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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000237
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(load all 5 compositions to match this statistic)
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%
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
Description
The number of small exceedances.
This is the number of indices i such that πi=i+1.
Matching statistic: St001484
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 91%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [.,.]
=> [1,0]
=> []
=> 0
[1,0,1,0]
=> [.,[.,.]]
=> [1,1,0,0]
=> []
=> 0
[1,1,0,0]
=> [[.,.],.]
=> [1,0,1,0]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,1,1]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,4,2,2,1]
=> ? = 2
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[[[[.,[.,.]],[.,.]],.],.]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,2]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[.,[[[.,[[.,.],.]],.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,1]
=> ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,1,1]
=> ? = 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> ? = 6
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [[[[.,[.,.]],[.,[[.,.],.]]],.],.]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,2,2]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[[[.,[.,[.,.]]],[[.,.],.]],.],.]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [[[.,.],[.,[[[.,.],[.,.]],.]]],.]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,1,1,1]
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [[[[.,[.,[[.,.],.]]],.],[.,.]],.]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,1]
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? = 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,.],[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2,1,1]
=> ? = 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2]
=> ? = 4
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],[[[[.,.],.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3]
=> ? = 3
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[.,[[.,.],[.,.]]],[.,[[.,.],.]]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,1,1]
=> ? = 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 2
[1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [[[.,.],.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,4,2,2,2,1]
=> ? = 2
[1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [5,5,3,2,2,2,1]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],[.,[[.,.],.]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,1]
=> ? = 3
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [[[.,[[.,[[.,.],.]],.]],.],[.,.]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,1]
=> ? = 3
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [[[.,[[.,.],[.,.]]],.],[[.,.],.]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,1,1]
=> ? = 2
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,1,1]
=> ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[.,.],[.,[.,.]]],.],[[.,.],.]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,1,1,1]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[[.,.],[.,[[.,.],[.,.]]]],[.,.]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [6,6,2,2,1,1,1]
=> ? = 0
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,[.,.]]]],[.,.]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [6,6,2,2,2,1,1]
=> ? = 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,1,1,1]
=> ? = 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ? = 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,1,1,1]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[.,.],[.,[.,[.,.]]]],[[.,.],.]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [6,5,5,1,1,1,1]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[[.,.],.]]],[[.,.],.]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,1,1,1]
=> ? = 2
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [[[.,[[.,.],.]],[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3,1]
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,2,2,1]
=> ? = 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,1,1,1]
=> ? = 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,1,1]
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1,1,1]
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,1,1]
=> ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,1,1]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,2,2,1]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ?
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ?
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ?
=> ? = 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ?
=> ? = 0
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [.,[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [5,5,4,4,2,2]
=> ? = 0
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [6,6,5,5,3,3,1,1]
=> ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ?
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[[[[.,.],.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ?
=> ? = 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
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 73%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[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]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 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]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[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]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 0
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 0
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [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]
=> ? = 0
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: St000214
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 91%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,2,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,3,4,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,4,5,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,3,2,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,2,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,4,5,2,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,2,1] => 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => [1,3,6,5,4,7,2] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7,3,6,5,4,2] => [1,3,5,6,4,7,2] => ? = 0
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,7,4,3,6,5,2] => [1,4,3,6,5,7,2] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,7,4,5,3,6,2] => [1,5,4,3,6,7,2] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => [1,5,6,4,3,7,2] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => [1,4,5,3,7,6,2] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,7,5,4,3,6,2] => [1,4,5,3,6,7,2] => ? = 0
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,7,5,4,6,3,2] => [1,4,6,5,3,7,2] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => [1,4,5,6,3,7,2] => ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [1,5,4,6,3,7,2] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => [2,1,5,6,4,7,3] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,5,7,6] => [4,3,2,1,5,7,6] => ? = 4
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => [2,7,6,5,4,3,1] => ? = 4
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => [2,3,7,6,5,4,1] => ? = 3
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,7,6,5,1] => ? = 2
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => [2,3,4,5,7,6,1] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,4,5,3,1,7] => [2,5,4,3,6,1,7] => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,2,5,4,3,1,7] => [2,4,5,3,6,1,7] => ? = 0
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ? = 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => [3,2,7,6,5,4,1] => ? = 4
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [5,3,2,4,6,7,1] => [3,2,4,7,6,5,1] => ? = 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,3,2,5,4,1,7] => [3,2,5,4,6,1,7] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,4,2,5,1,7] => [4,3,2,5,6,1,7] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => [4,5,3,2,6,1,7] => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [3,4,2,5,1,6,7] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => [3,4,2,5,1,7,6] => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => [3,4,2,6,5,1,7] => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,4,3,2,5,1,7] => [3,4,2,5,6,1,7] => ? = 0
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => [3,5,4,2,6,1,7] => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,5,3,4,2,1,7] => [3,4,5,2,6,1,7] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [1,2,3,8,7,6,5,4] => ? = 4
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,3,8,7,6,5,2] => [1,3,6,7,5,8,4,2] => ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => [1,6,5,7,4,3,8,2] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,6,5,4,3,8,7,2] => [1,4,5,3,7,8,6,2] => ? = 0
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,8,5,4,7,6,3,2] => [1,4,6,7,5,3,8,2] => ? = 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,8,7,4,5,6,3,2] => [1,4,5,6,7,3,8,2] => ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => [1,4,6,5,7,3,8,2] => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,8,6,5,4,7,3,2] => [1,5,4,7,6,3,8,2] => ? = 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,8,7,5,4,6,3,2] => [1,5,4,6,7,3,8,2] => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,8,7,5,6,4,3,2] => [1,6,5,4,7,3,8,2] => ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,5,6,4,7,3,8,2] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,3,6,8,7,5] => [2,1,4,3,7,8,6,5] => ? = 3
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,4,5,8,6,7,3] => [2,1,6,7,8,5,4,3] => ? = 3
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,1,4,6,5,3,8,7] => [2,1,5,6,4,3,8,7] => ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,4,7,6,8,3] => [2,1,4,6,8,7,5,3] => ? = 2
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,8,5,6,7,1] => [5,6,7,8,4,3,2,1] => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,3,6,4,5,1,8,7] => [4,5,6,3,2,1,8,7] => ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,3,7,4,5,6,8,1] => [4,5,6,8,7,3,2,1] => ? = 3
Description
The number of adjacencies of a permutation.
An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000445
Mp00140: Dyck paths —logarithmic height to pruning number⟶ 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: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [.,.]
=> [1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0]
=> [.,[.,.]]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0]
=> [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[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]
=> 0
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [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
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 0
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [[[.,[.,.]],.],[[[.,.],.],.]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[[.,.],.]]],[.,.]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[[.,.],[[.,[.,.]],.]],[.,.]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[[.,.],.]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [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
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[[.,[.,[.,.]]],.]],.]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [.,[[.,.],[[[.,.],[.,.]],[.,.]]]]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [.,[[[.,.],[[.,.],[.,.]]],[.,.]]]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[.,[[.,[[[.,.],[.,.]],.]],.]],.]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],.]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[[.,[[[.,.],.],[.,.]]],.]],.]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[.,[[[[.,[.,.]],[.,.]],.],.]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [[.,[[[.,.],[.,[[.,.],.]]],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],[[.,[[.,.],.]],.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[[.,.],[[.,.],.]]]],.]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[.,[[[.,.],.],[.,[[.,.],.]]]],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[.,[[[.,[[.,.],.]],.],[.,.]]],.]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [[[.,[.,[.,[[[.,.],.],.]]]],.],.]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [[[[.,[.,.]],[.,[[.,.],.]]],.],.]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[[[.,[.,[.,.]]],[[.,.],.]],.],.]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,[[.,[[.,.],.]],.]]],.]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [[[.,.],[.,[[[.,.],[.,.]],.]]],.]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [[[.,.],[.,[[.,.],[[.,.],.]]]],.]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [[[[.,[.,[[.,.],.]]],.],[.,.]],.]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000441
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 91%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 91%
Values
[1,0]
=> [.,.]
=> [.,.]
=> [1] => 0
[1,0,1,0]
=> [.,[.,.]]
=> [.,[.,.]]
=> [2,1] => 0
[1,1,0,0]
=> [[.,.],.]
=> [[.,.],.]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => 2
[1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,2,3,1] => ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],[.,.]],.]]
=> [.,[[[[.,.],[.,.]],[.,.]],.]]
=> [6,4,2,3,5,7,1] => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => ? = 0
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],.],[.,.]]]
=> [7,3,2,4,5,6,1] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],[.,.]],.]]
=> [6,3,2,4,5,7,1] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[[[.,.],[[.,.],.]],[.,.]]]
=> [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [7,5,3,2,4,6,1] => ? = 0
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [6,4,3,2,5,7,1] => ? = 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> [6,3,4,2,5,7,1] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,[[[.,.],[.,.]],[.,.]]],.]
=> [[.,.],[[[.,[.,.]],[.,.]],.]]
=> [6,4,3,5,7,1,2] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [[[[.,[.,[[.,.],.]]],.],.],.]
=> [[[[.,.],.],.],[.,[[.,.],.]]]
=> [6,7,5,1,2,3,4] => ? = 4
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[.,.],[.,[.,.]]],[.,.]],.]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [6,5,3,1,2,4,7] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [7,6,2,3,4,1,5] => ? = 2
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [7,6,5,2,3,1,4] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [5,6,7,3,2,1,4] => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,2,1,3] => ? = 0
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[[.,[[.,.],.]],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [6,7,3,2,1,4,5] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[[.,.],[.,[.,.]]],.],[.,.]]
=> [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [6,5,2,1,3,4,7] => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[[.,.],[.,[.,[.,.]]]],[.,.]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [6,5,4,2,1,3,7] => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[.,.],[.,[[.,.],.]]],[.,.]]
=> [[[.,[.,.]],[.,[[.,.],.]]],.]
=> [5,6,4,2,1,3,7] => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[[.,.],[[.,[.,.]],.]],[.,.]]
=> [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [6,4,5,2,1,3,7] => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[[.,[.,.]],[.,[.,.]]],[.,.]]
=> [[[.,[.,.]],[.,[.,.]]],[.,.]]
=> [7,5,4,2,1,3,6] => ? = 0
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [6,5,3,2,1,4,7] => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[[.,.],.]]
=> [[[.,[[.,.],.]],[.,[.,.]]],.]
=> [6,5,2,3,1,4,7] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],[.,.]]]
=> [[[.,[[.,[.,.]],.]],[.,.]],.]
=> [6,3,2,4,1,5,7] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [8,6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [8,7,6,4,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,4,2,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [8,6,7,4,5,2,3,1] => ? = 3
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [6,5,4,7,3,2,8,1] => ? = 0
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [.,[[.,.],[[[.,.],[.,.]],[.,.]]]]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [6,4,3,5,7,2,8,1] => ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> [7,3,4,2,5,6,8,1] => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [.,[[[.,.],[[.,.],[.,.]]],[.,.]]]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [6,5,7,3,2,4,8,1] => ? = 0
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [7,5,3,2,4,6,8,1] => ? = 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [7,5,4,3,2,6,8,1] => ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> [7,4,5,3,2,6,8,1] => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [.,[[[[.,[[.,.],.]],[.,.]],.],.]]
=> [6,3,4,2,5,7,8,1] => ? = 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [.,[[[.,[[.,.],[.,.]]],[.,.]],.]]
=> [7,5,3,4,2,6,8,1] => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> [7,3,4,5,2,6,8,1] => ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [7,4,3,5,2,6,8,1] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[.,[[.,[[[.,.],[.,.]],.]],.]],.]
=> [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,4,1,2] => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],.]
=> [[.,.],[[.,.],[[.,[[.,.],.]],.]]]
=> [6,7,5,8,3,4,1,2] => ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[[.,[[[.,.],.],[.,.]]],.]],.]
=> [[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,3,4,1,2] => ? = 3
Description
The number of successions of a permutation.
A succession of a permutation π is an index i such that π(i)+1=π(i+1). Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000502
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 73%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => {{1}}
=> ? = 0
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> 0
[1,1,0,0]
=> [2,1] => {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,8,7,2] => {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 0
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,3,8,7,6,5,2] => {{1},{2,4,8},{3},{5,7},{6}}
=> ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => {{1},{2,8},{3,4,7},{5,6}}
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,6,5,4,3,8,7,2] => {{1},{2,6,8},{3,5},{4},{7}}
=> ? = 0
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,8,5,4,7,6,3,2] => {{1},{2,8},{3,5,7},{4},{6}}
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,8,7,4,5,6,3,2] => {{1},{2,8},{3,7},{4},{5},{6}}
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => {{1},{2,8},{3,7},{4},{5,6}}
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,8,6,5,4,7,3,2] => {{1},{2,8},{3,6,7},{4,5}}
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,8,7,5,4,6,3,2] => {{1},{2,8},{3,7},{4,5},{6}}
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,8,7,5,6,4,3,2] => {{1},{2,8},{3,7},{4,5,6}}
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,3,6,8,7,5] => {{1,2},{3,4},{5,6,8},{7}}
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,6,8,5] => {{1,2},{3,4},{5,7,8},{6}}
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,8,7,6,5] => {{1,2},{3,4},{5,8},{6,7}}
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,4,5,8,6,7,3] => {{1,2},{3,4,5,8},{6},{7}}
=> ? = 3
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,1,4,6,5,3,8,7] => {{1,2},{3,4,6},{5},{7,8}}
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,6,3,8,7] => {{1,2},{3,5,6},{4},{7,8}}
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,4,7,6,8,3] => {{1,2},{3,5,7,8},{4},{6}}
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,6,5,4,3,8,7] => {{1,2},{3,6},{4,5},{7,8}}
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,8,5,4,7,6,3] => {{1,2},{3,8},{4,5},{6,7}}
=> ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,8,7,6,5,4,3] => {{1,2},{3,8},{4,7},{5,6}}
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
=> ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,4,1,6,7,8,5] => {{1,2,3,4},{5,6,7,8}}
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,5,1,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,1,7,8] => {{1,2,3,4,5,6},{7},{8}}
=> ? = 5
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,8,5,6,7,1] => {{1,2,3,4,8},{5},{6},{7}}
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,3,6,4,5,1,8,7] => {{1,2,3,6},{4},{5},{7,8}}
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,3,7,4,5,6,8,1] => {{1,2,3,7,8},{4},{5},{6}}
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,4,3,1,6,5,8,7] => {{1,2,4},{3},{5,6},{7,8}}
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,4,3,1,6,8,7,5] => {{1,2,4},{3},{5,6,8},{7}}
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,3,1,7,6,8,5] => {{1,2,4},{3},{5,7,8},{6}}
=> ? = 2
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,8,4,3,5,7,6,1] => {{1,2,8},{3,4},{5},{6,7}}
=> ? = 3
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 0
Description
The number of successions of a set partitions.
This is the number of indices i such that i and i+1 belonging to the same block.
Matching statistic: St000504
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 73%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 73%
Values
[1,0]
=> [1] => {{1}}
=> {{1}}
=> ? = 0 + 1
[1,0,1,0]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> {{1,3},{2}}
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [3,2,1] => {{1,3},{2}}
=> {{1},{2,3}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> {{1,3,4},{2}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> {{1,5},{2,4},{3}}
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> {{1,4},{2,5},{3}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> {{1,2,4,5},{3}}
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => {{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ?
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ?
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> {{1,5,6,7,8},{2},{3},{4}}
=> ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> {{1,4,6},{2},{3},{5},{7},{8}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => {{1},{2,3},{4},{5},{6},{7},{8}}
=> ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => {{1},{2,3},{4},{5},{6,7},{8}}
=> {{1,3,7},{2},{4},{5},{6},{8}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => {{1},{2,3},{4,5},{6},{7},{8}}
=> ?
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => {{1},{2,3},{4,5},{6,7},{8}}
=> {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => {{1},{2,3,4},{5,6,7},{8}}
=> {{1,3,4,6,7},{2},{5},{8}}
=> ? = 4 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,7,6,5,8] => {{1},{2,4},{3},{5,7},{6},{8}}
=> {{1},{2,4,7},{3},{5},{6},{8}}
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,8,7,2] => {{1},{2,4,6,8},{3},{5},{7}}
=> {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,3,8,7,6,5,2] => {{1},{2,4,8},{3},{5,7},{6}}
=> {{1},{2,4,7},{3,8},{5},{6}}
=> ? = 0 + 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => {{1},{2,8},{3,4,7},{5,6}}
=> {{1,4,6},{2,7},{3,8},{5}}
=> ? = 2 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,6,5,4,3,8,7,2] => {{1},{2,6,8},{3,5},{4},{7}}
=> {{1},{2,5,8},{3,6},{4},{7}}
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,8,5,4,7,6,3,2] => {{1},{2,8},{3,5,7},{4},{6}}
=> {{1},{2,5,7},{3},{4,8},{6}}
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,8,7,4,5,6,3,2] => {{1},{2,8},{3,7},{4},{5},{6}}
=> {{1},{2},{3},{4,7},{5,8},{6}}
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => {{1},{2,8},{3,7},{4},{5,6}}
=> {{1,6},{2},{3,7},{4,8},{5}}
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,8,6,5,4,7,3,2] => {{1},{2,8},{3,6,7},{4,5}}
=> {{1,5,7},{2,6},{3,8},{4}}
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,8,7,5,4,6,3,2] => {{1},{2,8},{3,7},{4,5},{6}}
=> {{1,5},{2},{3,7},{4,8},{6}}
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,8,7,5,6,4,3,2] => {{1},{2,8},{3,7},{4,5,6}}
=> {{1,5,6},{2,7},{3,8},{4}}
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => {{1},{2,8},{3,7},{4,6},{5}}
=> {{1},{2,6},{3,7},{4,8},{5}}
=> ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,8,7] => {{1,2},{3},{4},{5},{6},{7,8}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7] => {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 4 + 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,3,6,8,7,5] => {{1,2},{3,4},{5,6,8},{7}}
=> {{1,2,4,6},{3,8},{5},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,6,8,5] => {{1,2},{3,4},{5,7,8},{6}}
=> {{1,2,4,8},{3,7},{5},{6}}
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,8,7,6,5] => {{1,2},{3,4},{5,8},{6,7}}
=> {{1,2,4,7},{3,8},{5},{6}}
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,1,4,5,8,6,7,3] => {{1,2},{3,4,5,8},{6},{7}}
=> {{1,2,4,5},{3},{6,8},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,1,4,6,5,3,8,7] => {{1,2},{3,4,6},{5},{7,8}}
=> {{1,2,4,8},{3,6},{5},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,6,3,8,7] => {{1,2},{3,5,6},{4},{7,8}}
=> {{1,2,6,8},{3,5},{4},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,4,7,6,8,3] => {{1,2},{3,5,7,8},{4},{6}}
=> {{1,2,8},{3,5,7},{4},{6}}
=> ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,6,5,4,3,8,7] => {{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,1,8,5,4,7,6,3] => {{1,2},{3,8},{4,5},{6,7}}
=> {{1,2,5,7},{3},{4,8},{6}}
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,8,7,6,5,4,3] => {{1,2},{3,8},{4,7},{5,6}}
=> {{1,2,6},{3,7},{4,8},{5}}
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,5,7,8,6] => {{1,2,3},{4},{5},{6,7,8}}
=> {{1,2,3,7,8},{4},{5},{6}}
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,5,1,6,7,8] => {{1,2,3,4,5},{6},{7},{8}}
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,1,7,8] => {{1,2,3,4,5,6},{7},{8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,8,5,6,7,1] => {{1,2,3,4,8},{5},{6},{7}}
=> {{1,2,3,4},{5},{6},{7,8}}
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [2,3,6,4,5,1,8,7] => {{1,2,3,6},{4},{5},{7,8}}
=> {{1,2,3,8},{4},{5,6},{7}}
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,3,7,4,5,6,8,1] => {{1,2,3,7,8},{4},{5},{6}}
=> {{1,2,3,8},{4},{5},{6,7}}
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,4,3,1,6,5,8,7] => {{1,2,4},{3},{5,6},{7,8}}
=> {{1,2,6,8},{3,4},{5},{7}}
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,4,3,1,6,8,7,5] => {{1,2,4},{3},{5,6,8},{7}}
=> {{1,2,6},{3,4,8},{5},{7}}
=> ? = 2 + 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,3,1,7,6,8,5] => {{1,2,4},{3},{5,7,8},{6}}
=> {{1,2,8},{3,4,7},{5},{6}}
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,8,4,3,5,7,6,1] => {{1,2,8},{3,4},{5},{6,7}}
=> {{1,2,4,7},{3},{5},{6,8}}
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,5,8,7,6] => {{1,3},{2},{4},{5},{6,8},{7}}
=> {{1},{2,3,8},{4},{5},{6},{7}}
=> ? = 0 + 1
Description
The cardinality of the first block of a set partition.
The number of partitions of {1,…,n} into k blocks in which the first block has cardinality j+1 is given by \binom{n-1}{j}S(n-j-1,k-1), see [1, Theorem 1.1] and the references therein. Here, S(n,k) are the ''Stirling numbers of the second kind'' counting all set partitions of \{1,\ldots,n\} into k blocks [2].
Matching statistic: St001189
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 64%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 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]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[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]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [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]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [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
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [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]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [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]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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]
=> ? = 7
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001067
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 64%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 64%
Values
[1,0]
=> [1,0]
=> [.,.]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[[.,[[.,[[[.,.],.],.]],.]],.],.]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [[[.,.],[[[.,.],[[.,.],.]],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[[.,[[.,.],[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,[.,.]],.]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,[[.,.],.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[[.,[.,[[[.,.],.],[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [.,[[.,[[.,.],[[.,[.,.]],.]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [.,[[[.,.],[[.,.],[.,[.,.]]]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [.,[[[[.,.],[.,.]],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [.,[.,[[[.,.],.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 3
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 2
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001061The number of indices that are both descents and recoils of a permutation. St000247The number of singleton blocks of a set partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000731The number of double exceedences of a permutation. St000366The number of double descents of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000039The number of crossings of a permutation. St000732The number of double deficiencies of a permutation. St001429The number of negative entries in a signed permutation. St001948The number of augmented double ascents of a permutation. St000080The rank of the poset. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001115The number of even descents of a permutation. St000649The number of 3-excedences of a permutation. St000894The trace of an alternating sign matrix.
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