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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000246
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Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000246: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 0
[[.,.],.]
=> [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 2
[[[.,.],.],.]
=> [1,2,3] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 5
[[[[.,.],.],.],.]
=> [1,2,3,4] => 6
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 4
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 4
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 6
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 4
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 4
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 6
Description
The number of non-inversions of a permutation.
For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St001161
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 79%●distinct values known / distinct values provided: 66%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 79%●distinct values known / distinct values provided: 66%
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]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 5
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [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]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[[.,[.,[[.,[.,[.,.]]],[.,.]]]],.]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[[.,[[.,[.,.]],[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 13
[[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 18
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 22
[[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 25
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 28
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 36
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 36
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 45
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 28
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 36
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ?
=> ? = 27
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 27
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ?
=> ? = 45
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ?
=> ? = 35
[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 26
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 35
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 45
[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> ?
=> ?
=> ?
=> ? = 22
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> ?
=> ?
=> ?
=> ? = 18
[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 16
Description
The major index north count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
Matching statistic: St000947
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 79%●distinct values known / distinct values provided: 66%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 79%●distinct values known / distinct values provided: 66%
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]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 5
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [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]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[[.,[.,[[.,[.,[.,.]]],[.,.]]]],.]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[[.,[[.,[.,.]],[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 13
[[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 18
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 22
[[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 25
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 28
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 36
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 36
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 45
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 28
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 36
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ?
=> ? = 27
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 27
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ?
=> ? = 45
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ?
=> ? = 35
[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[[[[[[.,[[.,.],.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[.,[[[[[[[.,.],[.,.]],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 26
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 35
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 45
[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> ?
=> ?
=> ?
=> ? = 22
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> ?
=> ?
=> ?
=> ? = 18
Description
The major index east count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Matching statistic: St000378
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 74%●distinct values known / distinct values provided: 63%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 74%●distinct values known / distinct values provided: 63%
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]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[.,[.,[.,[.,.]]]]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
[.,[.,[.,[.,[.,.]]]]]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 4
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 6
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 5
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 6
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> ? = 11
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1,1]
=> ? = 11
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1]
=> ? = 11
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1]
=> ? = 12
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1]
=> ? = 13
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> ? = 5
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,1]
=> ? = 7
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 8
[[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2]
=> ? = 7
[[.,[.,[[[.,.],.],[.,.]]]],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1]
=> ? = 9
[[.,[[.,[[.,.],.]],[.,.]]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2]
=> ? = 10
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1,1]
=> ? = 9
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 12
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> ? = 13
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 16
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 17
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 17
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 18
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2,2,1]
=> ? = 6
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? = 6
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [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]
=> [4,4,3,2,2,1]
=> ? = 6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4]
=> ? = 12
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,2,1]
=> ? = 18
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,1]
=> ? = 18
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,2]
=> ? = 18
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1,1,1]
=> ? = 19
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1,1]
=> ? = 19
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 20
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,2,1,1]
=> ? = 4
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? = 8
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ? = 8
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,2,2,2,1]
=> ? = 6
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,2,1,1]
=> ? = 5
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,2,1,1]
=> ? = 7
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2,1,1]
=> ? = 9
[[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,4,2,2,2,2]
=> ? = 6
[[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [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]
=> [7,4,2,2,1,1]
=> ? = 8
[[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,1,1,1]
=> ? = 8
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2]
=> ? = 10
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,1,1]
=> ? = 8
[[[.,[.,.]],[.,[.,.]]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [5,4,4,1,1,1,1]
=> ? = 8
[[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,1,1,1]
=> ? = 9
[[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,1,1]
=> ? = 9
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2]
=> ? = 10
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [5,4,4,2]
=> ? = 12
[[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1]
=> ? = 18
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1,1,1,1]
=> ? = 8
[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4]
=> ? = 9
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,2,2,1]
=> ? = 10
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2,2]
=> ? = 8
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000012
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 73%●distinct values known / distinct values provided: 54%
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 73%●distinct values known / distinct values provided: 54%
Values
[.,.]
=> [1,0]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [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]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 6
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 6
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 8
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 10
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 12
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 18
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 18
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 18
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 19
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 19
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 20
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 21
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 8
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 16
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 9
[[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 6
[[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 8
[[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 8
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 10
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 12
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 13
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 9
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 10
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 8
[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 10
[[.,[.,[[.,.],[[.,.],[.,.]]]]],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 9
[[.,[.,[[.,[.,.]],[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 9
[[.,[.,[[[.,.],.],[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 10
[[.,[.,[[.,[.,[.,.]]],[.,.]]]],.]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 10
[[.,[[.,.],[.,[.,[[.,.],.]]]]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 9
[[.,[[.,[.,.]],[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 11
[[.,[[[.,.],.],[[[.,.],.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 13
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 22
[[[[[.,.],.],.],[[[.,.],.],.]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 16
[[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 18
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 22
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 13
[[[.,[[[[[.,.],.],.],.],.]],.],.]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 23
[[[[.,.],[[.,.],[[.,.],.]]],.],.]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 16
[[[[[.,.],.],[[[.,.],.],.]],.],.]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 19
[[[[[[[.,.],.],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 23
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 18
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 24
[[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 24
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000018
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 73%
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 73%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 0
[[.,.],.]
=> [1,2] => [2,1] => [2,1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [2,3,1] => 2
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [3,2,1] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,3,4,2] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [2,4,3,1] => 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 5
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 6
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,4,5,3] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,3,4,2,5] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,3,4,5,2] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,3,5,4,2] => 4
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,4,3,5,2] => 4
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,4,5,3,2] => 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 6
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,4,5,3] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [2,1,5,4,3] => 4
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [2,3,1,4,5] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [2,3,1,5,4] => 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 4
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [2,3,4,1,5] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [2,4,3,1,5] => 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [3,2,4,1,5] => 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [3,4,2,1,5] => 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 6
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [7,6,5,3,4,1,2] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 3
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [6,5,7,3,4,1,2] => [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 4
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 4
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 5
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => [3,2,1,6,5,4,7] => ? = 6
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [6,4,5,7,1,2,3] => [3,2,1,7,5,4,6] => [3,2,1,6,5,7,4] => ? = 7
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [5,4,6,7,1,2,3] => [3,2,1,7,6,4,5] => [3,2,1,6,7,5,4] => ? = 8
[[[[.,.],.],.],[[.,.],[.,.]]]
=> [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => [4,3,2,1,6,5,7] => ? = 7
[[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ? = 9
[[[.,.],[[.,.],.]],[[.,.],.]]
=> [6,7,3,4,1,2,5] => [5,2,1,4,3,7,6] => [3,2,5,4,1,7,6] => ? = 7
[[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [7,1,2,3,4,6,5] => [2,3,4,5,7,6,1] => ? = 7
[[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [5,4,6,3,2,1,7] => [7,1,2,3,6,4,5] => [2,3,4,6,7,5,1] => ? = 8
[[.,[.,[.,[[[.,.],.],.]]]],.]
=> [4,5,6,3,2,1,7] => [7,1,2,3,6,5,4] => [2,3,4,7,6,5,1] => ? = 9
[[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [6,5,3,4,2,1,7] => [7,1,2,4,3,5,6] => [2,3,5,4,6,7,1] => ? = 7
[[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [6,4,3,5,2,1,7] => [7,1,2,5,3,4,6] => [2,3,5,6,4,7,1] => ? = 8
[[.,[.,[[[.,.],.],[.,.]]]],.]
=> [6,3,4,5,2,1,7] => [7,1,2,5,4,3,6] => [2,3,6,5,4,7,1] => ? = 9
[[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => [7,1,2,6,3,4,5] => [2,3,5,6,7,4,1] => ? = 9
[[.,[.,[[.,[[.,.],.]],.]]],.]
=> [4,5,3,6,2,1,7] => [7,1,2,6,3,5,4] => [2,3,5,7,6,4,1] => ? = 10
[[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [6,4,3,2,5,1,7] => [7,1,5,2,3,4,6] => [2,4,5,6,3,7,1] => ? = 9
[[.,[[.,[[.,.],.]],[.,.]]],.]
=> [6,3,4,2,5,1,7] => [7,1,5,2,4,3,6] => [2,4,6,5,3,7,1] => ? = 10
[[.,[[.,[.,[.,[.,.]]]],.]],.]
=> [5,4,3,2,6,1,7] => [7,1,6,2,3,4,5] => [2,4,5,6,7,3,1] => ? = 10
[[.,[[[[[.,.],.],.],.],.]],.]
=> [2,3,4,5,6,1,7] => [7,1,6,5,4,3,2] => [2,7,6,5,4,3,1] => ? = 16
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [5,6,3,4,1,2,7] => [7,2,1,4,3,6,5] => [3,2,5,4,7,6,1] => ? = 9
[[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => [4,3,2,7,6,5,1] => ? = 12
[[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => [7,4,3,2,1,6,5] => [5,4,3,2,7,6,1] => ? = 13
[[[[[[.,.],.],.],.],[.,.]],.]
=> [6,1,2,3,4,5,7] => [7,5,4,3,2,1,6] => [6,5,4,3,2,7,1] => ? = 16
[[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [5,4,3,2,1,6,7] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 11
[[[.,[[[[.,.],.],.],.]],.],.]
=> [2,3,4,5,1,6,7] => [7,6,1,5,4,3,2] => [3,7,6,5,4,2,1] => ? = 17
[[[[[[.,.],.],.],[.,.]],.],.]
=> [5,1,2,3,4,6,7] => [7,6,4,3,2,1,5] => [6,5,4,3,7,2,1] => ? = 17
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 15
[[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => [7,6,5,1,4,3,2] => [4,7,6,5,3,2,1] => ? = 18
[[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => [7,6,5,2,1,4,3] => [5,4,7,6,3,2,1] => ? = 17
[[[[[[.,.],.],[.,.]],.],.],.]
=> [4,1,2,3,5,6,7] => [7,6,5,3,2,1,4] => [6,5,4,7,3,2,1] => ? = 18
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 18
[[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => [7,6,5,4,1,3,2] => [5,7,6,4,3,2,1] => ? = 19
[[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => [7,6,5,4,2,1,3] => [6,5,7,4,3,2,1] => ? = 19
[[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 20
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,2,3,4,5,7,8,6] => ? = 2
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [1,2,3,4,6,5,8,7] => ? = 2
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [1,2,3,5,4,6,8,7] => ? = 2
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,2,3,5,7,8,6,4] => ? = 6
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => ? = 2
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [1,2,4,3,5,7,6,8] => ? = 2
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => [1,2,5,4,3,8,7,6] => ? = 6
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => [1,3,2,4,5,6,8,7] => ? = 2
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => [1,3,2,4,6,5,7,8] => ? = 2
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => [1,4,3,2,5,8,7,6] => ? = 6
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => [1,5,6,4,3,2,8,7] => ? = 10
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 6
Description
The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000161
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000161: Binary trees ⟶ ℤResult quality: 63% ●values known / values provided: 70%●distinct values known / distinct values provided: 63%
St000161: Binary trees ⟶ ℤResult quality: 63% ●values known / values provided: 70%●distinct values known / distinct values provided: 63%
Values
[.,.]
=> [.,.]
=> 0
[.,[.,.]]
=> [[.,.],.]
=> 0
[[.,.],.]
=> [.,[.,.]]
=> 1
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> 0
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> 1
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> 2
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> 3
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> 0
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> 1
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> 2
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> 3
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> 1
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> 2
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> 2
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 3
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> 3
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> 4
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> 4
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> 5
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> 6
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> 4
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 4
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> 5
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> 6
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 3
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> 3
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> 5
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> 6
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> ? = 3
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],[.,.]],[.,.]]
=> ? = 4
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],[.,[.,.]]]
=> ? = 4
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> ? = 5
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> ? = 7
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> ? = 8
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> ? = 6
[[[[.,.],.],.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> ? = 7
[[[[.,.],.],.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ? = 9
[[[.,.],[[.,.],.]],[[.,.],.]]
=> [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> ? = 7
[[[[[.,.],.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> ? = 10
[[.,[.,[[[.,.],.],[.,.]]]],.]
=> [.,[[[[.,.],[.,[.,.]]],.],.]]
=> ? = 9
[[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [.,[[[.,.],[[[.,.],.],.]],.]]
=> ? = 9
[[.,[[.,[[.,.],.]],[.,.]]],.]
=> [.,[[[.,.],[[.,[.,.]],.]],.]]
=> ? = 10
[[.,[[[[[.,.],.],.],.],.]],.]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> ? = 16
[[[[.,.],.],[[[.,.],.],.]],.]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ? = 12
[[[[[.,.],.],.],[[.,.],.]],.]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ? = 13
[[[[[[.,.],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 16
[[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ? = 11
[[[.,[[[[.,.],.],.],.]],.],.]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ? = 17
[[[[[[.,.],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 17
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ? = 15
[[[[.,[[[.,.],.],.]],.],.],.]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ? = 18
[[[[[.,.],[[.,.],.]],.],.],.]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ? = 17
[[[[[[.,.],.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 18
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 18
[[[[[.,[[.,.],.]],.],.],.],.]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 19
[[[[[[.,.],[.,.]],.],.],.],.]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? = 19
[[[[[[.,[.,.]],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ? = 20
[[[[[[[.,.],.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 21
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [[[[[[.,[[.,.],.]],.],.],.],.],.]
=> ? = 2
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> ? = 2
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> ? = 2
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ? = 4
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [[[[.,[[.,[[.,.],.]],.]],.],.],.]
=> ? = 6
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> ? = 2
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> ? = 2
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> ? = 6
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> ? = 2
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> ? = 6
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [[[.,[.,.]],[.,[.,[[.,.],.]]]],.]
=> ? = 10
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 6
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> ? = 8
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> ? = 10
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> ? = 3
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> ? = 8
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> ? = 8
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 16
Description
The sum of the sizes of the right subtrees of a binary tree.
This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the $312$-avoiding permutation corresponding to the binary tree.
It is also the sum of all heights $j$ of the coordinates $(i,j)$ of the Dyck path corresponding to the binary tree.
Matching statistic: St000041
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 61% ●values known / values provided: 68%●distinct values known / distinct values provided: 61%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 61% ●values known / values provided: 68%●distinct values known / distinct values provided: 61%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [(1,2)]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 5
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 6
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 4
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 4
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 6
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 5
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 6
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,14),(12,13)]
=> ? = 4
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,14),(10,13),(11,12)]
=> ? = 7
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,14),(10,13),(11,12)]
=> ? = 8
[[[.,.],[[.,.],.]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,14),(6,9),(7,8),(10,13),(11,12)]
=> ? = 7
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12),(13,14),(15,16)]
=> ? = 2
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12),(13,14),(15,16)]
=> ? = 2
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14),(15,16)]
=> ? = 2
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12),(13,14),(15,16)]
=> ? = 2
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12),(13,14),(15,16)]
=> ? = 4
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9),(11,12),(13,14),(15,16)]
=> ? = 6
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11),(13,14),(15,16)]
=> ? = 2
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11),(13,14),(15,16)]
=> ? = 2
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16)]
=> ? = 2
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,14),(15,16)]
=> ? = 6
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,14),(12,13),(15,16)]
=> ? = 2
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,14),(12,13),(15,16)]
=> ? = 2
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13),(15,16)]
=> ? = 2
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13),(15,16)]
=> ? = 2
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,14),(10,11),(12,13),(15,16)]
=> ? = 4
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12),(15,16)]
=> ? = 6
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,14),(10,13),(11,12),(15,16)]
=> ? = 6
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,4),(2,3),(5,14),(6,13),(7,12),(8,9),(10,11),(15,16)]
=> ? = 10
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13),(15,16)]
=> ? = 6
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [(1,14),(2,7),(3,4),(5,6),(8,9),(10,11),(12,13),(15,16)]
=> ? = 8
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [(1,14),(2,11),(3,4),(5,6),(7,8),(9,10),(12,13),(15,16)]
=> ? = 10
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [(1,14),(2,11),(3,8),(4,5),(6,7),(9,10),(12,13),(15,16)]
=> ? = 12
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 2
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12),(13,16),(14,15)]
=> ? = 2
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12),(13,16),(14,15)]
=> ? = 2
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,16),(14,15)]
=> ? = 2
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11),(13,16),(14,15)]
=> ? = 2
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11),(13,16),(14,15)]
=> ? = 3
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,16),(14,15)]
=> ? = 4
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15)]
=> ? = 8
[[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9),(13,16),(14,15)]
=> ? = 8
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,16),(14,15)]
=> ? = 16
[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,16),(12,13),(14,15)]
=> ? = 4
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,16),(12,15),(13,14)]
=> ? = 6
[[[.,.],.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,16),(12,15),(13,14)]
=> ? = 6
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,16),(12,15),(13,14)]
=> ? = 4
[[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,16),(12,15),(13,14)]
=> ? = 5
[[[.,.],.],[[[.,.],.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,16),(12,15),(13,14)]
=> ? = 6
[[[.,.],.],[[[.,.],.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,16),(12,15),(13,14)]
=> ? = 7
[[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8),(11,16),(12,15),(13,14)]
=> ? = 9
[[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,16),(10,11),(12,13),(14,15)]
=> ? = 6
[[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,16),(10,13),(11,12),(14,15)]
=> ? = 8
[[[.,.],[.,.]],[[.,[[.,.],.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,16),(10,11),(12,15),(13,14)]
=> ? = 8
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,16),(10,15),(11,12),(13,14)]
=> ? = 10
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,16),(10,15),(11,14),(12,13)]
=> ? = 7
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,16),(10,15),(11,14),(12,13)]
=> ? = 8
Description
The number of nestings of a perfect matching.
This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$.
Matching statistic: St000081
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 73%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 73%
Values
[.,.]
=> [1] => [1] => ([],1)
=> 0
[.,[.,.]]
=> [2,1] => [1,2] => ([],2)
=> 0
[[.,.],.]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => ([],3)
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => ([(1,2)],3)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [1,7,6,3,2,5,4] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [7,6,5,3,4,1,2] => [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
=> ? = 2
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
=> ? = 3
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [6,5,7,3,4,1,2] => [2,1,4,3,7,5,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 4
[[.,.],[[[[[.,.],.],.],.],.]]
=> [3,4,5,6,7,1,2] => [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ([(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ? = 6
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [6,4,5,7,1,2,3] => [3,2,1,7,5,4,6] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 7
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [5,4,6,7,1,2,3] => [3,2,1,7,6,4,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8
[[[[.,.],.],.],[[.,.],[.,.]]]
=> [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7
[[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[[[.,.],[[.,.],.]],[[.,.],.]]
=> [6,7,3,4,1,2,5] => [5,2,1,4,3,7,6] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[[[[.,.],[[.,.],.]],.],[.,.]]
=> [7,3,4,1,2,5,6] => [6,5,2,1,4,3,7] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
[[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => [7,4,3,2,1,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13
[[[[[.,.],[[.,.],.]],.],.],.]
=> [3,4,1,2,5,6,7] => [7,6,5,2,1,4,3] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 17
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 0
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? = 2
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 6
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => ([(4,7),(5,6)],8)
=> ? = 2
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,3,2,4,1] => [1,4,2,3,7,5,6,8] => ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 4
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 6
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => ([(2,6),(2,7),(3,4),(3,5),(4,5),(6,7)],8)
=> ? = 6
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => ([(1,2),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [1,8,2,7,3,4,5,6] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 12
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 18
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [5,2,3,4,6,7,8,1] => [1,8,7,6,4,3,2,5] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 18
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 18
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 19
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 19
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 20
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 21
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 2
Description
The number of edges of a graph.
Matching statistic: St000067
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 59%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 59%
Values
[.,.]
=> [1,0]
=> [[1]]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 5
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 6
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 4
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> 4
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 6
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 5
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 6
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 0
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 2
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 3
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 4
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 4
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 6
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 7
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 8
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 6
[[[[.,.],.],.],[[.,.],[.,.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 7
[[[[.,.],.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 9
[[[.,.],[[.,.],.]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 7
[[[[[.,.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 10
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 6
[[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[.,[.,[.,[[.,[.,.]],.]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 8
[[.,[.,[.,[[[.,.],.],.]]]],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 9
[[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 7
[[.,[.,[[.,[.,.]],[.,.]]]],.]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 8
[[.,[.,[[[.,.],.],[.,.]]]],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 9
[[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 9
[[.,[.,[[.,[[.,.],.]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 10
[[.,[[.,[.,[.,.]]],[.,.]]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 9
[[.,[[.,[[.,.],.]],[.,.]]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 10
[[.,[[.,[.,[.,[.,.]]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 10
[[.,[[[[[.,.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 16
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 9
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 12
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 13
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,-1,1],[0,0,0,0,0,1,0]]
=> ? = 16
[[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 11
[[[.,[[[[.,.],.],.],.]],.],.]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,0,1],[1,-1,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 17
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[1,0,0,0,-1,1,0],[0,0,0,0,1,0,0]]
=> ? = 17
[[[[.,[.,[.,[.,.]]]],.],.],.]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,1,-1,1,-1,1,0],[1,-1,1,-1,1,-1,1],[0,1,-1,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 15
[[[[.,[[[.,.],.],.]],.],.],.]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 18
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,0,1],[0,1,0,-1,0,1,0],[1,0,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 17
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 18
[[[[[.,[.,[.,.]]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,1,-1,1,-1,1,0],[1,-1,1,-1,1,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 18
[[[[[.,[[.,.],.]],.],.],.],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 19
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,1,0,-1,1,0,0],[1,0,-1,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 19
[[[[[[.,[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 20
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 21
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 0
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? = 2
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 2
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? = 2
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
Description
The inversion number of the alternating sign matrix.
If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as
$$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$
When restricted to permutation matrices, this gives the usual inversion number of the permutation.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001397Number of pairs of incomparable elements in a finite poset. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000057The Shynar inversion number of a standard tableau. St000332The positive inversions of an alternating sign matrix. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000795The mad of a permutation. St001428The number of B-inversions of a signed permutation. St000833The comajor index of a permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000004The major index of a permutation. St000042The number of crossings of a perfect matching. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000448The number of pairs of vertices of a graph with distance 2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001718The number of non-empty open intervals in a poset. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001866The nesting alignments of a signed permutation. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001822The number of alignments of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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