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Your data matches 40 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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%
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
Description
The number of non-inversions of a permutation.
For a permutation of {1,…,n}, this is given by \operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi).
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: 95% ●values known / values provided: 97%●distinct values known / distinct values provided: 95%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 97%●distinct values known / distinct values provided: 95%
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
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [3,2,4,5,7,6,8,1] => [3,7,2,4,5,6,8,1] => ? => ? = 9
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [2,3,5,6,4,7,8,1] => [5,6,2,3,4,7,8,1] => ? => ? = 9
[.,[[[[[.,.],[.,.]],[.,.]],.],.]]
=> [2,4,3,6,5,7,8,1] => [4,6,2,3,5,7,8,1] => ? => ? = 9
[[.,[.,.]],[[[.,.],.],[.,[.,.]]]]
=> [2,1,4,5,8,7,6,3] => [8,7,2,4,5,6,1,3] => ? => ? = 9
[[[.,[.,.]],[.,[.,[[.,.],.]]]],.]
=> [2,1,6,7,5,4,3,8] => [6,7,5,2,4,1,3,8] => ? => ? = 10
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => 1111111110 => ? = 45
[[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]
=> ? => ? => ? => ? = 26
[[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? => ? => ? => ? = 26
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,11,10,9,8,7,6,5,4,3,2] => [11,10,9,8,7,6,5,4,3,1,2] => 1111111110 => ? = 45
[.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? => ? => ? => ? = 37
[.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]]
=> [10,11,9,8,7,6,5,4,3,2,1] => [10,11,9,8,7,6,5,4,3,2,1] => 0111111111 => ? = 54
[.,[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]]
=> ? => ? => ? => ? = 32
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
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 95%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 95%
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] => [1,3,2] => [[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] => [1,4,3,2] => [[1,2],[3],[4]]
=> 5
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [[1,3],[2],[4]]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 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] => [1,3,2,4] => [[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] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,4,3,2] => [[1,3],[2],[4],[5]]
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 8
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> 7
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,1,4,3] => [[1,2],[3,4],[5]]
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 6
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [[1,3,4],[2],[5]]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 7
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 6
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [[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] => [2,5,4,1,3] => [[1,2],[3,5],[4]]
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 4
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,2,5,1,4] => [[1,3],[2,5],[4]]
=> 4
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [5,1,4,2,3] => [[1,3,5],[2],[4]]
=> 4
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,5,2,4] => [[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] => [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,1,5,3,4] => [[1,3,5],[2,4]]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[.,[[[[.,.],[[[.,.],.],.]],.],.]]
=> [2,4,5,6,3,7,8,1] => [2,3,5,1,4,6,8,7] => ?
=> ? = 10
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [2,3,6,5,4,7,8,1] => [5,4,1,2,3,6,8,7] => ?
=> ? = 10
[.,[[[[[.,.],[.,.]],[.,.]],.],.]]
=> [2,4,3,6,5,7,8,1] => [2,5,1,3,4,6,8,7] => ?
=> ? = 9
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,2,3,5,7,8,6,4] => [5,8,4,7,1,2,3,6] => ?
=> ? = 6
[[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,2,4,3,6,5,8,7] => [3,5,8,1,2,4,6,7] => ?
=> ? = 3
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [4,7,6,5,3,2,1,8] => [7,6,1,5,4,3,2,8] => ?
=> ? = 18
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7,10,9] => [1,3,5,7,10,2,4,6,8,9] => ?
=> ? = 5
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,1] => [2,1,3,4,5,6,7,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 9
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> [3,4,2,5,6,7,8,9,1] => [1,3,2,4,5,6,7,9,8] => [[1,2,4,5,6,7,8],[3,9]]
=> ? = 10
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,9,10,1] => [2,1,3,4,5,6,7,8,10,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 10
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,11,1] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => [2,9,8,7,6,5,4,1,3] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? = 27
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 45
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> [7,6,8,5,4,3,2,1,9] => [2,1,8,7,6,5,4,3,9] => [[1,3,9],[2,4],[5],[6],[7],[8]]
=> ? = 26
[[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]
=> ? => ? => ?
=> ? = 26
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,8,7,9,6,5,4,3,2] => [3,2,9,8,7,6,5,1,4] => ?
=> ? = 26
[[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? => ? => ?
=> ? = 26
[[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,9,10,8,7,6,5,4,3,2] => [2,10,9,8,7,6,5,4,1,3] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? = 35
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,11,10,9,8,7,6,5,4,3,2] => [11,10,9,8,7,6,5,4,3,1,2] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 45
[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [2,1,9,8,7,6,5,4,3] => [9,8,7,6,5,1,4,2,3] => ?
=> ? = 22
[[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]]
=> [3,2,1,9,8,7,6,5,4] => [9,8,7,1,6,2,5,3,4] => ?
=> ? = 18
[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]]
=> [4,3,2,1,9,8,7,6,5] => [9,1,8,2,7,3,6,4,5] => [[1,3,5,7,9],[2],[4],[6],[8]]
=> ? = 16
[[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [2,1,10,9,8,7,6,5,4,3] => [10,9,8,7,6,5,1,4,2,3] => ?
=> ? = 29
[[.,[.,[.,.]]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [3,2,1,10,9,8,7,6,5,4] => [10,9,8,7,1,6,2,5,3,4] => ?
=> ? = 24
[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]
=> [4,3,2,1,10,9,8,7,6,5] => [10,9,1,8,2,7,3,6,4,5] => ?
=> ? = 21
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1,10,9,8,7,6] => [1,10,2,9,3,8,4,7,5,6] => ?
=> ? = 20
[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]
=> [3,9,8,7,6,5,4,2,1] => [9,8,7,6,5,1,4,3,2] => ?
=> ? = 30
[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [2,9,8,7,6,5,4,3,1] => [9,8,7,6,5,4,1,3,2] => [[1,8],[2],[3],[4],[5],[6],[7],[9]]
=> ? = 29
[.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? => ? => ?
=> ? = 37
[.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]]
=> [10,11,9,8,7,6,5,4,3,2,1] => [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 54
[.,[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]]
=> ? => ? => ?
=> ? = 32
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: 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: 76% ●values known / values provided: 88%●distinct values known / distinct values provided: 76%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 76% ●values known / values provided: 88%●distinct values known / distinct values provided: 76%
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
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [2,3,5,6,4,7,8,1] => [5,6,2,3,4,7,8,1] => ? => ? = 9
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1] => ? = 36
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [8,9,7,6,5,4,3,2,1] => [8,9,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1] => ? = 35
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> [7,8,9,6,5,4,3,2,1] => [7,8,9,6,5,4,3,2,1] => [3,1,1,1,1,1,1] => ? = 33
[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> [6,7,8,9,5,4,3,2,1] => [6,7,8,9,5,4,3,2,1] => [4,1,1,1,1,1] => ? = 30
[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> [5,6,7,8,9,4,3,2,1] => [5,6,7,8,9,4,3,2,1] => [5,1,1,1,1] => ? = 26
[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]
=> [4,5,6,7,8,9,3,2,1] => [4,5,6,7,8,9,3,2,1] => [6,1,1,1] => ? = 21
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [9] => ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [9,10,8,7,6,5,4,3,2,1] => [9,10,8,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => ? = 44
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> [8,9,10,7,6,5,4,3,2,1] => [8,9,10,7,6,5,4,3,2,1] => [3,1,1,1,1,1,1,1] => ? = 42
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> [7,8,9,10,6,5,4,3,2,1] => [7,8,9,10,6,5,4,3,2,1] => [4,1,1,1,1,1,1] => ? = 39
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> [6,7,8,9,10,5,4,3,2,1] => [6,7,8,9,10,5,4,3,2,1] => [5,1,1,1,1,1] => ? = 35
[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> [5,6,7,8,9,10,4,3,2,1] => [5,6,7,8,9,10,4,3,2,1] => [6,1,1,1,1] => ? = 30
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> [4,5,6,7,8,9,10,3,2,1] => [4,5,6,7,8,9,10,3,2,1] => [7,1,1,1] => ? = 24
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ? = 0
[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> [8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,9,2,1] => [1,1,1,1,1,2,1,1] => ? = 30
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,9,1] => [1,1,1,1,1,1,2,1] => ? = 29
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [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
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> [9,8,7,6,5,4,3,2,10,1] => [9,8,7,6,5,4,3,2,10,1] => [1,1,1,1,1,1,1,2,1] => ? = 37
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [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
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,9,1] => [2,6,1] => ? = 10
[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,5,6,9,8,7] => [9,8,1,2,3,4,5,6,7] => [1,1,7] => ? = 3
[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,9,8,7,6,5,4,3,2] => [9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,2] => ? = 28
[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]
=> [7,8,6,5,4,3,2,1,9] => [7,8,6,5,4,3,2,1,9] => [2,1,1,1,1,1,2] => ? = 27
[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,8,9,7,6,5,4,3,2] => [8,9,7,6,5,4,3,1,2] => [2,1,1,1,1,1,2] => ? = 27
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,10,9,8,7,6,5,4,3,2] => [10,9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,2] => ? = 36
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,2] => ? = 45
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1,9] => [1,2,1,1,1,1,2] => ? = 26
[[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]
=> ? => ? => ? => ? = 26
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,8,7,9,6,5,4,3,2] => [8,7,9,6,5,4,3,1,2] => [1,2,1,1,1,1,2] => ? = 26
[[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? => ? => ? => ? = 26
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,11,10,9,8,7,6,5,4,3,2] => [11,10,9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,1,2] => ? = 45
[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [2,1,9,8,7,6,5,4,3] => [9,8,7,6,5,2,4,1,3] => [1,1,1,1,1,2,2] => ? = 22
[[.,[.,[.,.]]],[.,[.,[.,[.,[.,.]]]]]]
=> [3,2,1,9,8,7,6,5,4] => [9,8,7,3,6,2,5,1,4] => [1,1,1,2,2,2] => ? = 18
[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,.]]]]]
=> [4,3,2,1,9,8,7,6,5] => [9,4,8,3,7,2,6,1,5] => [1,2,2,2,2] => ? = 16
[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,2,9,8,7,6,5,4,3] => [9,8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,3] => ? = 21
[[.,[.,[.,.]]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [3,2,1,10,9,8,7,6,5,4] => [10,9,8,7,3,6,2,5,1,4] => [1,1,1,1,2,2,2] => ? = 24
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1,10,9,8,7,6] => [5,10,4,9,3,8,2,7,1,6] => [2,2,2,2,2] => ? = 20
[.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]
=> [3,9,8,7,6,5,4,2,1] => [9,8,7,6,5,3,4,2,1] => ? => ? = 30
[.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [2,9,8,7,6,5,4,3,1] => [9,8,7,6,5,4,2,3,1] => [1,1,1,1,1,1,2,1] => ? = 29
[.,[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? => ? => ? => ? = 37
[[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,2,10,9,8,7,6,5,4,3] => [10,9,8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,3] => ? = 28
[[[[[[[[.,.],.],.],.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,5,6,7,10,9,8] => [10,9,1,2,3,4,5,6,7,8] => [1,1,8] => ? = 3
[[[[[[.,.],.],.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,4,5,9,8,7,6] => [9,8,7,1,2,3,4,5,6] => [1,1,1,6] => ? = 6
[[[[[.,.],.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,3,4,9,8,7,6,5] => [9,8,7,6,1,2,3,4,5] => [1,1,1,1,5] => ? = 10
[[[[.,.],.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,2,3,9,8,7,6,5,4] => [9,8,7,6,5,1,2,3,4] => [1,1,1,1,1,4] => ? = 15
[[[[[[[.,.],.],.],.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,4,5,6,10,9,8,7] => [10,9,8,1,2,3,4,5,6,7] => [1,1,1,7] => ? = 6
[[[[[[.,.],.],.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,3,4,5,10,9,8,7,6] => [10,9,8,7,1,2,3,4,5,6] => [1,1,1,1,6] => ? = 10
[[[[[.,.],.],.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,2,3,4,10,9,8,7,6,5] => [10,9,8,7,6,1,2,3,4,5] => [1,1,1,1,1,5] => ? = 15
[[[[.,.],.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,2,3,10,9,8,7,6,5,4] => [10,9,8,7,6,5,1,2,3,4] => [1,1,1,1,1,1,4] => ? = 21
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: 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: 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
[.,[.,.]]
=> [1,1,0,0]
=> [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
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [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
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [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
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [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
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [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
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [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
[.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],[[[.,.],.],.]],.],.]]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [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
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [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
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [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
[[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [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
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [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
[[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [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
[[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [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
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [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
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [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
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [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
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [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
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 35
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 33
[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 30
[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 26
[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 21
[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 15
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [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
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [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
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 44
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 42
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 39
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 35
[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 30
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 24
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 17
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [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
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 30
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ?
=> ?
=> ? = 29
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [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
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ?
=> ?
=> ?
=> ? = 37
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [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
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: 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
[.,[.,.]]
=> [1,1,0,0]
=> [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
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [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
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [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
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [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
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [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
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [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
[.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],[[[.,.],.],.]],.],.]]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [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
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [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
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [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
[[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[[[[.,[[.,.],.]],.],.],[[.,.],.]]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [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
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [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
[[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [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
[[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [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
[[.,[[[[[[.,.],.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [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
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [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
[[[[.,[[[[.,.],.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [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
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [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
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 35
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 33
[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 30
[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 26
[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 21
[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 15
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [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
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [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
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 44
[.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 42
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 39
[.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 35
[.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 30
[.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 24
[.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 17
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [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
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 30
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ?
=> ?
=> ? = 29
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [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
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ?
=> ?
=> ?
=> ? = 37
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: 54% ●values known / values provided: 73%●distinct values known / distinct values provided: 54%
St000012: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 73%●distinct values known / distinct values provided: 54%
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
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 27
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 25
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 24
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 22
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 24
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 24
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 18
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 23
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? = 19
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 16
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 23
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 13
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 22
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 18
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 16
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 22
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 13
[.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 11
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 9
[.,[[[[.,.],[[[.,.],.],.]],.],.]]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 10
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 10
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 9
[.,[[[[[.,.],[.,.]],[.,.]],.],.]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 9
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [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
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 13
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [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
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 8
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 8
[[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 6
[[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 9
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 16
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 8
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 21
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 20
[[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 19
[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 19
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 18
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 18
[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 18
[[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 12
[[.,[[.,[[[[.,.],.],.],.]],.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 10
[[.,[[[[.,[[.,.],.]],.],.],.]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 8
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 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 73%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 73%
Values
[.,.]
=> [.,.]
=> [1] => 0
[.,[.,.]]
=> [.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 3
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,3,2] => 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 6
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 5
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 4
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 4
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 3
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 10
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 9
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 8
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 8
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 7
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 7
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 6
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 6
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 5
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 7
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 6
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 5
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 5
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 6
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 5
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 4
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 4
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 3
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 4
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 3
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 3
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 2
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 4
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 3
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 2
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ? = 20
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ? = 19
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ? = 19
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => ? = 18
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => ? = 18
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ? = 17
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ? = 18
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => ? = 15
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => ? = 17
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ? = 17
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => ? = 11
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => ? = 16
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [5,4,3,2,7,6,1] => ? = 13
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [4,3,2,7,6,5,1] => ? = 12
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [3,2,5,4,7,6,1] => ? = 9
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ? = 16
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => ? = 10
[.,[[[.,.],[[.,[.,.]],.]],.]]
=> [.,[[[.,.],[[.,.],[.,.]]],.]]
=> [2,4,6,5,3,7,1] => ? = 10
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [.,[[[.,.],[[[.,.],.],.]],.]]
=> [2,4,5,6,3,7,1] => ? = 9
[.,[[[.,[[.,[.,.]],.]],.],.]]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => ? = 10
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => ? = 9
[.,[[[[.,.],[.,[.,.]]],.],.]]
=> [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [2,3,6,5,4,7,1] => ? = 9
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [.,[[[[.,.],.],[[.,.],.]],.]]
=> [2,3,5,6,4,7,1] => ? = 8
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [.,[[[[[.,.],.],[.,.]],.],.]]
=> [2,3,5,4,6,7,1] => ? = 7
[.,[[[[.,[.,[.,.]]],.],.],.]]
=> [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => ? = 9
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => ? = 8
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => ? = 7
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [[.,[[.,[.,.]],[.,.]]],[.,.]]
=> [3,2,5,4,1,7,6] => ? = 7
[[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 9
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 7
[[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ? = 8
[[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [3,2,1,6,5,7,4] => ? = 7
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 6
[[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [[[.,[.,[.,.]]],[.,.]],[.,.]]
=> [3,2,1,5,4,7,6] => ? = 5
[[[[.,.],.],[.,.]],[.,[.,.]]]
=> [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 4
[[[.,[[.,.],.]],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],[[.,.],.]]
=> [2,1,4,3,6,7,5] => ? = 4
[[[[.,.],[.,.]],[.,.]],[.,.]]
=> [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 3
[[[[[.,.],.],.],[.,.]],[.,.]]
=> [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 2
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => ? = 27
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => ? = 25
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => ? = 22
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => ? = 24
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => ? = 18
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => ? = 23
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [5,4,3,8,7,6,2,1] => ? = 19
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => ? = 23
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => ? = 13
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [6,5,4,3,2,8,7,1] => ? = 18
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [5,4,3,2,8,7,6,1] => ? = 16
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [2,8,7,6,5,4,3,1] => ? = 22
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: 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: 66% ●values known / values provided: 71%●distinct values known / distinct values provided: 66%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 71%●distinct values known / distinct values provided: 66%
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
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [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
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 17
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 13
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [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
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,1,1]
=> ? = 9
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 16
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2]
=> ? = 10
[.,[[[.,.],[[.,[.,.]],.]],.]]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,1]
=> ? = 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
[.,[[[[.,.],[.,[.,.]]],.],.]]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2]
=> ? = 9
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [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
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,1,1]
=> ? = 7
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 7
[[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> ? = 7
[[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [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
[[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> ? = 5
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2]
=> ? = 25
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,1]
=> ? = 24
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2]
=> ? = 22
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ? = 24
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 24
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ? = 18
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2,1]
=> ? = 23
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,2,1]
=> ? = 19
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2,1,1]
=> ? = 16
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> ? = 23
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3]
=> ? = 13
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> ? = 22
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> ? = 18
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ? = 16
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1]
=> ? = 22
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1,1]
=> ? = 13
[.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2,1,1]
=> ? = 11
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> [7,4,4,1]
=> ? = 9
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,2,2]
=> ? = 10
[.,[[[[[.,.],[.,.]],[.,.]],.],.]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,2,2,2]
=> ? = 9
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [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
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
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: St000161
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000161: Binary trees ⟶ ℤResult quality: 63% ●values known / values provided: 70%●distinct values known / distinct values provided: 63%
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
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 21
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ? = 20
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? = 19
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 19
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 18
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 18
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ? = 17
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ? = 18
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> ? = 15
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 17
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ? = 17
[.,[.,[[[[[.,.],.],.],.],.]]]
=> ? = 11
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 16
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ? = 13
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ? = 12
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> ? = 16
[.,[[[.,.],[[.,[.,.]],.]],.]]
=> ? = 10
[.,[[[.,.],[[[.,.],.],.]],.]]
=> ? = 9
[.,[[[[.,.],[.,[.,.]]],.],.]]
=> ? = 9
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> ? = 7
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> ? = 10
[[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ? = 9
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> ? = 7
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> ? = 6
[[.,[.,[[.,.],.]]],[.,[.,.]]]
=> ? = 8
[[.,[[.,.],[.,.]]],[.,[.,.]]]
=> ? = 7
[[[.,[.,.]],[.,.]],[.,[.,.]]]
=> ? = 5
[[[[.,.],.],[.,.]],[.,[.,.]]]
=> ? = 4
[[[.,[[.,.],.]],[.,.]],[.,.]]
=> ? = 4
[[[[.,.],[.,.]],[.,.]],[.,.]]
=> ? = 3
[[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 28
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 27
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ? = 25
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ? = 24
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ? = 22
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ? = 24
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ? = 24
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ? = 18
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ? = 23
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> ? = 19
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> ? = 16
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> ? = 23
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> ? = 13
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 22
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> ? = 18
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> ? = 16
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> ? = 22
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> ? = 13
[.,[[[.,[[.,.],.]],[[.,.],.]],.]]
=> ? = 11
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.
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. St000081The number of edges of a graph. St000067The inversion number of the alternating sign matrix. 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. St000795The mad of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. 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. 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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