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Your data matches 40 different statistics following compositions of up to 3 maps.
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Matching statistic: St000330
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(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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: 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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
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
[[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,6,5,4,7,3,8,2] => [6,5,4,7,3,8,1,2] => ? => ? = 13
[[[[.,.],.],[.,[.,.]]],[[.,.],.]]
=> [1,2,5,4,3,7,8,6] => [5,4,7,8,1,2,3,6] => ? => ? = 5
[[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> [1,4,6,5,3,7,2,8] => [6,4,5,3,7,1,2,8] => ? => ? = 9
[[[.,[.,.]],[[.,[.,[.,.]]],.]],.]
=> [2,1,6,5,4,7,3,8] => [6,5,2,4,7,1,3,8] => ? => ? = 8
[[[[.,[.,[[.,.],.]]],.],[.,.]],.]
=> [3,4,2,1,5,7,6,8] => [3,4,2,7,1,5,6,8] => ? => ? = 6
[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]]
=> [1,3,5,7,9,11,12,10,8,6,4,2] => [11,12,9,10,7,8,5,6,3,4,1,2] => 01010101010 => ? = 30
[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 00000100000 => ? = 6
[[[[[[[[.,.],[.,.]],.],.],.],.],.],[.,.]]
=> [1,3,2,4,5,6,7,8,10,9] => [3,10,1,2,4,5,6,7,8,9] => ? => ? = 2
Description
The sum of the positions of the ones in a binary word.
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: 85% ●values known / values provided: 95%●distinct values known / distinct values provided: 85%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 95%●distinct values known / distinct values provided: 85%
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
[[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> [5,6,4,3,2,7,1,8] => [5,6,4,3,2,7,1,8] => ? => ? = 15
[[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> [1,4,6,5,3,7,2,8] => [6,4,5,3,7,1,2,8] => ? => ? = 9
[[[[.,.],[[[.,[.,.]],.],.]],.],.]
=> [1,4,3,5,6,2,7,8] => [4,3,5,6,1,2,7,8] => ? => ? = 5
[[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> [1,4,3,5,2,6,7,8] => [4,3,5,1,2,6,7,8] => ? => ? = 4
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [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
[.,[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]]
=> [6,8,9,7,5,4,3,2,1] => [8,9,6,7,5,4,3,2,1] => [2,2,1,1,1,1,1] => ? = 32
[.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]]
=> [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
[[.,.],[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,3,4,5,6,7,8,9,2] => [3,4,5,6,7,8,9,1,2] => [7,2] => ? = 7
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ? = 0
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => ? = 28
[[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [1,8,7,6,5,4,3,2,9] => [8,7,6,5,4,3,1,2,9] => [1,1,1,1,1,1,3] => ? = 21
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> [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
[[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,7,9,8] => [2,9,1,3,4,5,6,7,8] => [2,7] => ? = 2
[[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,7,8,10,9] => [2,10,1,3,4,5,6,7,8,9] => [2,8] => ? = 2
[[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
=> [1,3,2,4,5,6,7,9,8] => [3,9,1,2,4,5,6,7,8] => [2,7] => ? = 2
[[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> [1,2,3,4,5,6,8,9,7] => [8,9,1,2,3,4,5,6,7] => [2,7] => ? = 2
[[[[[[.,.],.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,5,7,8,9,6] => [7,8,9,1,2,3,4,5,6] => [3,6] => ? = 3
[[[[[.,.],.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,4,6,7,8,9,5] => [6,7,8,9,1,2,3,4,5] => [4,5] => ? = 4
[[[[.,.],.],.],[[[[[.,.],.],.],.],.]]
=> [1,2,3,5,6,7,8,9,4] => [5,6,7,8,9,1,2,3,4] => [5,4] => ? = 5
[[[.,.],.],[[[[[[.,.],.],.],.],.],.]]
=> [1,2,4,5,6,7,8,9,3] => [4,5,6,7,8,9,1,2,3] => [6,3] => ? = 6
[[[[[[[.,.],.],.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,5,6,9,8,7] => [9,8,1,2,3,4,5,6,7] => [1,1,7] => ? = 3
[[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> [1,2,3,4,5,6,7,9,10,8] => [9,10,1,2,3,4,5,6,7,8] => [2,8] => ? = 2
[[[[[[.,.],.],.],.],.],[[.,[.,.]],.]]
=> [1,2,3,4,5,8,7,9,6] => [8,7,9,1,2,3,4,5,6] => [1,2,6] => ? = 4
[[[[[[[.,.],.],.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,5,6,8,9,10,7] => [8,9,10,1,2,3,4,5,6,7] => [3,7] => ? = 3
[[[[[.,.],.],.],.],[[[.,[.,.]],.],.]]
=> [1,2,3,4,7,6,8,9,5] => [7,6,8,9,1,2,3,4,5] => [1,3,5] => ? = 5
[[[[[[.,.],.],.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,4,5,7,8,9,10,6] => [7,8,9,10,1,2,3,4,5,6] => [4,6] => ? = 4
[[[[.,.],.],.],[[[[.,[.,.]],.],.],.]]
=> [1,2,3,6,5,7,8,9,4] => [6,5,7,8,9,1,2,3,4] => [1,4,4] => ? = 6
[[[[.,.],.],.],[[[[[[.,.],.],.],.],.],.]]
=> [1,2,3,5,6,7,8,9,10,4] => [5,6,7,8,9,10,1,2,3,4] => [6,4] => ? = 6
[[[.,.],.],[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,2,4,5,6,7,8,9,10,3] => [4,5,6,7,8,9,10,1,2,3] => [7,3] => ? = 7
[[.,.],[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,3,4,5,6,7,8,9,10,2] => [3,4,5,6,7,8,9,10,1,2] => [8,2] => ? = 8
[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [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
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,10,9,8,7,6,5,4,3,2] => [10,9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,2] => ? = 36
[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [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
[[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [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
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => [1,1,1,1,1,1,3] => ? = 21
[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,4] => ? = 15
[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => [1,1,1,1,5] => ? = 10
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: 74% ●values known / values provided: 80%●distinct values known / distinct values provided: 74%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 80%●distinct values known / distinct values provided: 74%
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
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 21
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 21
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 21
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [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
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[.,[.,.]],[[[.,.],.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[.,[.,[[[.,.],.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 20
[.,[.,[[.,[[[[.,.],.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [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
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 16
[.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 14
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [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
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[.,[.,[[.,.],.]]],[.,[.,.]]]]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[[.,.],.],.]],[[.,.],.]]]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[.,[[.,.],.]],.],[[.,.],.]]]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,.],[[[.,.],.],.]],[.,.]]]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],.],[[.,.],.]],[.,.]]]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[.,[[.,.],.]],[.,.]],[.,.]]]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [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
[.,[[[[.,[.,.]],.],[.,.]],[.,.]]]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[.,[[[.,.],.],.]],.],[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],[[.,.],.]],.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[.,.]],[.,.]],.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[[.,.],.]],.],.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 19
[.,[[.,[.,[[[[.,.],.],.],.]]],.]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 16
[.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 14
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: 74% ●values known / values provided: 80%●distinct values known / distinct values provided: 74%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 80%●distinct values known / distinct values provided: 74%
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
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 21
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 21
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 21
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [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
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[.,[.,.]],[[[.,.],.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[[.,.],[.,.]],[[.,.],.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[.,[.,[[[.,.],.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 20
[.,[.,[[.,[[[[.,.],.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 17
[.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [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
[.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 16
[.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 14
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,.],[.,.]],[[[.,.],.],.]]]
=> [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
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[.,[.,[[.,.],.]]],[.,[.,.]]]]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[[.,.],.],.]],[[.,.],.]]]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[.,[[.,.],.]],.],[[.,.],.]]]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
[.,[[[.,.],[[[.,.],.],.]],[.,.]]]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],.],[[.,.],.]],[.,.]]]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[.,[[.,.],.]],[.,.]],[.,.]]]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [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
[.,[[[[.,[.,.]],.],[.,.]],[.,.]]]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[.,[[[.,.],.],.]],.],[.,.]]]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 11
[.,[[[[.,.],[[.,.],.]],.],[.,.]]]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[.,.]],[.,.]],.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[[[.,[[.,.],.]],.],.],[.,.]]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 19
[.,[[.,[.,[[[[.,.],.],.],.]]],.]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 16
[.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 15
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 12
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: 56% ●values known / values provided: 59%●distinct values known / distinct values provided: 56%
St000012: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 59%●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
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 27
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 26
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 26
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 25
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 25
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 24
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 24
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 23
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 25
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 24
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? = 23
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 23
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [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
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 23
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 22
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 22
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 22
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 21
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 21
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 22
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> ? = 21
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 19
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 24
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 23
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 22
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 22
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 21
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 21
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 20
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 20
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 19
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> ? = 21
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 20
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 19
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 19
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [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
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 22
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 21
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 21
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 20
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 19
[.,[.,[[.,.],[[[.,.],.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 18
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 20
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 19
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 18
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 17
[.,[.,[[.,[.,.]],[.,[.,[.,.]]]]]]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 20
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: 49% ●values known / values provided: 49%●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: 49% ●values known / values provided: 49%●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
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [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
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,1]
=> ? = 17
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,1]
=> ? = 16
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2]
=> ? = 16
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,1]
=> ? = 15
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 15
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 14
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 15
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 14
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 13
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 13
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 17
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1]
=> ? = 16
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1]
=> ? = 15
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1]
=> ? = 15
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,1]
=> ? = 14
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? = 14
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2]
=> ? = 13
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1]
=> ? = 13
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> ? = 12
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2]
=> ? = 14
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2]
=> ? = 13
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> ? = 12
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? = 14
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,1,1]
=> ? = 12
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1,1]
=> ? = 13
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1,1]
=> ? = 12
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1,1]
=> ? = 11
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2]
=> ? = 11
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [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
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 12
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 11
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? = 11
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [6,4,4,1]
=> ? = 10
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2]
=> ? = 10
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 12
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2]
=> ? = 11
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 11
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [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
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 10
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [5,5,4,1]
=> ? = 9
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,1]
=> ? = 13
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,1,1]
=> ? = 12
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [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
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,1,1]
=> ? = 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
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,1,1]
=> ? = 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
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: St000041
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Values
[.,.]
=> [1,0]
=> [(1,2)]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 6
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 5
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 4
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 4
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 10
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 8
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 7
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 6
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> 7
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> 6
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 6
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 4
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> 4
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 4
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,8),(9,12),(10,11)]
=> ? = 10
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,10),(8,9),(11,12)]
=> ? = 10
[[.,.],[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [(1,2),(3,14),(4,13),(5,8),(6,7),(9,10),(11,12)]
=> ? = 10
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,2),(3,14),(4,13),(5,6),(7,8),(9,10),(11,12)]
=> ? = 9
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,14),(4,5),(6,13),(7,12),(8,9),(10,11)]
=> ? = 10
[[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,5),(6,13),(7,8),(9,12),(10,11)]
=> ? = 9
[[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,14),(4,5),(6,13),(7,10),(8,9),(11,12)]
=> ? = 9
[[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [(1,2),(3,14),(4,5),(6,13),(7,8),(9,10),(11,12)]
=> ? = 8
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,7),(5,6),(8,13),(9,12),(10,11)]
=> ? = 9
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [(1,2),(3,14),(4,7),(5,6),(8,13),(9,10),(11,12)]
=> ? = 8
[[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,14),(4,5),(6,7),(8,13),(9,12),(10,11)]
=> ? = 8
[[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,14),(4,5),(6,7),(8,13),(9,10),(11,12)]
=> ? = 7
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [(1,2),(3,14),(4,9),(5,8),(6,7),(10,13),(11,12)]
=> ? = 9
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [(1,2),(3,14),(4,9),(5,6),(7,8),(10,13),(11,12)]
=> ? = 8
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,14),(4,5),(6,9),(7,8),(10,13),(11,12)]
=> ? = 7
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [(1,2),(3,14),(4,7),(5,6),(8,9),(10,13),(11,12)]
=> ? = 7
[[.,.],[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,14),(4,5),(6,7),(8,9),(10,13),(11,12)]
=> ? = 6
[[.,.],[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [(1,2),(3,14),(4,11),(5,8),(6,7),(9,10),(12,13)]
=> ? = 9
[[.,.],[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,14),(4,5),(6,11),(7,10),(8,9),(12,13)]
=> ? = 8
[[.,.],[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,14),(4,5),(6,11),(7,8),(9,10),(12,13)]
=> ? = 7
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [(1,2),(3,14),(4,7),(5,6),(8,11),(9,10),(12,13)]
=> ? = 7
[[.,.],[[[[.,.],.],[.,.]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,14),(4,5),(6,7),(8,11),(9,10),(12,13)]
=> ? = 6
[[.,.],[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [(1,2),(3,14),(4,5),(6,9),(7,8),(10,11),(12,13)]
=> ? = 6
[[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,4),(2,3),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ? = 10
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ? = 9
[[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,4),(2,3),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ? = 9
[[.,[.,.]],[.,[[[.,.],.],.]]]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ? = 8
[[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ? = 8
[[.,[.,.]],[[.,.],[[.,.],.]]]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,14),(6,7),(8,13),(9,10),(11,12)]
=> ? = 7
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,14),(6,9),(7,8),(10,13),(11,12)]
=> ? = 7
[[.,[.,.]],[[[.,.],.],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,14),(6,7),(8,9),(10,13),(11,12)]
=> ? = 6
[[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,4),(2,3),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ? = 8
[[.,[.,.]],[[.,[[.,.],.]],.]]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,14),(6,11),(7,8),(9,10),(12,13)]
=> ? = 7
[[.,[.,.]],[[[.,.],[.,.]],.]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,14),(6,7),(8,11),(9,10),(12,13)]
=> ? = 6
[[.,[.,.]],[[[.,[.,.]],.],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,14),(6,9),(7,8),(10,11),(12,13)]
=> ? = 6
[[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,14),(6,7),(8,9),(10,11),(12,13)]
=> ? = 5
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ? = 8
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ? = 7
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,13),(9,10),(11,12)]
=> ? = 6
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,9),(7,8),(10,13),(11,12)]
=> ? = 6
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,9),(10,13),(11,12)]
=> ? = 5
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,7),(8,11),(9,10),(12,13)]
=> ? = 5
[[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,14),(8,13),(9,10),(11,12)]
=> ? = 8
[[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,14),(8,9),(10,13),(11,12)]
=> ? = 7
[[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [(1,6),(2,5),(3,4),(7,14),(8,11),(9,10),(12,13)]
=> ? = 7
[[.,[.,[.,.]]],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [(1,6),(2,5),(3,4),(7,14),(8,9),(10,11),(12,13)]
=> ? = 6
[[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 8
[[.,[[.,.],.]],[.,[[.,.],.]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,10),(11,12)]
=> ? = 7
[[.,[[.,.],.]],[[.,.],[.,.]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,9),(10,13),(11,12)]
=> ? = 6
[[.,[[.,.],.]],[[.,[.,.]],.]]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,11),(9,10),(12,13)]
=> ? = 6
Description
The number of nestings of a perfect matching.
This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$.
Matching statistic: St000081
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 82%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 82%
Values
[.,.]
=> [1] => ([],1)
=> 0
[.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1
[[.,.],.]
=> [1,2] => ([],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 1
[[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 6
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 17
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 14
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [3,5,4,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [4,3,5,7,6,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [4,3,6,5,7,2,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [2,5,4,7,6,3,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [3,2,7,6,5,4,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [3,2,6,7,5,4,1] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [4,3,2,7,6,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 11
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [3,4,2,7,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 11
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [3,4,2,6,7,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 10
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [2,4,3,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [2,4,3,6,7,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [3,2,4,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [3,2,4,6,7,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [5,4,3,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 13
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [4,5,3,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [3,5,4,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [4,3,5,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [3,4,5,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [2,5,4,3,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [2,4,5,3,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [2,3,5,4,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [4,3,2,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [3,4,2,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [2,4,3,5,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [3,2,4,5,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[.,[[.,[[.,[.,.]],[.,.]]],.]]
=> [4,3,6,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 12
[.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [3,2,6,5,4,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [3,2,5,6,4,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [4,3,2,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
[.,[[[.,[[.,.],.]],[.,.]],.]]
=> [3,4,2,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> [2,4,3,6,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[.,[[[[.,[.,.]],.],[.,.]],.]]
=> [3,2,4,6,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [3,2,5,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,5,4,7,6,3,2] => ([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,4,3,7,6,5,2] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,4,3,6,7,5,2] => ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8
[[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => ([(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 9
[[.,.],[[.,[[.,.],.]],[.,.]]]
=> [1,4,5,3,7,6,2] => ([(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 8
[[.,.],[[[.,.],[.,.]],[.,.]]]
=> [1,3,5,4,7,6,2] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[[.,.],[[[.,[.,.]],.],[.,.]]]
=> [1,4,3,5,7,6,2] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[[.,.],[[[.,[.,.]],[.,.]],.]]
=> [1,4,3,6,5,7,2] => ([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 11
[[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [2,1,6,7,5,4,3] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10
Description
The number of edges of a graph.
Matching statistic: St001397
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 56%
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 56%
Values
[.,.]
=> [1] => ([],1)
=> 0
[.,[.,.]]
=> [2,1] => ([],2)
=> 1
[[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 3
[.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 2
[[.,.],[.,.]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 6
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 5
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> 10
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 9
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> 8
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 8
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 7
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 7
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> 6
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 7
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 6
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 6
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 5
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 4
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 4
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
=> ? = 17
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
=> ? = 14
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 15
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [3,4,6,7,5,2,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ? = 13
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [4,5,3,7,6,2,1] => ([(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ? = 14
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [4,6,5,3,7,2,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 15
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 14
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [3,4,6,5,7,2,1] => ([(2,3),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 12
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 13
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 11
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [2,4,6,7,5,3,1] => ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
=> ? = 12
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [2,4,5,7,6,3,1] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? = 11
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [2,4,6,5,7,3,1] => ([(1,2),(1,5),(3,6),(4,6),(5,3),(5,4)],7)
=> ? = 11
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? = 10
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [3,2,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 13
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [3,2,6,7,5,4,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
=> ? = 12
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> ? = 11
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [2,3,6,7,5,4,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? = 11
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? = 10
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => ([(1,5),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
=> ? = 10
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7)
=> ? = 9
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [4,3,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 12
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [3,4,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? = 11
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [3,4,2,6,7,5,1] => ([(1,5),(1,6),(2,4),(4,5),(4,6),(6,3)],7)
=> ? = 10
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => ([(1,5),(4,3),(5,6),(6,2),(6,4)],7)
=> ? = 8
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [3,5,4,2,7,6,1] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 11
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [3,4,5,2,7,6,1] => ([(1,5),(1,6),(2,3),(3,4),(4,5),(4,6)],7)
=> ? = 10
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [2,4,5,3,7,6,1] => ([(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,2)],7)
=> ? = 9
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [2,3,5,4,7,6,1] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3)],7)
=> ? = 8
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [3,4,2,5,7,6,1] => ([(1,6),(2,3),(3,6),(6,4),(6,5)],7)
=> ? = 9
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => ([(1,5),(4,6),(5,4),(6,2),(6,3)],7)
=> ? = 7
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [3,5,6,4,2,7,1] => ([(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? = 12
[.,[[.,[[[.,.],.],[.,.]]],.]]
=> [3,4,6,5,2,7,1] => ([(1,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 11
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 12
[.,[[.,[[[.,.],[.,.]],.]],.]]
=> [3,5,4,6,2,7,1] => ([(1,5),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ? = 11
[.,[[[.,.],[[.,.],[.,.]]],.]]
=> [2,4,6,5,3,7,1] => ([(1,4),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3)],7)
=> ? = 10
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [2,4,5,6,3,7,1] => ([(1,3),(1,5),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 9
[.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [3,2,6,5,4,7,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 10
[.,[[[[.,.],.],[.,[.,.]]],.]]
=> [2,3,6,5,4,7,1] => ([(1,5),(2,6),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
=> ? = 9
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [2,3,5,6,4,7,1] => ([(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> ? = 8
[.,[[[.,[[.,.],.]],[.,.]],.]]
=> [3,4,2,6,5,7,1] => ([(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 9
[.,[[[.,[[.,.],[.,.]]],.],.]]
=> [3,5,4,2,6,7,1] => ([(1,6),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ? = 10
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 9
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [2,3,5,4,6,7,1] => ([(1,5),(2,6),(3,6),(5,2),(5,3),(6,4)],7)
=> ? = 7
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 8
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [2,4,3,5,6,7,1] => ([(1,3),(1,4),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 7
[.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 6
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,4,6,7,5,3,2] => ([(0,3),(0,4),(0,6),(5,2),(6,1),(6,5)],7)
=> ? = 11
[[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,4,5,6,7,3,2] => ([(0,2),(0,3),(0,6),(4,5),(5,1),(6,4)],7)
=> ? = 9
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,3,6,7,5,4,2] => ([(0,4),(0,6),(5,3),(6,1),(6,2),(6,5)],7)
=> ? = 10
Description
Number of pairs of incomparable elements in a finite poset.
For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000161The sum of the sizes of the right subtrees of a binary tree. St000246The number of non-inversions of a permutation. St000018The number of inversions of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000795The mad of a permutation. St000833The comajor index of a permutation. 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. St000332The positive inversions of an alternating sign matrix. 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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