2,5,3,6,7,4,8] => [1,2,5,7,3,6,4,8] => ?
=> ? = 6
[1,2,3,6,4,8,5,7] => [1,2,3,5,7,4,8,6] => [1,2,3,7,5,8,4,6] => ?
=> ? = 6
[2,1,4,3,6,5,10,9,8,7] => ? => ? => ?
=> ? = 4
[2,1,6,5,4,3,10,9,8,7] => ? => ? => ?
=> ? = 3
[2,1,4,3,10,9,8,7,6,5] => ? => ? => ?
=> ? = 3
[7,5,8,1,2,3,4,6] => [4,5,6,7,2,8,1,3] => [8,7,1,4,5,6,2,3] => ?
=> ? = 5
[3,8,2,7,1,4,5,6] => [5,3,1,6,7,8,4,2] => [3,5,4,8,2,6,7,1] => ?
=> ? = 4
[7,8,4,6,1,2,3,5] => [5,6,7,3,8,4,1,2] => [4,1,8,7,5,6,2,3] => ?
=> ? = 4
[6,8,4,7,3,5,1,2] => [7,8,5,3,6,1,4,2] => [6,4,5,1,8,3,7,2] => ?
=> ? = 2
[7,8,3,6,2,5,1,4] => [7,5,3,8,6,4,1,2] => [4,1,6,5,8,7,3,2] => ?
=> ? = 2
[7,8,4,6,2,5,1,3] => [7,5,8,3,6,4,1,2] => [4,1,6,8,7,2,5,3] => ?
=> ? = 2
[7,8,4,6,3,5,1,2] => [7,8,5,3,6,4,1,2] => [4,1,6,5,8,2,7,3] => ?
=> ? = 2
[8,1,2,5,3,4,6,7] => [2,3,5,6,4,7,8,1] => [8,2,3,6,5,4,7,1] => ?
=> ? = 6
[8,1,4,5,2,3,6,7] => [2,5,6,3,4,7,8,1] => [8,2,6,3,5,4,7,1] => ?
=> ? = 5
[9,1,2,5,3,4,6,7,8] => [2,3,5,6,4,7,8,9,1] => [9,2,3,6,5,4,7,8,1] => ?
=> ? = 7
[2,4,1,6,3,8,5,9,7] => [3,1,5,2,7,4,9,6,8] => [3,5,1,7,2,9,4,6,8] => ?
=> ? = 4
[3,5,2,7,4,9,6,10,8,1] => [10,3,1,5,2,7,4,9,6,8] => [3,10,5,7,2,9,1,6,4,8] => ?
=> ? = 1
Description
The length of the partition.
Matching statistic: St000340
Mp00066: Permutations inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00122: Dyck paths Elizalde-Deutsch bijectionDyck paths
St000340: Dyck paths ⟶ ℤResult quality: 67% values known / values provided: 73%distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 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] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 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] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 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]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[6,7,5,4,3,2,1,8] => [7,6,5,4,3,1,2,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[6,5,7,4,3,2,1,8] => [7,6,5,4,2,1,3,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[5,6,7,4,3,2,1,8] => [7,6,5,4,1,2,3,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[6,5,4,7,3,2,1,8] => [7,6,5,3,2,1,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[6,7,4,5,2,3,1,8] => [7,5,6,3,4,1,2,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[5,6,4,7,2,3,1,8] => [7,5,6,3,1,2,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[6,7,3,4,2,5,1,8] => [7,5,3,4,6,1,2,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[5,6,3,4,2,7,1,8] => [7,5,3,4,1,2,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[4,5,3,6,2,7,1,8] => [7,5,3,1,2,4,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[3,4,5,6,2,7,1,8] => [7,5,1,2,3,4,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[3,4,2,5,6,7,1,8] => [7,3,1,2,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[2,3,4,5,6,7,1,8] => [7,1,2,3,4,5,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[6,7,3,4,5,1,2,8] => [6,7,3,4,5,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 3 - 1
[7,6,4,3,1,2,5,8] => [5,6,4,3,7,2,1,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 4 - 1
[7,3,1,2,4,5,6,8] => [3,4,2,5,6,7,1,8] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 4 - 1
[6,4,1,2,3,5,7,8] => [3,4,5,2,6,1,7,8] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 6 - 1
[3,4,2,5,1,6,7,8] => [5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[2,3,4,5,1,6,7,8] => [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 6 - 1
[2,3,1,4,5,6,7,8] => [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
[1,8,7,6,5,4,3,2] => [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
[1,7,8,6,5,4,3,2] => [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
[1,6,8,7,5,4,3,2] => [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
[1,7,6,8,5,4,3,2] => [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
[1,6,7,8,5,4,3,2] => [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
[1,5,8,7,6,4,3,2] => [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
[1,7,6,5,8,4,3,2] => [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
[1,3,5,7,8,6,4,2] => [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
[1,3,5,6,7,8,4,2] => [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
[1,3,4,5,7,8,6,2] => [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
[1,4,3,6,5,8,7,2] => [1,8,3,2,5,4,7,6] => [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
[1,3,4,5,6,8,7,2] => [1,8,2,3,4,5,7,6] => [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
[1,3,4,5,6,7,8,2] => [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
[2,1,8,7,6,5,4,3] => [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
[2,1,4,5,8,7,6,3] => [2,1,8,3,4,7,6,5] => [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
[2,1,4,5,7,8,6,3] => [2,1,8,3,4,7,5,6] => ?
=> ?
=> ? = 2 - 1
[1,2,8,7,6,5,4,3] => [1,2,8,7,6,5,4,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 - 1
[3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [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
[1,3,2,8,7,6,5,4] => [1,3,2,8,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,2,3,8,7,6,5,4] => [1,2,3,8,7,6,5,4] => [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
[1,2,3,5,7,8,6,4] => [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
[1,2,3,5,6,7,8,4] => [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
[4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[3,4,2,1,7,8,6,5] => [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
[2,4,3,1,7,6,8,5] => [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
[3,2,4,1,6,8,7,5] => [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
[2,3,4,1,6,7,8,5] => [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
[1,4,3,2,8,7,6,5] => [1,4,3,2,8,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 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.
Mp00061: Permutations to increasing treeBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 69% values known / values provided: 69%distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 1
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 1
[7,5,6,8,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[7,8,5,3,4,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[7,6,5,4,3,8,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[6,7,4,5,8,1,2,3] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
[7,5,6,8,2,3,1,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
[7,8,5,6,1,2,3,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[6,7,8,3,4,1,2,5] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
[8,5,6,3,4,1,2,7] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[8,2,3,4,1,5,6,7] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 4
[3,4,5,6,2,7,1,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[3,4,2,5,6,7,1,8] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[6,7,3,4,5,1,2,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
[5,6,3,4,1,2,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[3,4,2,5,1,6,7,8] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 4
[3,4,2,6,7,5,8,1] => [[[.,[.,.]],[[.,[.,.]],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,7,6,8,5,4,3,2] => [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2
[1,7,6,5,8,4,3,2] => [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[1,3,5,6,7,8,4,2] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,4,5,8,7,6,3] => [[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[2,1,4,5,7,8,6,3] => [[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[1,2,8,7,6,5,4,3] => [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,3,2,8,7,6,5,4] => [.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[3,4,2,1,7,8,6,5] => [[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[2,4,3,1,7,6,8,5] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[2,3,4,1,6,7,8,5] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,4,3,2,8,7,6,5] => [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[1,2,4,3,8,7,6,5] => [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4
[1,2,3,4,8,7,6,5] => [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[1,2,5,4,3,8,7,6] => [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 4
[1,3,2,5,4,8,7,6] => [.,[[.,.],[[.,.],[[[.,.],.],.]]]]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 4
[1,2,3,5,4,8,7,6] => [.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 5
[1,4,3,2,5,8,7,6] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
[3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[2,1,6,5,4,3,8,7] => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[1,2,4,3,6,5,8,7] => [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5
[1,2,3,4,6,5,8,7] => [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[4,7,6,5,3,2,1,8] => [[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2
[2,1,6,7,5,4,3,8] => [[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 3
[1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[1,3,4,2,6,7,5,8] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 4
[3,2,1,4,7,6,5,8] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[1,2,3,5,4,7,6,8] => [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[1,2,4,3,5,7,6,8] => [.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[3,2,1,6,5,4,7,8] => [[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,2,4,3,6,5,7,8] => [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 6
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.
Mp00061: Permutations to increasing treeBinary trees
Mp00016: Binary trees left-right symmetryBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 67% values known / values provided: 67%distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> [.,.]
=> [1,0]
=> 2 = 1 + 1
[1,2] => [.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> 3 = 2 + 1
[2,1] => [[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,2] => [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[6,7,4,5,8,1,2,3] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[4,5,6,7,8,1,2,3] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[7,8,5,6,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[6,7,5,8,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[7,5,6,8,3,1,2,4] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[6,5,7,8,2,1,3,4] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[6,7,8,3,4,1,2,5] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[6,7,8,1,2,3,4,5] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[8,7,3,2,1,4,5,6] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[8,5,6,3,4,1,2,7] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[8,5,6,1,2,3,4,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[8,6,4,2,1,3,5,7] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[8,2,3,4,1,5,6,7] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[8,3,2,1,4,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[8,2,3,1,4,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[8,2,1,3,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[6,7,3,4,5,1,2,8] => [[[.,[.,.]],[.,[.,.]]],[.,[.,.]]]
=> [[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[7,6,4,3,1,2,5,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[7,3,1,2,4,5,6,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[6,4,1,2,3,5,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[1,3,5,7,8,6,4,2] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 + 1
[1,3,5,6,7,8,4,2] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,3,4,5,7,8,6,2] => [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,4,3,6,5,8,7,2] => [.,[[[.,.],[[.,.],[[.,.],.]]],.]]
=> [[.,[[[.,[.,.]],[.,.]],[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[2,1,4,5,8,7,6,3] => [[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [[.,[[[.,[.,[.,.]]],.],.]],[.,.]]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[2,1,4,5,7,8,6,3] => [[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [[.,[[[.,[[.,.],.]],.],.]],[.,.]]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,2,3,8,7,6,5,4] => [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
[3,4,2,1,7,8,6,5] => [[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[2,4,3,1,7,6,8,5] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[3,2,4,1,6,8,7,5] => [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[2,3,4,1,6,7,8,5] => [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[1,4,3,2,8,7,6,5] => [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 + 1
[2,1,4,3,8,7,6,5] => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,2,4,3,8,7,6,5] => [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 4 + 1
[3,2,5,4,1,8,7,6] => [[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 + 1
[2,1,5,4,3,8,7,6] => [[.,.],[[[.,.],.],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,2,5,4,3,8,7,6] => [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 4 + 1
[1,3,2,5,4,8,7,6] => [.,[[.,.],[[.,.],[[[.,.],.],.]]]]
=> [[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 1
[1,2,3,5,4,8,7,6] => [.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 5 + 1
[1,4,3,2,5,8,7,6] => [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
[3,2,1,4,5,8,7,6] => [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 4 + 1
[2,1,6,5,4,3,8,7] => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[4,3,2,1,6,5,8,7] => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,2,4,3,6,5,8,7] => [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 5 + 1
[1,2,4,3,5,6,8,7] => [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 6 + 1
[2,1,6,7,5,4,3,8] => [[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [[[.,.],[.,[.,[[.,.],.]]]],[.,.]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,4,3,2,7,6,5,8] => [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4 + 1
Description
The position of the first down step of a Dyck path.
Mp00061: Permutations to increasing treeBinary trees
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
St000678: Dyck paths ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 67%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[4,3,2,5,6,7,8,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[7,6,5,4,3,8,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[8,7,6,4,3,2,1,5] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[8,7,3,2,1,4,5,6] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[8,6,4,2,1,3,5,7] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[8,3,2,1,4,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[8,2,1,3,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[6,5,4,7,3,2,1,8] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[7,6,4,3,1,2,5,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[7,3,1,2,4,5,6,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[6,4,1,2,3,5,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[6,5,8,7,4,3,2,1] => [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,8,7,6,5,3,2,1] => [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[5,4,3,8,7,6,2,1] => [[[[[.,.],.],[[[.,.],.],.]],.],.]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 1
[4,3,6,5,8,7,2,1] => [[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
[2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
[3,2,8,7,6,5,4,1] => [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 1
[4,3,2,8,7,6,5,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 1
[4,3,2,7,6,5,8,1] => [[[[.,.],.],[[[.,.],.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> ? = 1
[3,2,4,5,7,6,8,1] => [[[.,.],[.,[.,[[.,.],[.,.]]]]],.]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 1
[2,3,6,5,4,7,8,1] => [[.,[.,[[[.,.],.],[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
[2,4,3,6,5,7,8,1] => [[.,[[.,.],[[.,.],[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 1
[2,3,5,4,6,7,8,1] => [[.,[.,[[.,.],[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[1,6,8,7,5,4,3,2] => [.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,7,6,8,5,4,3,2] => [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 2
[1,5,8,7,6,4,3,2] => [.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,7,6,5,8,4,3,2] => [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 2
[1,4,3,6,5,8,7,2] => [.,[[[.,.],[[.,.],[[.,.],.]]],.]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 2
[1,3,4,5,6,8,7,2] => [.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
[2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 2
[2,1,4,5,8,7,6,3] => [[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,2,8,7,6,5,4,3] => [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 3
[3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 2
Description
The number of up steps after the last double rise of a Dyck path.
Mp00061: Permutations to increasing treeBinary trees
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00066: Permutations inversePermutations
St000546: Permutations ⟶ ℤResult quality: 66% values known / values provided: 66%distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> [1] => [1] => 0 = 1 - 1
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 1 = 2 - 1
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0 = 1 - 1
[3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => 2 = 3 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => 1 = 2 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => 0 = 1 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 2 = 3 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 0 = 1 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1 = 2 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 2 = 3 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 0 = 1 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 4 = 5 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => 3 = 4 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,3,1] => 3 = 4 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,3,1] => 3 = 4 - 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] => [5,3,4,2,1] => 3 = 4 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,3,4,1,2] => 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => 2 = 3 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => 1 = 2 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => 1 = 2 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,3,4,2,1] => 3 = 4 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,3,4,1,2] => 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => 2 = 3 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,2,3,1,4] => 1 = 2 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => 2 = 3 - 1
[2,4,5,6,1,7,3] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [6,7,4,3,2,1,5] => [6,5,4,3,7,1,2] => ? = 2 - 1
[2,5,3,4,6,7,1] => [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => [6,4,5,3,2,1,7] => ? = 1 - 1
[2,6,3,4,5,7,1] => [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => [6,4,5,3,2,1,7] => ? = 1 - 1
[2,7,6,5,1,4,3] => [[.,[[[.,.],.],.]],[[.,.],.]]
=> [6,7,2,3,4,1,5] => [6,3,4,5,7,1,2] => ? = 2 - 1
[3,2,7,6,5,1,4] => [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => [5,6,2,3,4,7,1] => ? = 2 - 1
[3,4,5,6,7,1,2] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [7,5,4,3,2,1,6] => [6,5,4,3,2,7,1] => ? = 2 - 1
[3,5,6,2,7,4,1] => [[[.,[.,[.,.]]],[[.,.],.]],.]
=> [5,6,3,2,1,4,7] => [5,4,3,6,1,2,7] => ? = 1 - 1
[3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
=> [7,4,5,2,1,3,6] => [5,4,6,2,3,7,1] => ? = 2 - 1
[3,7,2,6,5,4,1] => [[[.,[.,.]],[[[.,.],.],.]],.]
=> [4,5,6,2,1,3,7] => [5,4,6,1,2,3,7] => ? = 1 - 1
[3,7,6,2,5,4,1] => [[[.,[[.,.],.]],[[.,.],.]],.]
=> [5,6,2,3,1,4,7] => [5,3,4,6,1,2,7] => ? = 1 - 1
[3,7,6,5,1,4,2] => [[.,[[[.,.],.],.]],[[.,.],.]]
=> [6,7,2,3,4,1,5] => [6,3,4,5,7,1,2] => ? = 2 - 1
[3,7,6,5,2,4,1] => [[[.,[[[.,.],.],.]],[.,.]],.]
=> [6,2,3,4,1,5,7] => [5,2,3,4,6,1,7] => ? = 1 - 1
[4,2,7,6,5,1,3] => [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => [5,6,2,3,4,7,1] => ? = 2 - 1
[4,3,7,2,6,5,1] => [[[[.,.],[.,.]],[[.,.],.]],.]
=> [5,6,3,1,2,4,7] => [4,5,3,6,1,2,7] => ? = 1 - 1
[4,6,3,7,5,2,1] => [[[[.,[.,.]],[[.,.],.]],.],.]
=> [4,5,2,1,3,6,7] => [4,3,5,1,2,6,7] => ? = 1 - 1
[4,7,2,6,5,3,1] => [[[.,[.,.]],[[[.,.],.],.]],.]
=> [4,5,6,2,1,3,7] => [5,4,6,1,2,3,7] => ? = 1 - 1
[6,2,7,1,5,4,3] => [[[.,.],[.,.]],[[[.,.],.],.]]
=> [5,6,7,3,1,2,4] => [5,6,4,7,1,2,3] => ? = 2 - 1
[6,3,7,5,4,2,1] => [[[[.,.],[[[.,.],.],.]],.],.]
=> [3,4,5,1,2,6,7] => [4,5,1,2,3,6,7] => ? = 1 - 1
[6,5,4,1,7,3,2] => [[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 2 - 1
[6,7,1,2,3,4,5] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => [6,5,7,4,3,2,1] => ? = 5 - 1
[6,7,1,5,4,3,2] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [4,5,6,7,2,1,3] => [6,5,7,1,2,3,4] => ? = 2 - 1
[6,7,2,5,4,3,1] => [[[.,[.,.]],[[[.,.],.],.]],.]
=> [4,5,6,2,1,3,7] => [5,4,6,1,2,3,7] => ? = 1 - 1
[6,7,5,4,1,3,2] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [6,7,2,1,3,4,5] => [4,3,5,6,7,1,2] => ? = 2 - 1
[7,1,2,3,4,5,6] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 6 - 1
[7,1,2,4,3,5,6] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => [6,7,5,3,4,2,1] => ? = 5 - 1
[7,1,3,4,2,5,6] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => [6,7,4,3,5,2,1] => ? = 4 - 1
[7,1,3,4,5,2,6] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => [6,7,4,3,2,5,1] => ? = 3 - 1
[7,1,4,2,3,5,6] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [7,6,5,3,4,1,2] => [6,7,4,5,3,2,1] => ? = 5 - 1
[7,1,4,3,2,5,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [7,6,3,4,5,1,2] => [6,7,3,4,5,2,1] => ? = 4 - 1
[7,1,4,3,5,2,6] => [[.,.],[[[.,.],[.,.]],[.,.]]]
=> [7,5,3,4,6,1,2] => [6,7,3,4,2,5,1] => ? = 3 - 1
[7,1,4,5,2,3,6] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => [6,7,4,3,5,2,1] => ? = 4 - 1
[7,1,5,2,3,4,6] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [7,6,5,3,4,1,2] => [6,7,4,5,3,2,1] => ? = 5 - 1
[7,1,5,3,2,4,6] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [7,6,3,4,5,1,2] => [6,7,3,4,5,2,1] => ? = 4 - 1
[7,2,1,3,4,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 5 - 1
[7,2,3,1,4,5,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => [5,6,4,7,3,2,1] => ? = 4 - 1
[7,2,3,4,1,5,6] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [7,6,4,3,1,2,5] => [5,6,4,3,7,2,1] => ? = 3 - 1
[7,2,3,4,5,1,6] => [[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [7,5,4,3,1,2,6] => [5,6,4,3,2,7,1] => ? = 2 - 1
[7,2,6,1,5,4,3] => [[[.,.],[.,.]],[[[.,.],.],.]]
=> [5,6,7,3,1,2,4] => [5,6,4,7,1,2,3] => ? = 2 - 1
[7,2,6,5,1,4,3] => [[[.,.],[[.,.],.]],[[.,.],.]]
=> [6,7,3,4,1,2,5] => [5,6,3,4,7,1,2] => ? = 2 - 1
[7,2,6,5,4,1,3] => [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => [5,6,2,3,4,7,1] => ? = 2 - 1
[7,3,1,2,4,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 5 - 1
[7,3,2,1,4,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 4 - 1
[7,3,2,1,6,5,4] => [[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 2 - 1
[7,3,2,4,1,5,6] => [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => [4,5,6,3,7,2,1] => ? = 3 - 1
[7,3,2,6,1,5,4] => [[[[.,.],.],[.,.]],[[.,.],.]]
=> [6,7,4,1,2,3,5] => [4,5,6,3,7,1,2] => ? = 2 - 1
[7,3,4,1,2,5,6] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => [5,6,4,7,3,2,1] => ? = 4 - 1
[7,3,6,1,5,4,2] => [[[.,.],[.,.]],[[[.,.],.],.]]
=> [5,6,7,3,1,2,4] => [5,6,4,7,1,2,3] => ? = 2 - 1
[7,3,6,5,4,1,2] => [[[.,.],[[[.,.],.],.]],[.,.]]
=> [7,3,4,5,1,2,6] => [5,6,2,3,4,7,1] => ? = 2 - 1
[7,4,1,2,3,5,6] => [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 5 - 1
[7,4,2,1,3,5,6] => [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 4 - 1
Description
The number of global descents of a permutation. The global descents are the integers in the set C(π)={i[n1]:1ji<kn:π(j)>π(k)}. In particular, if iC(π) then i is a descent. For the number of global ascents, see [[St000234]].
Matching statistic: St000925
Mp00066: Permutations inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000925: Set partitions ⟶ ℤResult quality: 58% values known / values provided: 65%distinct values known / distinct values provided: 58%
Values
[1] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 2
[2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[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] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[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] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 4
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 4
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 3
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 3
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 4
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
[1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 2
[6,7,8,5,4,1,2,3] => [6,7,8,5,4,1,2,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 3
[6,7,4,5,8,1,2,3] => [6,7,8,3,4,1,2,5] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 3
[4,5,6,7,8,1,2,3] => [6,7,8,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 3
[7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[7,8,5,6,2,3,1,4] => [7,5,6,8,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[6,7,5,8,2,3,1,4] => [7,5,6,8,3,1,2,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[7,5,6,8,2,3,1,4] => [7,5,6,8,2,3,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[7,8,5,6,3,1,2,4] => [6,7,5,8,3,4,1,2] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,7},{6}}
=> ? = 3
[6,7,5,8,3,1,2,4] => [6,7,5,8,3,1,2,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,7},{6}}
=> ? = 3
[7,5,6,8,3,1,2,4] => [6,7,5,8,2,3,1,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,7},{6}}
=> ? = 3
[6,5,7,8,2,1,3,4] => [6,5,7,8,2,1,3,4] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 3
[7,8,5,6,1,2,3,4] => [5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 4
[5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 4
[8,7,6,4,3,2,1,5] => [7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[6,7,8,3,4,1,2,5] => [6,7,4,5,8,1,2,3] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,7},{6}}
=> ? = 3
[6,7,8,1,2,3,4,5] => [4,5,6,7,8,1,2,3] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5},{6},{7}}
=> ? = 5
[7,8,4,5,2,3,1,6] => [7,5,6,3,4,8,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 2
[7,8,3,4,2,5,1,6] => [7,5,3,4,6,8,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 2
[8,7,3,2,1,4,5,6] => [5,4,3,6,7,8,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3,4,5},{6},{7}}
=> ? = 4
[7,8,1,2,3,4,5,6] => [3,4,5,6,7,8,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 6
[8,5,6,3,4,1,2,7] => [6,7,4,5,2,3,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> ? = 3
[8,3,4,5,6,1,2,7] => [6,7,2,3,4,5,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> {{1,8},{2,3,4,5,7},{6}}
=> ? = 3
[8,5,6,1,2,3,4,7] => [4,5,6,7,2,3,8,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> {{1,8},{2,3,7},{4},{5},{6}}
=> ? = 5
[8,6,4,2,1,3,5,7] => [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? = 4
[8,2,3,4,1,5,6,7] => [5,2,3,4,6,7,8,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4,5},{6},{7}}
=> ? = 4
[8,4,1,2,3,5,6,7] => [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> {{1,8},{2,5},{3},{4},{6},{7}}
=> ? = 6
[8,3,2,1,4,5,6,7] => [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4},{5},{6},{7}}
=> ? = 5
[8,2,3,1,4,5,6,7] => [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3,4},{5},{6},{7}}
=> ? = 5
[8,3,1,2,4,5,6,7] => [3,4,2,5,6,7,8,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,4},{3},{5},{6},{7}}
=> ? = 6
[8,2,1,3,4,5,6,7] => [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2,3},{4},{5},{6},{7}}
=> ? = 6
[8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 7
[6,7,3,4,5,1,2,8] => [6,7,3,4,5,1,2,8] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> ? = 3
[7,6,4,3,1,2,5,8] => [5,6,4,3,7,2,1,8] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> {{1,2,7},{3,4,6},{5},{8}}
=> ? = 4
[7,3,1,2,4,5,6,8] => [3,4,2,5,6,7,1,8] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> {{1,7},{2,4},{3},{5},{6},{8}}
=> ? = 6
[5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7,8] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,6},{5},{7},{8}}
=> ? = 4
[6,4,1,2,3,5,7,8] => [3,4,5,2,6,1,7,8] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> {{1,6},{2,5},{3},{4},{7},{8}}
=> ? = 6
[3,4,2,5,1,6,7,8] => [5,3,1,2,4,6,7,8] => [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
[2,3,4,5,1,6,7,8] => [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
[3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 6
[2,3,1,4,5,6,7,8] => [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
[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},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[2,1,4,5,7,8,6,3] => [2,1,8,3,4,7,5,6] => ?
=> ?
=> ? = 2
[3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [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
[1,3,2,8,7,6,5,4] => [1,3,2,8,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 3
[1,2,3,8,7,6,5,4] => [1,2,3,8,7,6,5,4] => [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
[1,2,3,5,7,8,6,4] => [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
[1,2,3,5,6,7,8,4] => [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
[4,3,2,1,8,7,6,5] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 2
[3,4,2,1,7,8,6,5] => [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
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: St000675
Mp00061: Permutations to increasing treeBinary trees
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00296: Dyck paths Knuth-KrattenthalerDyck paths
St000675: Dyck paths ⟶ ℤResult quality: 58% values known / values provided: 63%distinct values known / distinct values provided: 58%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[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]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 1
[8,6,7,4,5,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 1
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 1
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 1
[8,6,7,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 1
[6,7,4,5,2,3,8,1] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[4,3,2,5,6,7,8,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 2
[7,8,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> ? = 2
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[6,7,5,8,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[7,5,6,8,3,4,1,2] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[7,8,4,5,3,6,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[7,8,5,3,4,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2
[7,6,5,4,3,8,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 2
[6,7,4,5,3,8,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[5,6,4,7,3,8,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[7,6,8,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> ? = 2
[7,8,5,6,2,3,1,4] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[6,7,5,8,2,3,1,4] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[7,5,6,8,2,3,1,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[7,8,5,6,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[6,7,5,8,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[7,5,6,8,3,1,2,4] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[6,5,7,8,2,1,3,4] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[8,7,6,4,3,2,1,5] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 2
[7,8,4,5,2,3,1,6] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[7,8,3,4,2,5,1,6] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[8,7,3,2,1,4,5,6] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4
[8,5,6,3,4,1,2,7] => [[[[.,.],[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[8,6,4,2,1,3,5,7] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4
[8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
[8,3,2,1,4,5,6,7] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 6
Description
The number of centered multitunnels of a Dyck path. This is the number of factorisations D=ABC of a Dyck path, such that B is a Dyck path and A and B have the same length.
Mp00069: Permutations complementPermutations
Mp00065: Permutations permutation posetPosets
St000069: Posets ⟶ ℤResult quality: 57% values known / values provided: 57%distinct values known / distinct values provided: 92%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2,1] => ([],2)
=> 2
[2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [3,2,1] => ([],3)
=> 3
[1,3,2] => [3,1,2] => ([(1,2)],3)
=> 2
[2,1,3] => [2,3,1] => ([(1,2)],3)
=> 2
[2,3,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [4,3,2,1] => ([],4)
=> 4
[1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 3
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 3
[1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 3
[1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
=> 3
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 3
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[3,2,4,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 5
[1,2,3,5,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 4
[1,2,4,3,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 4
[1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 3
[1,2,5,3,4] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 4
[1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 3
[1,3,2,4,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 4
[1,3,2,5,4] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 3
[1,3,4,2,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,2,4] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,3,5,4,2] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,2,3,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 4
[1,4,2,5,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 3
[1,4,3,5,2] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,5,2,3] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[2,1,4,3,7,6,5] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 3
[2,1,5,4,7,6,3] => [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 2
[2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[2,4,3,1,7,6,5] => [6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 2
[2,5,4,1,7,6,3] => [6,3,4,7,1,2,5] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 2
[2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ([(0,6),(1,5),(3,4),(4,2),(5,3),(5,6)],7)
=> ? = 2
[2,7,6,5,1,4,3] => [6,1,2,3,7,4,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7)
=> ? = 2
[3,2,1,7,6,5,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 2
[3,5,2,1,7,6,4] => [5,3,6,7,1,2,4] => ([(0,5),(1,5),(1,6),(2,3),(3,6),(5,4)],7)
=> ? = 2
[3,7,2,6,5,4,1] => [5,1,6,2,3,4,7] => ([(0,6),(1,4),(1,6),(2,5),(3,2),(4,3),(6,5)],7)
=> ? = 1
[3,7,6,2,5,4,1] => [5,1,2,6,3,4,7] => ([(0,6),(1,4),(2,5),(3,2),(4,3),(4,6),(6,5)],7)
=> ? = 1
[3,7,6,5,2,4,1] => [5,1,2,3,6,4,7] => ([(0,6),(1,4),(2,5),(3,2),(3,6),(4,3),(6,5)],7)
=> ? = 1
[3,7,6,5,4,1,2] => [5,1,2,3,4,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 2
[4,2,7,6,5,1,3] => [4,6,1,2,3,7,5] => ([(0,3),(0,6),(1,4),(2,5),(2,6),(3,5),(4,2)],7)
=> ? = 2
[4,3,7,2,6,5,1] => [4,5,1,6,2,3,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,2),(6,5)],7)
=> ? = 1
[4,7,2,6,5,3,1] => [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
[4,7,6,5,3,1,2] => [4,1,2,3,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 2
[5,2,7,6,4,3,1] => [3,6,1,2,4,5,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,5),(6,2)],7)
=> ? = 1
[5,3,2,7,6,4,1] => [3,5,6,1,2,4,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,2),(6,5)],7)
=> ? = 1
[5,4,3,2,1,7,6] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[5,7,6,4,3,1,2] => [3,1,2,4,5,7,6] => ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ? = 2
[6,2,7,1,5,4,3] => [2,6,1,7,3,4,5] => ([(0,5),(0,6),(1,3),(1,6),(3,5),(4,2),(6,4)],7)
=> ? = 2
[6,2,7,5,4,3,1] => [2,6,1,3,4,5,7] => ([(0,6),(1,3),(1,6),(2,5),(3,5),(4,2),(6,4)],7)
=> ? = 1
[6,3,1,7,5,4,2] => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(1,6),(3,5),(4,2),(4,5),(6,3)],7)
=> ? = 2
[6,3,7,5,4,2,1] => [2,5,1,3,4,6,7] => ([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> ? = 1
[6,5,3,2,1,7,4] => [2,3,5,6,7,1,4] => ([(0,6),(1,5),(3,4),(4,2),(5,3),(5,6)],7)
=> ? = 2
[6,5,4,2,1,7,3] => [2,3,4,6,7,1,5] => ([(0,6),(1,4),(3,2),(4,5),(5,3),(5,6)],7)
=> ? = 2
[6,5,4,3,7,1,2] => [2,3,4,5,1,7,6] => ([(0,5),(0,6),(1,4),(2,5),(2,6),(3,2),(4,3)],7)
=> ? = 2
[6,5,4,7,3,1,2] => [2,3,4,1,5,7,6] => ([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7)
=> ? = 2
[7,2,6,5,1,4,3] => [1,6,2,3,7,4,5] => ([(0,2),(0,5),(2,6),(3,1),(4,3),(4,6),(5,4)],7)
=> ? = 2
[7,2,6,5,4,1,3] => [1,6,2,3,4,7,5] => ([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
=> ? = 2
[7,3,1,6,5,4,2] => [1,5,7,2,3,4,6] => ([(0,4),(0,5),(2,6),(3,2),(4,3),(5,1),(5,6)],7)
=> ? = 2
[7,3,2,6,1,5,4] => [1,5,6,2,7,3,4] => ([(0,4),(0,5),(2,6),(3,1),(4,2),(5,3),(5,6)],7)
=> ? = 2
[7,3,6,1,5,4,2] => [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
[7,3,6,5,4,1,2] => [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
[7,4,1,6,5,3,2] => [1,4,7,2,3,5,6] => ([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
=> ? = 2
[7,4,2,1,6,5,3] => [1,4,6,7,2,3,5] => ([(0,4),(0,5),(2,6),(3,1),(4,2),(5,3),(5,6)],7)
=> ? = 2
[7,5,1,6,4,3,2] => [1,3,7,2,4,5,6] => ([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
=> ? = 2
[7,6,2,5,4,1,3] => [1,2,6,3,4,7,5] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 2
[7,6,5,2,4,1,3] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(5,7),(6,7),(7,4)],8)
=> ? = 1
[7,6,5,4,8,3,2,1] => [2,3,4,5,1,6,7,8] => ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 1
[4,5,6,7,8,3,2,1] => [5,4,3,2,1,6,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,5)],8)
=> ? = 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(5,7),(6,7),(7,1)],8)
=> ? = 1
[7,6,5,4,3,8,2,1] => [2,3,4,5,6,1,7,8] => ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 1
[3,4,5,6,7,8,2,1] => [6,5,4,3,2,1,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
=> ? = 1
[8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2),(5,7),(6,7)],8)
=> ? = 1
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => ([(0,3),(0,4),(1,6),(2,5),(2,7),(3,5),(3,7),(4,2),(5,6),(7,1)],8)
=> ? = 1
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(5,4),(6,4),(7,4)],8)
=> ? = 1
[7,6,5,4,3,2,8,1] => [2,3,4,5,6,7,1,8] => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ? = 1
Description
The number of maximal elements of a poset.
The following 74 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000237The number of small exceedances. St000054The first entry of the permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000912The number of maximal antichains in a poset. St000068The number of minimal elements in a poset. St000167The number of leaves of an ordered tree. 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. St000025The number of initial rises of a Dyck path. St000105The number of blocks in the set partition. 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. St000234The number of global ascents of a 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. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000843The decomposition number of a perfect matching. 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. St000989The number of final rises of a permutation. St000740The last entry of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000702The number of weak deficiencies of a permutation. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000991The number of right-to-left minima of a permutation. St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St000084The number of subtrees. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000286The number of connected components of the complement of a graph. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. 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. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000203The number of external nodes of a binary tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. 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. St000181The number of connected components of the Hasse diagram for the poset. St001863The number of weak excedances of a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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.