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Matching statistic: St000161
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St000161: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0
[.,[.,.]]
=> 1
[[.,.],.]
=> 0
[.,[.,[.,.]]]
=> 3
[.,[[.,.],.]]
=> 2
[[.,.],[.,.]]
=> 1
[[.,[.,.]],.]
=> 1
[[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> 6
[.,[.,[[.,.],.]]]
=> 5
[.,[[.,.],[.,.]]]
=> 4
[.,[[.,[.,.]],.]]
=> 4
[.,[[[.,.],.],.]]
=> 3
[[.,.],[.,[.,.]]]
=> 3
[[.,.],[[.,.],.]]
=> 2
[[.,[.,.]],[.,.]]
=> 2
[[[.,.],.],[.,.]]
=> 1
[[.,[.,[.,.]]],.]
=> 3
[[.,[[.,.],.]],.]
=> 2
[[[.,.],[.,.]],.]
=> 1
[[[.,[.,.]],.],.]
=> 1
[[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> 10
[.,[.,[.,[[.,.],.]]]]
=> 9
[.,[.,[[.,.],[.,.]]]]
=> 8
[.,[.,[[.,[.,.]],.]]]
=> 8
[.,[.,[[[.,.],.],.]]]
=> 7
[.,[[.,.],[.,[.,.]]]]
=> 7
[.,[[.,.],[[.,.],.]]]
=> 6
[.,[[.,[.,.]],[.,.]]]
=> 6
[.,[[[.,.],.],[.,.]]]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> 7
[.,[[.,[[.,.],.]],.]]
=> 6
[.,[[[.,.],[.,.]],.]]
=> 5
[.,[[[.,[.,.]],.],.]]
=> 5
[.,[[[[.,.],.],.],.]]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> 6
[[.,.],[.,[[.,.],.]]]
=> 5
[[.,.],[[.,.],[.,.]]]
=> 4
[[.,.],[[.,[.,.]],.]]
=> 4
[[.,.],[[[.,.],.],.]]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> 4
[[.,[.,.]],[[.,.],.]]
=> 3
[[[.,.],.],[.,[.,.]]]
=> 3
[[[.,.],.],[[.,.],.]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> 4
[[.,[[.,.],.]],[.,.]]
=> 3
[[[.,.],[.,.]],[.,.]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> 2
[[[[.,.],.],.],[.,.]]
=> 1
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: St000008
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 64%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 64%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [1,1] => 1
[[.,.],.]
=> [1,2] => [1,2] => [2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1] => 3
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [2,1] => 2
[[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => [1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [2,1,1] => 5
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,3,1] => [1,2,1] => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [1,2,1] => 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => [1,1,2] => 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,4,1,2] => [2,2] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,4,1,3] => [2,2] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1] => 10
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [2,1,1,1] => 9
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,3,4,2,1] => [1,2,1,1] => 8
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [1,2,1,1] => 8
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => 7
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,2,3,1] => [1,1,2,1] => 7
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [4,5,2,3,1] => [2,2,1] => 6
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,5,2,4,1] => [2,2,1] => 6
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,3,1] => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [1,1,2,1] => 7
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => [2,2,1] => 6
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [4,2,3,5,1] => [1,3,1] => 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => [1,3,1] => 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => [4,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,2] => 6
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [4,5,3,1,2] => [2,1,2] => 5
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [5,3,4,1,2] => [1,2,2] => 4
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [4,3,5,1,2] => [1,2,2] => 4
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [3,4,5,1,2] => [3,2] => 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [5,2,4,1,3] => [1,2,2] => 4
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,4,5,1,3] => [3,2] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,5,1,2,3] => [2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,5,1,4] => [1,2,2] => 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,5,1,4] => [3,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,5,1,2,4] => [2,3] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,5,1,3,4] => [2,3] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,4] => 1
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [2,3,5,4,7,8,6,1] => [5,7,8,2,3,4,6,1] => ? => ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [2,4,3,5,7,8,6,1] => [4,7,8,2,3,5,6,1] => ? => ? = 10
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => [5,8,3,4,6,7,1,2] => ? => ? = 8
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
=> [2,1,4,6,7,5,8,3] => [6,7,2,4,5,8,1,3] => ? => ? = 8
[[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [2,3,1,6,7,5,8,4] => [6,7,2,3,5,8,1,4] => ? => ? = 8
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => [7,8,3,6,1,2,4,5] => ? => ? = 6
[[[[.,.],[.,.]],.],[[.,.],[.,.]]]
=> [1,3,2,4,6,8,7,5] => [8,3,6,7,1,2,4,5] => ? => ? = 5
[[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,4,3,6,9,8,7,5] => [9,8,4,6,7,1,2,3,5] => ? => ? = 8
[.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,5,4,7,9,8,6,1] => [9,5,7,8,2,3,4,6,1] => ? => ? = 13
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,4,7,10,9,8,6] => [10,9,3,5,7,8,1,2,4,6] => ? => ? = 9
[[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,3,6,9,8,7,5] => [9,8,2,4,6,7,1,3,5] => ? => ? = 9
[.,[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,4,7,9,8,6,1] => [9,3,5,7,8,2,4,6,1] => ? => ? = 14
[.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,5,8,10,9,7,1] => [10,4,6,8,9,2,3,5,7,1] => ? => ? = 15
[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,6,4,8,11,10,9,7] => [11,10,3,5,6,8,9,1,2,4,7] => ? => ? = 10
[[[[[.,.],.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,4,7,10,9,8,6] => [10,9,5,7,8,1,2,3,4,6] => ? => ? = 8
[[[.,.],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,4,2,6,9,8,7,5] => [9,8,3,4,6,7,1,2,5] => ? => ? = 9
[[[.,.],[.,.]],[[[.,.],[.,[.,.]]],.]]
=> [1,3,2,5,8,7,6,9,4] => [8,7,3,5,6,9,1,2,4] => ? => ? = 9
[.,[[[.,.],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,5,3,7,9,8,6,1] => [9,4,5,7,8,2,3,6,1] => ? => ? = 14
[.,[[[.,.],[.,.]],[[[.,.],[.,.]],.]]]
=> [2,4,3,6,8,7,9,5,1] => [8,4,6,7,9,2,3,5,1] => ? => ? = 14
[.,[[[[[.,.],.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,4,6,5,8,10,9,7,1] => [10,6,8,9,2,3,4,5,7,1] => ? => ? = 14
[.,[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,3,6,7,5,9,11,10,8,1] => [11,4,6,7,9,10,2,3,5,8,1] => ? => ? = 17
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => ? = 28
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => [1,1,1,1,1,1,1,1,2] => ? = 36
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,3,5,7,9,10,8,6,4,2] => [9,10,7,8,5,6,3,4,1,2] => [2,2,2,2,2] => ? = 20
[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,3,5,8,9,7,10,6,4,2] => [8,9,7,10,5,6,3,4,1,2] => [2,2,2,2,2] => ? = 20
[[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,3,6,7,5,9,10,8,4,2] => [6,7,9,10,5,8,3,4,1,2] => ? => ? = 18
[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,3,6,8,9,7,5,10,4,2] => [8,9,6,7,5,10,3,4,1,2] => [2,2,2,2,2] => ? = 20
[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [1,3,7,8,6,9,5,10,4,2] => [7,8,6,9,5,10,3,4,1,2] => [2,2,2,2,2] => ? = 20
[[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]
=> [1,4,5,3,7,9,10,8,6,2] => [9,10,4,5,7,8,3,6,1,2] => ? => ? = 16
[[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]
=> [1,4,5,3,8,9,7,10,6,2] => [8,9,4,5,7,10,3,6,1,2] => ? => ? = 16
[[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]
=> [1,4,6,7,5,3,9,10,8,2] => [6,7,4,5,9,10,3,8,1,2] => ? => ? = 16
[[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]
=> [1,5,6,4,7,3,9,10,8,2] => [5,6,4,7,9,10,3,8,1,2] => ? => ? = 16
[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [1,4,6,8,9,7,5,3,10,2] => [8,9,6,7,4,5,3,10,1,2] => [2,2,2,2,2] => ? = 20
[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [1,4,7,8,6,9,5,3,10,2] => [7,8,6,9,4,5,3,10,1,2] => [2,2,2,2,2] => ? = 20
[[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [1,5,6,4,8,9,7,3,10,2] => [5,6,8,9,4,7,3,10,1,2] => ? => ? = 18
[[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [1,5,7,8,6,4,9,3,10,2] => [7,8,5,6,4,9,3,10,1,2] => [2,2,2,2,2] => ? = 20
[[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]
=> [1,6,7,5,8,4,9,3,10,2] => [6,7,5,8,4,9,3,10,1,2] => [2,2,2,2,2] => ? = 20
[[.,[[.,.],.]],[[.,.],[[.,.],[[.,.],.]]]]
=> [2,3,1,5,7,9,10,8,6,4] => [9,10,7,8,2,3,5,6,1,4] => ? => ? = 14
[[.,[[.,.],.]],[[.,.],[[.,[[.,.],.]],.]]]
=> [2,3,1,5,8,9,7,10,6,4] => [8,9,7,10,2,3,5,6,1,4] => ? => ? = 14
[[.,[[.,.],.]],[[.,[[.,.],.]],[[.,.],.]]]
=> [2,3,1,6,7,5,9,10,8,4] => [6,7,9,10,2,3,5,8,1,4] => ? => ? = 12
[[.,[[.,.],.]],[[.,[[.,.],[[.,.],.]]],.]]
=> [2,3,1,6,8,9,7,5,10,4] => [8,9,6,7,2,3,5,10,1,4] => ? => ? = 14
[[.,[[.,.],.]],[[.,[[.,[[.,.],.]],.]],.]]
=> [2,3,1,7,8,6,9,5,10,4] => [7,8,6,9,2,3,5,10,1,4] => ? => ? = 14
[[.,[[.,.],[[.,.],.]]],[[.,.],[[.,.],.]]]
=> [2,4,5,3,1,7,9,10,8,6] => [4,5,9,10,2,3,7,8,1,6] => ? => ? = 12
[[.,[[.,.],[[.,.],.]]],[[.,[[.,.],.]],.]]
=> [2,4,5,3,1,8,9,7,10,6] => [4,5,8,9,2,3,7,10,1,6] => ? => ? = 12
[[.,[[.,[[.,.],.]],.]],[[.,.],[[.,.],.]]]
=> [3,4,2,5,1,7,9,10,8,6] => [3,4,9,10,2,5,7,8,1,6] => ? => ? = 12
[[.,[[.,[[.,.],.]],.]],[[.,[[.,.],.]],.]]
=> [3,4,2,5,1,8,9,7,10,6] => [3,4,8,9,2,5,7,10,1,6] => ? => ? = 12
[[.,[[.,.],[[.,.],[[.,.],.]]]],[[.,.],.]]
=> [2,4,6,7,5,3,1,9,10,8] => [6,7,4,5,2,3,9,10,1,8] => ? => ? = 14
[[.,[[.,.],[[.,[[.,.],.]],.]]],[[.,.],.]]
=> [2,5,6,4,7,3,1,9,10,8] => [5,6,4,7,2,3,9,10,1,8] => ? => ? = 14
[[.,[[.,[[.,.],.]],[[.,.],.]]],[[.,.],.]]
=> [3,4,2,6,7,5,1,9,10,8] => [3,4,6,7,2,5,9,10,1,8] => ? => ? = 12
[[.,[[.,[[.,.],[[.,.],.]]],.]],[[.,.],.]]
=> [3,5,6,4,2,7,1,9,10,8] => [5,6,3,4,2,7,9,10,1,8] => ? => ? = 14
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000391
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 75%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 75%
Values
[.,.]
=> [1] => [1] => => ? = 0
[.,[.,.]]
=> [2,1] => [2,1] => 1 => 1
[[.,.],.]
=> [1,2] => [1,2] => 0 => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 11 => 3
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 01 => 2
[[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => 10 => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 10 => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 111 => 6
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 011 => 5
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,3,1] => 101 => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 101 => 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 001 => 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => 110 => 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,4,1,2] => 010 => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,4,1,3] => 010 => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => 100 => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 110 => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 010 => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => 100 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1111 => 10
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => 0111 => 9
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,3,4,2,1] => 1011 => 8
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => 1011 => 8
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => 0011 => 7
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,2,3,1] => 1101 => 7
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [4,5,2,3,1] => 0101 => 6
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,5,2,4,1] => 0101 => 6
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,4,1] => 1001 => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => 1101 => 7
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => 0101 => 6
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [4,2,3,5,1] => 1001 => 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => 1001 => 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => 0001 => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,1,2] => 1110 => 6
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [4,5,3,1,2] => 0110 => 5
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [5,3,4,1,2] => 1010 => 4
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [4,3,5,1,2] => 1010 => 4
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [3,4,5,1,2] => 0010 => 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [5,2,4,1,3] => 1010 => 4
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,4,5,1,3] => 0010 => 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [5,4,1,2,3] => 1100 => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,5,1,2,3] => 0100 => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,5,1,4] => 1010 => 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,5,1,4] => 0010 => 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,5,1,2,4] => 0100 => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,5,1,3,4] => 0100 => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1110 => 6
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [2,4,3,7,8,6,5,1] => [7,8,4,6,2,3,5,1] => ? => ? = 13
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [2,4,3,7,6,8,5,1] => [7,4,6,8,2,3,5,1] => ? => ? = 12
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [2,3,5,4,7,8,6,1] => [5,7,8,2,3,4,6,1] => ? => ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [2,4,3,5,7,8,6,1] => [4,7,8,2,3,5,6,1] => ? => ? = 10
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,4,3,6,8,7,5,2] => [8,4,6,7,3,5,1,2] => ? => ? = 11
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,4,5,3,7,8,6,2] => [4,5,7,8,3,6,1,2] => ? => ? = 10
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => [5,8,3,4,6,7,1,2] => ? => ? = 8
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [1,3,4,6,7,5,8,2] => [6,7,3,4,5,8,1,2] => ? => ? = 8
[[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [2,1,4,7,8,6,5,3] => [7,8,6,2,4,5,1,3] => ? => ? = 11
[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [2,1,5,4,8,7,6,3] => [8,5,7,2,4,6,1,3] => ? => ? = 10
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
=> [2,1,4,5,6,8,7,3] => [8,2,4,5,6,7,1,3] => ? => ? = 7
[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]
=> [2,1,5,7,6,4,8,3] => [7,5,6,2,4,8,1,3] => ? => ? = 10
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
=> [2,1,4,6,7,5,8,3] => [6,7,2,4,5,8,1,3] => ? => ? = 8
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [3,2,1,6,8,7,5,4] => [8,3,6,7,2,5,1,4] => ? => ? = 11
[[.,[.,[.,.]]],[.,[[[.,.],.],.]]]
=> [3,2,1,6,7,8,5,4] => [3,6,7,8,2,5,1,4] => ? => ? = 10
[[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [2,3,1,6,7,5,8,4] => [6,7,2,3,5,8,1,4] => ? => ? = 8
[[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [2,3,1,5,7,6,8,4] => [7,2,3,5,6,8,1,4] => ? => ? = 7
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => [4,7,8,3,6,1,2,5] => ? => ? = 8
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => [7,8,3,6,1,2,4,5] => ? => ? = 6
[[[[.,.],[.,.]],.],[[.,.],[.,.]]]
=> [1,3,2,4,6,8,7,5] => [8,3,6,7,1,2,4,5] => ? => ? = 5
[[[[.,[.,.]],.],.],[.,[[.,.],.]]]
=> [2,1,3,4,7,8,6,5] => [7,8,2,6,1,3,4,5] => ? => ? = 6
[[.,[[.,.],[[.,.],.]]],[[.,.],.]]
=> [2,4,5,3,1,7,8,6] => [4,5,2,3,7,8,1,6] => ? => ? = 8
[[.,[[.,[[.,.],.]],.]],[[.,.],.]]
=> [3,4,2,5,1,7,8,6] => [3,4,2,5,7,8,1,6] => ? => ? = 8
[[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [1,2,5,4,3,7,8,6] => [5,4,7,8,1,2,3,6] => ? => ? = 5
[[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => [5,4,6,8,1,2,3,7] => ? => ? = 5
[[.,[[.,[.,.]],[[.,.],[.,.]]]],.]
=> [3,2,5,7,6,4,1,8] => [7,3,5,6,2,4,1,8] => ? => ? = 11
[[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [3,4,2,6,7,5,1,8] => [3,4,6,7,2,5,1,8] => ? => ? = 10
[[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [1,2,4,6,7,5,3,8] => [6,7,4,5,1,2,3,8] => ? => ? = 6
[[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [1,2,5,4,7,6,3,8] => [5,7,4,6,1,2,3,8] => ? => ? = 6
[[[[.,.],[.,.]],[.,[[.,.],.]]],.]
=> [1,3,2,6,7,5,4,8] => [6,7,3,5,1,2,4,8] => ? => ? = 6
[[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,2,3,5,6,4,7,8] => [5,6,1,2,3,4,7,8] => ? => ? = 2
[[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,4,3,6,9,8,7,5] => [9,8,4,6,7,1,2,3,5] => ? => ? = 8
[.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,5,4,7,9,8,6,1] => [9,5,7,8,2,3,4,6,1] => ? => ? = 13
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,4,7,10,9,8,6] => [10,9,3,5,7,8,1,2,4,6] => ? => ? = 9
[[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,3,6,9,8,7,5] => [9,8,2,4,6,7,1,3,5] => ? => ? = 9
[.,[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,4,7,9,8,6,1] => [9,3,5,7,8,2,4,6,1] => ? => ? = 14
[.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,5,8,10,9,7,1] => [10,4,6,8,9,2,3,5,7,1] => ? => ? = 15
[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,6,4,8,11,10,9,7] => [11,10,3,5,6,8,9,1,2,4,7] => ? => ? = 10
[[[[[.,.],.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,4,7,10,9,8,6] => [10,9,5,7,8,1,2,3,4,6] => ? => ? = 8
[[[.,.],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,4,2,6,9,8,7,5] => [9,8,3,4,6,7,1,2,5] => ? => ? = 9
[[[.,.],[.,.]],[[[.,.],[.,[.,.]]],.]]
=> [1,3,2,5,8,7,6,9,4] => [8,7,3,5,6,9,1,2,4] => ? => ? = 9
[.,[[[.,.],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,5,3,7,9,8,6,1] => [9,4,5,7,8,2,3,6,1] => ? => ? = 14
[.,[[[.,.],[.,.]],[[[.,.],[.,.]],.]]]
=> [2,4,3,6,8,7,9,5,1] => [8,4,6,7,9,2,3,5,1] => ? => ? = 14
[.,[[[[[.,.],.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,4,6,5,8,10,9,7,1] => [10,6,8,9,2,3,4,5,7,1] => ? => ? = 14
[.,[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,3,6,7,5,9,11,10,8,1] => [11,4,6,7,9,10,2,3,5,8,1] => ? => ? = 17
[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,3,5,8,9,7,10,6,4,2] => [8,9,7,10,5,6,3,4,1,2] => ? => ? = 20
[[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,3,6,7,5,9,10,8,4,2] => [6,7,9,10,5,8,3,4,1,2] => ? => ? = 18
[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,3,6,8,9,7,5,10,4,2] => [8,9,6,7,5,10,3,4,1,2] => ? => ? = 20
[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [1,3,7,8,6,9,5,10,4,2] => [7,8,6,9,5,10,3,4,1,2] => ? => ? = 20
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 72%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 72%
Values
[.,.]
=> [1] => [1] => [[1]]
=> 0
[.,[.,.]]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[[.,.],.]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2
[[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 6
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 5
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 10
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 8
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 7
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,5,2,4,1] => [[1,2],[3,4],[5]]
=> 6
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 7
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 6
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 6
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 4
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 4
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 4
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,5,1,4] => [[1,3],[2,5],[4]]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [2,4,3,8,7,6,5,1] => [8,7,4,6,2,3,5,1] => ?
=> ? = 14
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [2,4,3,7,8,6,5,1] => [7,8,4,6,2,3,5,1] => ?
=> ? = 13
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [2,4,3,7,6,8,5,1] => [7,4,6,8,2,3,5,1] => ?
=> ? = 12
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [2,3,5,4,7,8,6,1] => [5,7,8,2,3,4,6,1] => ?
=> ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [2,4,3,5,7,8,6,1] => [4,7,8,2,3,5,6,1] => ?
=> ? = 10
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,4,3,6,8,7,5,2] => [8,4,6,7,3,5,1,2] => ?
=> ? = 11
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,4,5,3,7,8,6,2] => [4,5,7,8,3,6,1,2] => ?
=> ? = 10
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => [5,8,3,4,6,7,1,2] => ?
=> ? = 8
[[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [1,3,4,6,7,5,8,2] => [6,7,3,4,5,8,1,2] => ?
=> ? = 8
[[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [1,3,5,4,7,6,8,2] => [5,7,3,4,6,8,1,2] => ?
=> ? = 8
[[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [2,1,4,7,8,6,5,3] => [7,8,6,2,4,5,1,3] => ?
=> ? = 11
[[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [2,1,4,6,8,7,5,3] => [8,6,7,2,4,5,1,3] => ?
=> ? = 10
[[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [2,1,5,4,8,7,6,3] => [8,5,7,2,4,6,1,3] => ?
=> ? = 10
[[.,[.,.]],[[[[.,.],.],.],[.,.]]]
=> [2,1,4,5,6,8,7,3] => [8,2,4,5,6,7,1,3] => ?
=> ? = 7
[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]
=> [2,1,5,7,6,4,8,3] => [7,5,6,2,4,8,1,3] => ?
=> ? = 10
[[.,[.,.]],[[[.,.],[[.,.],.]],.]]
=> [2,1,4,6,7,5,8,3] => [6,7,2,4,5,8,1,3] => ?
=> ? = 8
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [3,2,1,6,8,7,5,4] => [8,3,6,7,2,5,1,4] => ?
=> ? = 11
[[.,[.,[.,.]]],[.,[[[.,.],.],.]]]
=> [3,2,1,6,7,8,5,4] => [3,6,7,8,2,5,1,4] => ?
=> ? = 10
[[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [2,3,1,6,7,5,8,4] => [6,7,2,3,5,8,1,4] => ?
=> ? = 8
[[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [2,3,1,5,7,6,8,4] => [7,2,3,5,6,8,1,4] => ?
=> ? = 7
[[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,8,7,6,4] => [8,7,3,5,6,1,2,4] => ?
=> ? = 8
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => [4,7,8,3,6,1,2,5] => ?
=> ? = 8
[[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => [7,8,3,6,1,2,4,5] => ?
=> ? = 6
[[[[.,.],[.,.]],.],[[.,.],[.,.]]]
=> [1,3,2,4,6,8,7,5] => [8,3,6,7,1,2,4,5] => ?
=> ? = 5
[[[[.,[.,.]],.],.],[.,[[.,.],.]]]
=> [2,1,3,4,7,8,6,5] => [7,8,2,6,1,3,4,5] => ?
=> ? = 6
[[.,[[.,.],[[.,.],.]]],[[.,.],.]]
=> [2,4,5,3,1,7,8,6] => [4,5,2,3,7,8,1,6] => ?
=> ? = 8
[[.,[[.,[[.,.],.]],.]],[[.,.],.]]
=> [3,4,2,5,1,7,8,6] => [3,4,2,5,7,8,1,6] => ?
=> ? = 8
[[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [1,2,5,4,3,8,7,6] => [5,8,4,7,1,2,3,6] => ?
=> ? = 6
[[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => [5,4,6,8,1,2,3,7] => ?
=> ? = 5
[[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => [4,3,6,8,1,2,5,7] => ?
=> ? = 5
[[.,[[.,[.,.]],[[.,.],[.,.]]]],.]
=> [3,2,5,7,6,4,1,8] => [7,3,5,6,2,4,1,8] => ?
=> ? = 11
[[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [3,4,2,6,7,5,1,8] => [3,4,6,7,2,5,1,8] => ?
=> ? = 10
[[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [1,2,4,6,7,5,3,8] => [6,7,4,5,1,2,3,8] => ?
=> ? = 6
[[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [1,2,5,4,7,6,3,8] => [5,7,4,6,1,2,3,8] => ?
=> ? = 6
[[[[.,.],[.,.]],[.,[[.,.],.]]],.]
=> [1,3,2,6,7,5,4,8] => [6,7,3,5,1,2,4,8] => ?
=> ? = 6
[[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,3,2,5,7,6,4,8] => [7,3,5,6,1,2,4,8] => ?
=> ? = 5
[[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,2,3,5,6,4,7,8] => [5,6,1,2,3,4,7,8] => ?
=> ? = 2
[[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,4,3,6,9,8,7,5] => [9,8,4,6,7,1,2,3,5] => ?
=> ? = 8
[.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,5,4,7,9,8,6,1] => [9,5,7,8,2,3,4,6,1] => ?
=> ? = 13
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,4,7,10,9,8,6] => [10,9,3,5,7,8,1,2,4,6] => ?
=> ? = 9
[[[.,[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,3,6,9,8,7,5] => [9,8,2,4,6,7,1,3,5] => ?
=> ? = 9
[.,[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,4,7,9,8,6,1] => [9,3,5,7,8,2,4,6,1] => ?
=> ? = 14
[.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,5,8,10,9,7,1] => [10,4,6,8,9,2,3,5,7,1] => ?
=> ? = 15
[[[[.,.],[.,.]],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,2,5,6,4,8,11,10,9,7] => [11,10,3,5,6,8,9,1,2,4,7] => ?
=> ? = 10
[[[[[.,.],.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,4,7,10,9,8,6] => [10,9,5,7,8,1,2,3,4,6] => ?
=> ? = 8
[[[.,.],[[.,.],.]],[[.,.],[.,[.,.]]]]
=> [1,3,4,2,6,9,8,7,5] => [9,8,3,4,6,7,1,2,5] => ?
=> ? = 9
[[[.,.],[.,.]],[[[.,.],[.,[.,.]]],.]]
=> [1,3,2,5,8,7,6,9,4] => [8,7,3,5,6,9,1,2,4] => ?
=> ? = 9
[.,[[[.,.],[[.,.],.]],[[.,.],[.,.]]]]
=> [2,4,5,3,7,9,8,6,1] => [9,4,5,7,8,2,3,6,1] => ?
=> ? = 14
[.,[[[.,.],[.,.]],[[[.,.],[.,.]],.]]]
=> [2,4,3,6,8,7,9,5,1] => [8,4,6,7,9,2,3,5,1] => ?
=> ? = 14
[.,[[[[[.,.],.],.],[.,.]],[[.,.],[.,.]]]]
=> [2,3,4,6,5,8,10,9,7,1] => [10,6,8,9,2,3,4,5,7,1] => ?
=> ? = 14
Description
The (standard) major index of a standard tableau.
A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St001161
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 58%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 58%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[.,.],[.,.]]
=> [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
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[.,[.,[[.,.],.]]]
=> [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
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [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
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,.],[.,[.,.]]]
=> [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
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [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
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 10
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [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
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [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
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [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
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [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
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
=> [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[.,[.,[.,.]]],[.,[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [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
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [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
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [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
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],.],.],[[[.,.],.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [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
[[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[.,.]],[[.,.],.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],.],[[.,.],.]],[.,.]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [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
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [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
[[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [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
[[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [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
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ?
=> ? = 8
[.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
=> ?
=> ? = 13
[[[[.,.],[.,.]],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> ?
=> ? = 9
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
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 58%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 58%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[.,.],[.,.]]
=> [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
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[.,[.,[[.,.],.]]]
=> [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
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [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
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,.],[.,[.,.]]]
=> [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
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [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
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 10
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [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
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [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
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [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
[[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [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
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,[.,.]],[[.,.],[[[.,.],.],.]]]
=> [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[.,[.,[.,.]]],[.,[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [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
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [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
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [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
[[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 8
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],.],.],[[[.,.],.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [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
[[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[.,.]],[[.,.],.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],.],[[.,.],.]],[.,.]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [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
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[[.,[[.,.],.]],.],.],.],[.,.]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [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
[[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [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
[[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [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
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ?
=> ? = 8
[.,[[[[.,.],.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0,1,0]
=> ?
=> ? = 13
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: St000012
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 56%
St000012: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 56%
Values
[.,.]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 6
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 5
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 4
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 4
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 8
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 6
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 14
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 13
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 12
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 12
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 11
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 10
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 10
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 12
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 14
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 16
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 16
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 14
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 12
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 12
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 10
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 10
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 8
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 10
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 9
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 12
[[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 11
[[.,[.,[.,.]]],[.,[[[.,.],.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 10
[[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 10
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 9
[[.,[.,[.,.]]],[[[.,.],.],[.,.]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 8
[[.,[[.,.],.]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 8
[[.,[[.,.],.]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 8
[[.,[[.,.],.]],[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 7
[[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 8
[[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 7
[[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 12
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 10
[[.,[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 8
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [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]
=> ? = 8
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 8
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 6
[[.,[[.,.],[[.,.],.]]],[[.,.],.]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 8
[[.,[[.,[[.,.],.]],.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 8
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 10
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 8
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 7
[[[[.,[[[.,.],.],.]],.],.],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
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: St000378
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 69%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 69%
Values
[.,.]
=> [1,0]
=> [1,0]
=> []
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[[.,.],[.,.]]
=> [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
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
[.,[[.,.],[.,.]]]
=> [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
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 3
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 8
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 7
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 6
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 7
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 6
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [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
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [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
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 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
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [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
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [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
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [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
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> ? = 10
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2]
=> ? = 9
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1,1,1]
=> ? = 11
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,5,3,1,1,1]
=> ? = 10
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,1,1]
=> ? = 9
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [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
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,3,1,1,1]
=> ? = 8
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1,1,1]
=> ? = 9
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1,1]
=> ? = 8
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,5,2,1,1,1]
=> ? = 8
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2]
=> ? = 10
[.,[[[.,[[.,[.,.]],.]],.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1]
=> ? = 10
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [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
[.,[[[[.,[.,[.,.]]],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [5,5,2,1]
=> ? = 9
[[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [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
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1,1]
=> ? = 11
[[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1]
=> ? = 12
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1]
=> ? = 11
[[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> ? = 11
[[.,[[[.,.],.],.]],[.,[.,.]]]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> ? = 6
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? = 6
[[[.,[[.,.],.]],.],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> ? = 5
[[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,1,1]
=> ? = 8
[[[.,[.,.]],[.,[.,.]]],[.,.]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> ? = 5
[[[[.,.],.],[.,[.,.]]],[.,.]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? = 4
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,3,2,1]
=> ? = 14
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,2]
=> ? = 13
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,1,1]
=> ? = 12
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2]
=> ? = 12
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> [7,5,5,2]
=> ? = 11
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3,2]
=> ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2]
=> ? = 10
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1,1,1,1]
=> ? = 10
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,1,1]
=> ? = 12
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,1]
=> ? = 14
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1,1]
=> ? = 16
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,1]
=> ? = 16
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,1,1]
=> ? = 14
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,1]
=> ? = 12
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,1]
=> ? = 12
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2]
=> ? = 10
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,3]
=> ? = 10
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3]
=> ? = 8
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1]
=> ? = 8
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [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
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [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
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,2]
=> ? = 18
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [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
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: St000246
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 72%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000246: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 72%
Values
[.,.]
=> [.,.]
=> [1] => 0
[.,[.,.]]
=> [[.,.],.]
=> [1,2] => 1
[[.,.],.]
=> [.,[.,.]]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,2,3] => 3
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 6
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 5
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 4
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 4
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 10
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 9
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 8
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 8
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 7
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 7
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 6
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 6
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 5
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 7
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 6
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 5
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 5
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 4
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 6
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 5
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 4
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 4
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 3
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 4
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 4
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [[[[.,.],.],[[.,.],[.,.]]],.]
=> [6,4,5,1,2,3,7] => ? = 10
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [[[.,[.,.]],[[.,.],[.,.]]],.]
=> [6,4,5,2,1,3,7] => ? = 9
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [[[.,.],[[[.,.],[.,.]],.]],.]
=> [5,3,4,6,1,2,7] => ? = 11
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [[[.,.],[[.,[.,[.,.]]],.]],.]
=> [5,4,3,6,1,2,7] => ? = 10
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [[[.,.],[[.,[.,.]],[.,.]]],.]
=> [6,4,3,5,1,2,7] => ? = 9
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [6,5,3,4,1,2,7] => ? = 8
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [[[.,.],[.,[[.,[.,.]],.]]],.]
=> [5,4,6,3,1,2,7] => ? = 9
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [[[.,.],[.,[[.,.],[.,.]]]],.]
=> [6,4,5,3,1,2,7] => ? = 8
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [[[.,.],[.,[.,[[.,.],.]]]],.]
=> [5,6,4,3,1,2,7] => ? = 8
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [6,5,4,3,1,2,7] => ? = 7
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> [[.,[[.,.],[[.,.],[.,.]]]],.]
=> [6,4,5,2,3,1,7] => ? = 8
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => ? = 7
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [5,6,3,4,2,1,7] => ? = 8
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [6,4,5,3,2,1,7] => ? = 7
[[.,.],[[[[[.,.],.],.],.],.]]
=> [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [7,5,4,3,2,1,6] => ? = 5
[[.,[[[.,.],.],.]],[.,[.,.]]]
=> [[[.,.],.],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,1,2,3] => ? = 6
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> [[.,[.,.]],[[[.,.],.],[.,.]]]
=> [7,4,5,6,2,1,3] => ? = 5
[[[.,[[.,.],.]],.],[.,[.,.]]]
=> [[[.,.],.],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,1,2,3] => ? = 5
[[[[.,[.,.]],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,1,2,3] => ? = 4
[[[[[.,.],.],.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 2
[[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [[.,.],[[[[.,.],.],[.,.]],.]]
=> [6,3,4,5,7,1,2] => ? = 8
[[.,[[[[.,.],.],.],.]],[.,.]]
=> [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [6,5,4,3,7,1,2] => ? = 5
[[[.,.],[.,[[.,.],.]]],[.,.]]
=> [[.,.],[[[.,[.,.]],.],[.,.]]]
=> [7,4,3,5,6,1,2] => ? = 6
[[[.,.],[[.,[.,.]],.]],[.,.]]
=> [[.,.],[[.,[[.,.],.]],[.,.]]]
=> [7,4,5,3,6,1,2] => ? = 5
[[[.,[.,.]],[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],[[.,.],.]]]
=> [6,7,3,4,5,1,2] => ? = 5
[[[[.,.],.],[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [7,6,3,4,5,1,2] => ? = 4
[[[.,[[.,[.,.]],.]],.],[.,.]]
=> [[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => ? = 5
[[[.,[[[.,.],.],.]],.],[.,.]]
=> [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,3,1,2] => ? = 4
[[[[.,.],[.,[.,.]]],.],[.,.]]
=> [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [7,4,5,6,3,1,2] => ? = 4
[[[[.,.],[[.,.],.]],.],[.,.]]
=> [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 3
[[[[.,[.,.]],[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[[[[[.,.],.],[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[[[[.,[.,[.,.]]],.],.],[.,.]]
=> [[.,.],[.,[.,[[[.,.],.],.]]]]
=> [5,6,7,4,3,1,2] => ? = 4
[[[[.,[[.,.],.]],.],.],[.,.]]
=> [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,3,1,2] => ? = 3
[[[[[.,.],[.,.]],.],.],[.,.]]
=> [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[[[[[.,[.,.]],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[[.,[[[.,.],[.,.]],[.,.]]],.]
=> [.,[[[.,.],[[.,.],[.,.]]],.]]
=> [6,4,5,2,3,7,1] => ? = 7
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> [7,4,5,6,2,3,1] => ? = 4
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],[[.,.],[.,.]]],.]
=> [7,5,6,1,2,3,4,8] => ? = 14
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],[[.,.],[.,.]]],.]
=> [7,5,6,2,1,3,4,8] => ? = 13
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [7,5,6,3,1,2,4,8] => ? = 12
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],[[.,.],[.,.]]],.]
=> [7,5,6,2,3,1,4,8] => ? = 12
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[[.,.],[.,.]]],.]
=> [7,5,6,3,2,1,4,8] => ? = 11
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [[[.,[.,.]],[[.,.],[.,[.,.]]]],.]
=> [7,6,4,5,2,1,3,8] => ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [[[.,[.,.]],[.,[[.,.],[.,.]]]],.]
=> [7,5,6,4,2,1,3,8] => ? = 10
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> [7,5,6,3,4,1,2,8] => ? = 10
[.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [[[.,.],[.,[[[.,.],[.,.]],.]]],.]
=> [6,4,5,7,3,1,2,8] => ? = 12
[.,[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [[.,[[[.,.],[[.,.],[.,.]]],.]],.]
=> [6,4,5,2,3,7,1,8] => ? = 14
[.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [[.,[[.,[[[.,.],[.,.]],.]],.]],.]
=> [5,3,4,6,2,7,1,8] => ? = 16
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [4,5,3,6,2,7,1,8] => ? = 16
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: St000018
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 75%
St000018: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 75%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => 3
[.,[[.,.],.]]
=> [2,3,1] => 2
[[.,.],[.,.]]
=> [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 6
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 5
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 10
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 9
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 8
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 8
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 7
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 7
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 6
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 6
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 7
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 6
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 6
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 5
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 4
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 4
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 4
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [2,4,3,7,6,5,1] => ? = 10
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [2,4,3,6,7,5,1] => ? = 9
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [3,5,4,2,7,6,1] => ? = 11
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [3,4,5,2,7,6,1] => ? = 10
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [2,4,5,3,7,6,1] => ? = 9
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [3,2,5,4,7,6,1] => ? = 9
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [2,3,5,4,7,6,1] => ? = 8
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [3,4,2,5,7,6,1] => ? = 9
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [2,4,3,5,7,6,1] => ? = 8
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [3,2,4,5,7,6,1] => ? = 8
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => ? = 7
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> [2,4,3,6,5,7,1] => ? = 8
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [2,3,4,6,5,7,1] => ? = 7
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [2,4,5,3,6,7,1] => ? = 8
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [3,2,5,4,6,7,1] => ? = 8
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [2,3,5,4,6,7,1] => ? = 7
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [2,4,3,5,6,7,1] => ? = 7
[[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 5
[[.,[[[.,.],.],.]],[.,[.,.]]]
=> [2,3,4,1,7,6,5] => ? = 6
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [1,4,3,2,7,6,5] => ? = 6
[[[.,.],[.,[.,.]]],[[.,.],.]]
=> [1,4,3,2,6,7,5] => ? = 5
[[[.,[[.,.],.]],.],[.,[.,.]]]
=> [2,3,1,4,7,6,5] => ? = 5
[[[[.,[.,.]],.],.],[.,[.,.]]]
=> [2,1,3,4,7,6,5] => ? = 4
[[[[[.,.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,7,6,5] => ? = 3
[[[[[.,.],.],.],.],[[.,.],.]]
=> [1,2,3,4,6,7,5] => ? = 2
[[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [2,5,4,3,1,7,6] => ? = 8
[[[.,.],[.,[[.,.],.]]],[.,.]]
=> [1,4,5,3,2,7,6] => ? = 6
[[[.,.],[[.,[.,.]],.]],[.,.]]
=> [1,4,3,5,2,7,6] => ? = 5
[[[.,[.,.]],[.,[.,.]]],[.,.]]
=> [2,1,5,4,3,7,6] => ? = 5
[[[[.,.],.],[.,[.,.]]],[.,.]]
=> [1,2,5,4,3,7,6] => ? = 4
[[[[[.,.],.],.],[.,.]],[.,.]]
=> [1,2,3,5,4,7,6] => ? = 2
[[[.,[[.,[.,.]],.]],.],[.,.]]
=> [3,2,4,1,5,7,6] => ? = 5
[[[[.,.],[.,[.,.]]],.],[.,.]]
=> [1,4,3,2,5,7,6] => ? = 4
[[[[.,.],[[.,.],.]],.],[.,.]]
=> [1,3,4,2,5,7,6] => ? = 3
[[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 3
[[[[[.,.],.],[.,.]],.],[.,.]]
=> [1,2,4,3,5,7,6] => ? = 2
[[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => ? = 4
[[[[.,[[.,.],.]],.],.],[.,.]]
=> [2,3,1,4,5,7,6] => ? = 3
[[[[[.,.],[.,.]],.],.],[.,.]]
=> [1,3,2,4,5,7,6] => ? = 2
[[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 2
[[.,[[[.,.],[.,.]],[.,.]]],.]
=> [2,4,3,6,5,1,7] => ? = 7
[[[[.,.],[.,[.,.]]],[.,.]],.]
=> [1,4,3,2,6,5,7] => ? = 4
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [2,4,3,8,7,6,5,1] => ? = 14
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [2,4,3,7,8,6,5,1] => ? = 13
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,8,7,5,1] => ? = 12
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [2,4,3,7,6,8,5,1] => ? = 12
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [2,4,3,6,7,8,5,1] => ? = 11
[.,[[[[.,.],.],[.,.]],[[.,.],.]]]
=> [2,3,5,4,7,8,6,1] => ? = 10
[.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [2,4,3,5,7,8,6,1] => ? = 10
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [2,4,3,6,5,8,7,1] => ? = 10
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 $\pi$.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000041The number of nestings of a perfect matching. St001397Number of pairs of incomparable elements in a finite poset. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000057The Shynar inversion number of a standard tableau. St000081The number of edges of a graph. St000795The mad of a permutation. St000332The positive inversions of an alternating sign matrix. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000833The comajor index of a permutation. St001428The number of B-inversions of a signed permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000004The major index of a permutation. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000042The number of crossings of a perfect matching. St000233The number of nestings of a set partition. St000305The inverse major index of a permutation. St000496The rcs statistic of a set partition. St001311The cyclomatic number 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. St000450The number of edges minus the number of vertices plus 2 of a graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St000359The number of occurrences of the pattern 23-1. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001433The flag major index 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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