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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000237
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,3,1] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St001484
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> []
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> []
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[7,8,5,6,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,6,7,8,2,3,4,1] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[7,5,6,2,3,4,8,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,6,7,2,3,4,8,1] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[7,5,2,3,4,6,8,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,2,3,4,6,7,8,1] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[8,7,6,5,4,1,2,3] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 5
[8,7,6,4,5,1,2,3] => [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3]
=> ? = 3
[7,8,5,6,2,3,1,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[7,5,6,8,2,3,1,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,6,7,8,2,3,1,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[7,8,5,6,2,1,3,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[7,5,6,8,2,1,3,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,6,7,8,2,1,3,4] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[8,7,6,5,1,2,3,4] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
[8,7,6,4,1,2,3,5] => [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3]
=> ? = 3
[7,8,5,2,3,4,1,6] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[7,8,5,2,1,3,4,6] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[7,5,6,2,3,4,1,8] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,6,7,2,3,4,1,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[5,2,3,4,6,7,1,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[7,5,6,2,3,1,4,8] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,6,7,2,3,1,4,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[7,5,6,2,1,3,4,8] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,6,7,2,1,3,4,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[7,5,2,3,4,1,6,8] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[7,5,2,3,1,4,6,8] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[5,6,2,3,4,1,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[5,2,3,4,6,1,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[5,6,2,1,3,4,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[5,2,3,4,1,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[5,2,3,1,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[5,2,1,3,4,6,7,8] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[3,4,7,6,5,8,2,1] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 2
[2,5,4,7,8,6,3,1] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[2,5,4,7,6,8,3,1] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[3,4,2,7,8,6,5,1] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 2
[3,2,4,7,8,6,5,1] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 2
[2,5,4,3,7,8,6,1] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[3,4,7,6,5,2,8,1] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 2
[2,5,4,7,6,3,8,1] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[3,4,2,7,6,5,8,1] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 2
[3,2,4,7,6,5,8,1] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 2
[2,5,4,3,7,6,8,1] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[2,1,5,7,8,6,4,3] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[2,1,5,7,6,8,4,3] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[2,1,5,4,7,8,6,3] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[2,1,5,7,6,4,8,3] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[2,1,5,4,7,6,8,3] => [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[3,4,2,1,7,8,6,5] => [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 2
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000932
Mp00069: Permutations —complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [.,.]
=> [1,0]
=> ? = 0
[1,2] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,3,4,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,4,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,3,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,1,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,4,3,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,2,4] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,4,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,5,3,2] => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,4,5,6,7,8,2,1] => [6,5,4,3,2,1,7,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,8,5,6,2,3,4,1] => [2,1,4,3,7,6,5,8] => ?
=> ?
=> ? = 0
[5,6,7,8,2,3,4,1] => [4,3,2,1,7,6,5,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,5,6,7,2,8,1] => [6,5,4,3,2,7,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,5,6,2,3,4,8,1] => [2,4,3,7,6,5,1,8] => ?
=> ?
=> ? = 0
[5,6,7,2,3,4,8,1] => [4,3,2,7,6,5,1,8] => ?
=> ?
=> ? = 0
[7,5,2,3,4,6,8,1] => [2,4,7,6,5,3,1,8] => ?
=> ?
=> ? = 0
[3,4,5,6,2,7,8,1] => [6,5,4,3,7,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,4,5,2,6,7,8,1] => [6,5,4,7,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,4,6,7,8,1] => [4,7,6,5,3,2,1,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,2,5,6,7,8,1] => [6,5,7,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,4,5,6,7,8,1] => [6,7,5,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[5,6,7,8,2,3,1,4] => [4,3,2,1,7,6,8,5] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[5,6,7,8,2,1,3,4] => [4,3,2,1,7,8,6,5] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,5,6,7,2,1,8] => [6,5,4,3,2,7,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,6,7,2,3,4,1,8] => [4,3,2,7,6,5,8,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,5,6,2,7,1,8] => [6,5,4,3,7,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,4,5,2,6,7,1,8] => [6,5,4,7,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,4,6,7,1,8] => [4,7,6,5,3,2,8,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,2,5,6,7,1,8] => [6,5,7,4,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,4,5,6,7,1,8] => [6,7,5,4,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,3,4,5,6,7,1,8] => [7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,5,6,2,3,1,4,8] => [2,4,3,7,6,8,5,1] => ?
=> ?
=> ? = 0
[5,6,7,2,3,1,4,8] => [4,3,2,7,6,8,5,1] => ?
=> ?
=> ? = 0
[7,5,6,2,1,3,4,8] => [2,4,3,7,8,6,5,1] => ?
=> ?
=> ? = 0
[5,6,7,2,1,3,4,8] => [4,3,2,7,8,6,5,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,5,2,3,4,1,6,8] => [2,4,7,6,5,8,3,1] => ?
=> ?
=> ? = 0
[7,5,2,3,1,4,6,8] => [2,4,7,6,8,5,3,1] => ?
=> ?
=> ? = 0
[7,2,3,4,1,5,6,8] => [2,7,6,5,8,4,3,1] => ?
=> ?
=> ? = 0
[3,4,5,6,2,1,7,8] => [6,5,4,3,7,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,6,2,3,4,1,7,8] => [4,3,7,6,5,8,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,5,2,6,1,7,8] => [6,5,4,7,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,4,6,1,7,8] => [4,7,6,5,3,8,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,2,5,6,1,7,8] => [6,5,7,4,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,4,5,6,1,7,8] => [6,7,5,4,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,3,4,5,6,1,7,8] => [7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[5,6,2,1,3,4,7,8] => [4,3,7,8,6,5,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,5,2,1,6,7,8] => [6,5,4,7,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,4,1,6,7,8] => [4,7,6,5,8,3,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,2,5,1,6,7,8] => [6,5,7,4,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,4,5,1,6,7,8] => [6,7,5,4,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,3,4,5,1,6,7,8] => [7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[5,2,3,1,4,6,7,8] => [4,7,6,8,5,3,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[5,2,1,3,4,6,7,8] => [4,7,8,6,5,3,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,2,1,5,6,7,8] => [6,5,7,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,4,1,5,6,7,8] => [6,7,5,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,3,4,1,5,6,7,8] => [7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[3,2,1,4,5,6,7,8] => [6,7,8,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001067
Mp00069: Permutations —complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 70%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 70%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,3,4,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,4,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,3,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,1,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,4,3,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,2,4] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,4,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[3,4,5,6,7,8,2,1] => [6,5,4,3,2,1,7,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,8,5,6,2,3,4,1] => [2,1,4,3,7,6,5,8] => ?
=> ?
=> ? = 0
[5,6,7,8,2,3,4,1] => [4,3,2,1,7,6,5,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,8,2,3,4,5,6,1] => [2,1,7,6,5,4,3,8] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[3,4,5,6,7,2,8,1] => [6,5,4,3,2,7,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,5,6,2,3,4,8,1] => [2,4,3,7,6,5,1,8] => ?
=> ?
=> ? = 0
[5,6,7,2,3,4,8,1] => [4,3,2,7,6,5,1,8] => ?
=> ?
=> ? = 0
[7,5,2,3,4,6,8,1] => [2,4,7,6,5,3,1,8] => ?
=> ?
=> ? = 0
[7,2,3,4,5,6,8,1] => [2,7,6,5,4,3,1,8] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[3,4,5,6,2,7,8,1] => [6,5,4,3,7,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,4,5,2,6,7,8,1] => [6,5,4,7,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,4,6,7,8,1] => [4,7,6,5,3,2,1,8] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,2,5,6,7,8,1] => [6,5,7,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,4,5,6,7,8,1] => [6,7,5,4,3,2,1,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[8,7,6,5,4,1,2,3] => [1,2,3,4,5,8,7,6] => [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
[8,7,6,4,5,1,2,3] => [1,2,3,5,4,8,7,6] => [.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[7,8,5,6,2,3,1,4] => [2,1,4,3,7,6,8,5] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[5,6,7,8,2,3,1,4] => [4,3,2,1,7,6,8,5] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,8,5,6,2,1,3,4] => [2,1,4,3,7,8,6,5] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,5,6,8,2,1,3,4] => [2,4,3,1,7,8,6,5] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[5,6,7,8,2,1,3,4] => [4,3,2,1,7,8,6,5] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[8,7,6,5,1,2,3,4] => [1,2,3,4,8,7,6,5] => [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[8,7,6,4,1,2,3,5] => [1,2,3,5,8,7,6,4] => [.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[7,8,5,2,3,4,1,6] => [2,1,4,7,6,5,8,3] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,8,2,3,4,5,1,6] => [2,1,7,6,5,4,8,3] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[7,8,5,2,1,3,4,6] => [2,1,4,7,8,6,5,3] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,8,2,3,4,1,5,6] => [2,1,7,6,5,8,4,3] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[7,8,2,3,1,4,5,6] => [2,1,7,6,8,5,4,3] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[7,8,2,1,3,4,5,6] => [2,1,7,8,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[8,1,2,3,4,5,6,7] => [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
[3,4,5,6,7,2,1,8] => [6,5,4,3,2,7,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[7,5,6,2,3,4,1,8] => [2,4,3,7,6,5,8,1] => [[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[5,6,7,2,3,4,1,8] => [4,3,2,7,6,5,8,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,2,3,4,5,6,1,8] => [2,7,6,5,4,3,8,1] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[3,4,5,6,2,7,1,8] => [6,5,4,3,7,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,4,5,2,6,7,1,8] => [6,5,4,7,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[5,2,3,4,6,7,1,8] => [4,7,6,5,3,2,8,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[3,4,2,5,6,7,1,8] => [6,5,7,4,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[3,2,4,5,6,7,1,8] => [6,7,5,4,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,3,4,5,6,7,1,8] => [7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,5,6,2,3,1,4,8] => [2,4,3,7,6,8,5,1] => ?
=> ?
=> ? = 0
[5,6,7,2,3,1,4,8] => [4,3,2,7,6,8,5,1] => ?
=> ?
=> ? = 0
[7,5,6,2,1,3,4,8] => [2,4,3,7,8,6,5,1] => ?
=> ?
=> ? = 0
[5,6,7,2,1,3,4,8] => [4,3,2,7,8,6,5,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,5,2,3,4,1,6,8] => [2,4,7,6,5,8,3,1] => ?
=> ?
=> ? = 0
[7,2,3,4,5,1,6,8] => [2,7,6,5,4,8,3,1] => [[.,.],[[[[[.,.],.],.],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000445
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,1,0,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[4,1,2,3,7,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,2,7,3,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,2,7,6,3,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,2,7,6,5,3] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,7,2,3,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,7,2,6,3,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,7,2,6,5,3] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,7,6,2,3,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,7,6,2,5,3] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,1,7,6,5,2,3] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
[4,2,1,3,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,1,3,6,7,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,1,6,3,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,1,6,3,7,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,1,6,5,3,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,1,6,5,7,3] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,1,6,7,3,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,1,6,7,5,3] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,3,1,6,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,3,1,6,7,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,3,6,1,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,3,6,1,7,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,3,6,5,1,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,3,6,5,7,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,3,6,7,1,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,3,6,7,5,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,1,3,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,1,3,7,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,1,5,3,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,1,5,7,3] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,1,7,3,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,1,7,5,3] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,3,1,5,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,3,1,7,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,3,5,1,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,3,5,7,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,3,7,1,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,3,7,5,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,5,1,3,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,5,1,7,3] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,5,3,1,7] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,5,3,7,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,5,7,1,3] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,5,7,3,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,7,1,3,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,7,1,5,3] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,7,3,1,5] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,7,3,5,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,7,5,1,3] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
[4,2,6,7,5,3,1] => [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St001640
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 70%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [2,1] => 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => ? = 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => ? = 1
[1,5,3,2,4,6,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,2,4,7,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,2,6,4,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,2,6,7,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,2,7,4,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,2,7,6,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,4,2,6,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,4,2,7,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,4,6,2,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,4,6,7,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,4,7,2,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,4,7,6,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,6,2,4,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,6,2,7,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,6,4,2,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,6,4,7,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,6,7,2,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,6,7,4,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,3,7,2,4,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,7,2,6,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,7,4,2,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,7,4,6,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,7,6,2,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,3,7,6,4,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,6,3,2,4,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,6,3,2,7,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,6,3,4,2,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,6,3,4,7,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,6,3,7,2,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,6,3,7,4,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,6,7,3,2,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,6,7,3,4,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ? = 0
[1,5,7,3,2,4,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,7,3,2,6,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,7,3,4,2,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,7,3,4,6,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,7,3,6,2,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,7,3,6,4,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,7,6,3,2,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,5,7,6,3,4,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,3,6,4,7] => ? = 1
[1,6,2,4,3,5,7] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,6,3,7,5] => ? = 0
[1,6,2,4,3,7,5] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,6,3,7,5] => ? = 0
[1,6,2,4,5,3,7] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,6,3,7,5] => ? = 0
[1,6,2,4,5,7,3] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,6,3,7,5] => ? = 0
[1,6,2,4,7,3,5] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,6,3,7,5] => ? = 0
[1,6,2,4,7,5,3] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,6,3,7,5] => ? = 0
[1,6,2,5,4,3,7] => [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => ? = 2
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000441
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [.,.]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [.,[.,.]]
=> [2,1] => 0
[2,1] => [[.,.],.]
=> [[.,.],.]
=> [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[2,3,1] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[3,1,2] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ? = 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => ? = 1
[1,5,3,2,4,6,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,2,4,7,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,2,6,4,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,2,6,7,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,2,7,4,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,2,7,6,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,4,2,6,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,4,2,7,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,4,6,2,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,4,6,7,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,4,7,2,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,4,7,6,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,6,2,4,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,6,2,7,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,6,4,2,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,6,4,7,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,6,7,2,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,6,7,4,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,3,7,2,4,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,7,2,6,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,7,4,2,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,7,4,6,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,7,6,2,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,3,7,6,4,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,6,3,2,4,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,6,3,2,7,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,6,3,4,2,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,6,3,4,7,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,6,3,7,2,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,6,3,7,4,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,6,7,3,2,4] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,6,7,3,4,2] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,5,7,3,2,4,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,7,3,2,6,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,7,3,4,2,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,7,3,4,6,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,7,3,6,2,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,7,3,6,4,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,7,6,3,2,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,5,7,6,3,4,2] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,6,2,4,3,5,7] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => ? = 0
[1,6,2,4,3,7,5] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => ? = 0
[1,6,2,4,5,3,7] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => ? = 0
[1,6,2,4,5,7,3] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => ? = 0
[1,6,2,4,7,3,5] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => ? = 0
[1,6,2,4,7,5,3] => [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => ? = 0
[1,6,2,5,4,3,7] => [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => ? = 2
[1,6,2,5,4,7,3] => [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => ? = 2
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St001223
Mp00069: Permutations —complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,3,4,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,4,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,3,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,1,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,4,3,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,1,2,4] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,4,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [[[[[.,[[.,.],.]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1
[1,5,3,2,4,6,7] => [7,3,5,6,4,2,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,2,4,7,6] => [7,3,5,6,4,1,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,2,6,4,7] => [7,3,5,6,2,4,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,2,6,7,4] => [7,3,5,6,2,1,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,2,7,4,6] => [7,3,5,6,1,4,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,2,7,6,4] => [7,3,5,6,1,2,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,4,2,6,7] => [7,3,5,4,6,2,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,4,2,7,6] => [7,3,5,4,6,1,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,4,6,2,7] => [7,3,5,4,2,6,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,4,6,7,2] => [7,3,5,4,2,1,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,4,7,2,6] => [7,3,5,4,1,6,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,4,7,6,2] => [7,3,5,4,1,2,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,6,2,4,7] => [7,3,5,2,6,4,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,6,2,7,4] => [7,3,5,2,6,1,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,6,4,2,7] => [7,3,5,2,4,6,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,6,4,7,2] => [7,3,5,2,4,1,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,6,7,2,4] => [7,3,5,2,1,6,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,6,7,4,2] => [7,3,5,2,1,4,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,3,7,2,4,6] => [7,3,5,1,6,4,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,7,2,6,4] => [7,3,5,1,6,2,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,7,4,2,6] => [7,3,5,1,4,6,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,7,4,6,2] => [7,3,5,1,4,2,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,7,6,2,4] => [7,3,5,1,2,6,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,3,7,6,4,2] => [7,3,5,1,2,4,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,6,3,2,4,7] => [7,3,2,5,6,4,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,6,3,2,7,4] => [7,3,2,5,6,1,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,6,3,4,2,7] => [7,3,2,5,4,6,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,6,3,4,7,2] => [7,3,2,5,4,1,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,6,3,7,2,4] => [7,3,2,5,1,6,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,6,3,7,4,2] => [7,3,2,5,1,4,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,6,7,3,2,4] => [7,3,2,1,5,6,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,6,7,3,4,2] => [7,3,2,1,5,4,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,5,7,3,2,4,6] => [7,3,1,5,6,4,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,7,3,2,6,4] => [7,3,1,5,6,2,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,7,3,4,2,6] => [7,3,1,5,4,6,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,7,3,4,6,2] => [7,3,1,5,4,2,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,7,3,6,2,4] => [7,3,1,5,2,6,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,7,3,6,4,2] => [7,3,1,5,2,4,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,7,6,3,2,4] => [7,3,1,2,5,6,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,5,7,6,3,4,2] => [7,3,1,2,5,4,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 1
[1,6,2,4,3,5,7] => [7,2,6,4,5,3,1] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
[1,6,2,4,3,7,5] => [7,2,6,4,5,1,3] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
[1,6,2,4,5,3,7] => [7,2,6,4,3,5,1] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
[1,6,2,4,5,7,3] => [7,2,6,4,3,1,5] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
[1,6,2,4,7,3,5] => [7,2,6,4,1,5,3] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
[1,6,2,4,7,5,3] => [7,2,6,4,1,3,5] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 0
[1,6,2,5,4,3,7] => [7,2,6,3,4,5,1] => [[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
Matching statistic: St001233
Mp00069: Permutations —complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[1,2,3] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,4,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,4,3,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1,3,4] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,1,4,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,3,4,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,4,1,3] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,2,4] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,4,2] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,1,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,4,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,4,1,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,4,2,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,3,1,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2,4,5] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,3,2,5,4] => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,3,4,5,2] => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,3,5,2,4] => [5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,4,2,5,3] => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,4,3,2,5] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [[[[[.,[[.,.],.]],.],.],.],.]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,5,3,2,4,6,7] => [7,3,5,6,4,2,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,2,4,7,6] => [7,3,5,6,4,1,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,2,6,4,7] => [7,3,5,6,2,4,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,2,6,7,4] => [7,3,5,6,2,1,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,2,7,4,6] => [7,3,5,6,1,4,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,2,7,6,4] => [7,3,5,6,1,2,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,4,2,6,7] => [7,3,5,4,6,2,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,4,2,7,6] => [7,3,5,4,6,1,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,4,6,2,7] => [7,3,5,4,2,6,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,4,6,7,2] => [7,3,5,4,2,1,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,4,7,2,6] => [7,3,5,4,1,6,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,4,7,6,2] => [7,3,5,4,1,2,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,6,2,4,7] => [7,3,5,2,6,4,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,6,2,7,4] => [7,3,5,2,6,1,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,6,4,2,7] => [7,3,5,2,4,6,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,6,4,7,2] => [7,3,5,2,4,1,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,6,7,2,4] => [7,3,5,2,1,6,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,6,7,4,2] => [7,3,5,2,1,4,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,3,7,2,4,6] => [7,3,5,1,6,4,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,7,2,6,4] => [7,3,5,1,6,2,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,7,4,2,6] => [7,3,5,1,4,6,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,7,4,6,2] => [7,3,5,1,4,2,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,7,6,2,4] => [7,3,5,1,2,6,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,3,7,6,4,2] => [7,3,5,1,2,4,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,6,3,2,4,7] => [7,3,2,5,6,4,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,6,3,2,7,4] => [7,3,2,5,6,1,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,6,3,4,2,7] => [7,3,2,5,4,6,1] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,6,3,4,7,2] => [7,3,2,5,4,1,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,6,3,7,2,4] => [7,3,2,5,1,6,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,6,3,7,4,2] => [7,3,2,5,1,4,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,6,7,3,2,4] => [7,3,2,1,5,6,4] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,6,7,3,4,2] => [7,3,2,1,5,4,6] => [[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 0
[1,5,7,3,2,4,6] => [7,3,1,5,6,4,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,7,3,2,6,4] => [7,3,1,5,6,2,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,7,3,4,2,6] => [7,3,1,5,4,6,2] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,7,3,4,6,2] => [7,3,1,5,4,2,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,7,3,6,2,4] => [7,3,1,5,2,6,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,7,3,6,4,2] => [7,3,1,5,2,4,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,7,6,3,2,4] => [7,3,1,2,5,6,4] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,5,7,6,3,4,2] => [7,3,1,2,5,4,6] => [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,6,2,4,3,5,7] => [7,2,6,4,5,3,1] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,6,2,4,3,7,5] => [7,2,6,4,5,1,3] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,6,2,4,5,3,7] => [7,2,6,4,3,5,1] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,6,2,4,5,7,3] => [7,2,6,4,3,1,5] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,6,2,4,7,3,5] => [7,2,6,4,1,5,3] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,6,2,4,7,5,3] => [7,2,6,4,1,3,5] => [[[.,.],[[[.,.],[.,.]],.]],.]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,6,2,5,4,3,7] => [7,2,6,3,4,5,1] => [[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 2
Description
The number of indecomposable 2-dimensional modules with projective dimension one.
Matching statistic: St001948
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Mp00064: Permutations —reverse⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 0
[2,3,1] => [3,1,2] => [2,1,3] => 0
[3,1,2] => [2,3,1] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 0
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 2
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [3,1,2,4] => [4,2,1,3] => 0
[2,3,4,1] => [4,1,2,3] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [2,3,1,4] => 1
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 0
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 1
[3,2,4,1] => [4,2,1,3] => [3,1,2,4] => 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 0
[3,4,2,1] => [4,3,1,2] => [2,1,3,4] => 1
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => 1
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => 2
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 1
[1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => 0
[1,3,4,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => 0
[1,3,5,2,4] => [1,4,2,5,3] => [3,5,2,4,1] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [3,4,2,5,1] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [5,2,4,3,1] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [4,2,5,3,1] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1
[1,4,3,5,2] => [1,5,3,2,4] => [4,2,3,5,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2,5,4,1] => 0
[1,4,5,3,2] => [1,5,4,2,3] => [3,2,4,5,1] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ? = 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => ? = 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => ? = 0
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => ? = 0
[1,2,3,6,4,5] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => ? = 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [4,5,6,3,2,1] => ? = 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => ? = 0
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => ? = 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => ? = 0
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => ? = 0
[1,2,4,6,3,5] => [1,2,5,3,6,4] => [4,6,3,5,2,1] => ? = 1
[1,2,4,6,5,3] => [1,2,6,3,5,4] => [4,5,3,6,2,1] => ? = 1
[1,2,5,3,4,6] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => ? = 0
[1,2,5,3,6,4] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => ? = 0
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => ? = 1
[1,2,5,4,6,3] => [1,2,6,4,3,5] => [5,3,4,6,2,1] => ? = 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,3,6,5,2,1] => ? = 0
[1,2,5,6,4,3] => [1,2,6,5,3,4] => [4,3,5,6,2,1] => ? = 1
[1,2,6,3,4,5] => [1,2,4,5,6,3] => [3,6,5,4,2,1] => ? = 1
[1,2,6,3,5,4] => [1,2,4,6,5,3] => [3,5,6,4,2,1] => ? = 2
[1,2,6,4,3,5] => [1,2,5,4,6,3] => [3,6,4,5,2,1] => ? = 1
[1,2,6,4,5,3] => [1,2,6,4,5,3] => [3,5,4,6,2,1] => ? = 1
[1,2,6,5,3,4] => [1,2,5,6,4,3] => [3,4,6,5,2,1] => ? = 2
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [3,4,5,6,2,1] => ? = 3
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => ? = 0
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [5,6,4,2,3,1] => ? = 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => ? = 0
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [5,4,6,2,3,1] => ? = 0
[1,3,2,6,4,5] => [1,3,2,5,6,4] => [4,6,5,2,3,1] => ? = 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [4,5,6,2,3,1] => ? = 2
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => ? = 0
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [5,6,3,2,4,1] => ? = 1
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [6,4,3,2,5,1] => ? = 0
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => ? = 0
[1,3,4,6,2,5] => [1,5,2,3,6,4] => [4,6,3,2,5,1] => ? = 1
[1,3,4,6,5,2] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ? = 1
[1,3,5,2,4,6] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => ? = 0
[1,3,5,2,6,4] => [1,4,2,6,3,5] => [5,3,6,2,4,1] => ? = 0
[1,3,5,4,2,6] => [1,5,2,4,3,6] => [6,3,4,2,5,1] => ? = 0
[1,3,5,4,6,2] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ? = 0
[1,3,5,6,2,4] => [1,5,2,6,3,4] => [4,3,6,2,5,1] => ? = 0
[1,3,5,6,4,2] => [1,6,2,5,3,4] => [4,3,5,2,6,1] => ? = 0
[1,3,6,2,4,5] => [1,4,2,5,6,3] => [3,6,5,2,4,1] => ? = 1
[1,3,6,2,5,4] => [1,4,2,6,5,3] => [3,5,6,2,4,1] => ? = 2
[1,3,6,4,2,5] => [1,5,2,4,6,3] => [3,6,4,2,5,1] => ? = 1
[1,3,6,4,5,2] => [1,6,2,4,5,3] => [3,5,4,2,6,1] => ? = 1
[1,3,6,5,2,4] => [1,5,2,6,4,3] => [3,4,6,2,5,1] => ? = 2
[1,3,6,5,4,2] => [1,6,2,5,4,3] => [3,4,5,2,6,1] => ? = 2
[1,4,2,3,5,6] => [1,3,4,2,5,6] => [6,5,2,4,3,1] => ? = 0
Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.
A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
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