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Your data matches 32 different statistics following compositions of up to 3 maps.
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Matching statistic: St001280
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> 0
[3,2,1] => [2,1]
=> [1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [3,1]
=> [1]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> 1
[4,1,3,2] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000291
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> 1 => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 11 => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> 01 => 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> 01 => 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> 01 => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 0
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 0
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> 001 => 0
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> 001 => 0
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 0
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 0
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> 001 => 0
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> 001 => 0
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> 001 => 0
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 0
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> 001 => 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 001 => 0
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> 001 => 0
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 011 => 0
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 0
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 0
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 0
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 0
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 0
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 0
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 0
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 0
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
[1,4,5,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 0
[1,5,2,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 0
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 0
[1,5,3,2,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0011 => 0
[1,5,3,4,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0
[1,5,4,2,3] => [4,1]
=> [[1,2,3,4],[5]]
=> 0001 => 0
[1,5,4,3,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0111 => 0
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
[2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0101 => 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 0
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 0
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? = 1
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 1
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? = 1
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? = 1
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 4
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 4
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 1
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 1
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 2
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 1
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 2
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 1
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 1
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 1
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 1
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 2
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 1
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> 00001001010 => ? = 3
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? => ? = 2
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? => ? = 1
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 1
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 1
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 3
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,3,4,5,6,7,8],[9,10],[11,12]]
=> ? => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 1
[2,7,8,9,10,12,1,3,4,5,6,11] => [3,3,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12]]
=> ? => ? = 4
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
=> 00001001010 => ? = 3
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12]]
=> ? => ? = 2
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[3,4,6,8,11,12,1,2,5,7,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 2
[3,4,7,8,10,12,1,2,5,6,9,11] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 2
[3,4,7,8,11,12,1,2,5,6,9,10] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[3,4,7,9,11,12,1,2,5,6,8,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 2
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12]]
=> ? => ? = 1
[3,5,6,9,11,12,1,2,4,7,8,10] => [6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> 00000100010 => ? = 2
[3,5,6,10,11,12,1,2,4,7,8,9] => [8,4]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12]]
=> ? => ? = 1
[3,5,7,8,10,12,1,2,4,6,9,11] => [9,3]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12]]
=> ? => ? = 1
[3,5,7,8,11,12,1,2,4,6,9,10] => [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 2
Description
The number of descents of a binary word.
Matching statistic: St000292
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> 1 => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 11 => 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> 10 => 0
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> 10 => 0
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> 10 => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 111 => 0
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 0
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 0
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 0
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 0
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 0
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> 100 => 0
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 110 => 0
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> 010 => 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1111 => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
[1,5,2,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
[1,5,2,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
[1,5,3,2,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1100 => 0
[1,5,3,4,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0
[1,5,4,2,3] => [4,1]
=> [[1,3,4,5],[2]]
=> 1000 => 0
[1,5,4,3,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1110 => 0
[2,1,3,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 1010 => 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 0
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 0
[11,1,2,3,4,5,6,10,7,8,9] => [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? = 1
[10,1,2,3,4,5,11,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 1
[11,1,2,3,4,10,5,6,7,8,9] => [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 1
[11,10,1,2,3,4,5,6,7,8,9] => [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 1
[10,9,8,7,6,5,4,3,2,1,11] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 4
[1,11,10,9,8,7,6,5,4,3,2] => [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 4
[2,4,6,8,11,12,1,3,5,7,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[2,4,6,9,10,12,1,3,5,7,8,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 1
[2,4,6,10,11,12,1,3,5,7,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 1
[2,4,7,8,11,12,1,3,5,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 2
[2,4,7,9,11,12,1,3,5,6,8,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[2,4,8,9,10,12,1,3,5,6,7,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 1
[2,4,8,9,11,12,1,3,5,6,7,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 2
[2,4,8,10,11,12,1,3,5,6,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 1
[2,5,6,8,10,12,1,3,4,7,9,11] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 1
[2,5,6,9,10,12,1,3,4,7,8,11] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 2
[2,5,6,9,11,12,1,3,4,7,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 1
[2,5,7,8,11,12,1,3,4,6,9,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[2,5,7,9,10,12,1,3,4,6,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 1
[2,5,7,9,11,12,1,3,4,6,8,10] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 2
[2,5,7,10,11,12,1,3,4,6,8,9] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 1
[2,5,8,9,10,12,1,3,4,6,7,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,5,8,9,11,12,1,3,4,6,7,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> 01010010000 => ? = 3
[2,5,8,10,11,12,1,3,4,6,7,9] => [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? => ? = 2
[2,5,9,10,11,12,1,3,4,6,7,8] => [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? => ? = 1
[2,6,7,8,10,12,1,3,4,5,9,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 1
[2,6,7,9,10,12,1,3,4,5,8,11] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,7,9,11,12,1,3,4,5,8,10] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 1
[2,6,8,9,10,12,1,3,4,5,7,11] => [6,2,2,2]
=> [[1,2,9,10,11,12],[3,4],[5,6],[7,8]]
=> ? => ? = 3
[2,6,8,9,11,12,1,3,4,5,7,10] => [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ? => ? = 2
[2,6,8,10,11,12,1,3,4,5,7,9] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 1
[2,7,8,9,10,12,1,3,4,5,6,11] => [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? => ? = 4
[2,7,8,9,11,12,1,3,4,5,6,10] => [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> 01010010000 => ? = 3
[2,7,8,10,11,12,1,3,4,5,6,9] => [7,3,2]
=> [[1,2,5,9,10,11,12],[3,4,8],[6,7]]
=> ? => ? = 2
[2,7,9,10,11,12,1,3,4,5,6,8] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[3,4,6,8,10,12,1,2,5,7,9,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[3,4,6,8,11,12,1,2,5,7,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 2
[3,4,7,8,10,12,1,2,5,6,9,11] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 2
[3,4,7,8,11,12,1,2,5,6,9,10] => [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? => ? = 3
[3,4,7,9,10,12,1,2,5,6,8,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[3,4,7,9,11,12,1,2,5,6,8,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 2
[3,4,7,10,11,12,1,2,5,6,8,9] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[3,4,8,9,11,12,1,2,5,6,7,10] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[3,5,6,9,10,12,1,2,4,7,8,11] => [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> ? => ? = 1
[3,5,6,9,11,12,1,2,4,7,8,10] => [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> 01000100000 => ? = 2
[3,5,6,10,11,12,1,2,4,7,8,9] => [8,4]
=> [[1,2,3,4,9,10,11,12],[5,6,7,8]]
=> ? => ? = 1
[3,5,7,8,10,12,1,2,4,6,9,11] => [9,3]
=> [[1,2,3,7,8,9,10,11,12],[4,5,6]]
=> ? => ? = 1
[3,5,7,8,11,12,1,2,4,6,9,10] => [6,3,3]
=> [[1,2,3,10,11,12],[4,5,6],[7,8,9]]
=> ? => ? = 2
Description
The number of ascents of a binary word.
Matching statistic: St000052
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,4,1,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,3,2,1] => [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,5,4,2,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[7,6,8,4,5,3,2,1] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[7,8,3,4,5,6,2,1] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,6,3,4,5,8,2,1] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[7,5,3,4,6,8,2,1] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[7,8,3,4,2,5,6,1] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,5,3,4,7,2,8,1] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,2,3,4,5,7,8,1] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,2,3,4,6,7,8,1] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[4,2,3,5,6,7,8,1] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,7,8,4,5,3,1,2] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,7,3,4,5,6,1,2] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,5,3,4,6,7,1,2] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,6,7,4,5,1,2,3] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,7,3,4,2,5,1,6] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[7,8,3,4,5,1,2,6] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[7,8,3,4,2,1,5,6] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,7,3,4,1,2,5,6] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,6,3,4,5,1,2,7] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[8,5,3,4,6,1,2,7] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,6,3,4,2,1,5,7] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,2,3,4,5,1,6,7] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,5,3,4,1,2,6,7] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,2,3,4,1,5,6,7] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[8,2,3,1,4,5,6,7] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,5,7,4,3,2,1,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,7,5,4,2,3,1,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[5,6,7,4,2,3,1,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,7,3,4,2,5,1,8] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[6,7,3,2,4,5,1,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[5,6,3,4,2,7,1,8] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[5,4,3,6,2,7,1,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[5,2,3,4,6,7,1,8] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,2,3,5,6,7,1,8] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,2,4,5,6,7,1,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[7,6,5,4,2,1,3,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[7,6,3,2,4,1,5,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,7,3,4,1,2,5,8] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[6,7,3,2,1,4,5,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[7,5,3,2,1,4,6,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[7,2,3,4,1,5,6,8] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,4,3,1,2,5,6,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[7,2,3,1,4,5,6,8] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[7,2,1,3,4,5,6,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[5,4,6,3,2,1,7,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[5,6,3,4,2,1,7,8] => [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,4,3,6,2,1,7,8] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[3,4,5,6,2,1,7,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[5,6,3,2,4,1,7,8] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[5,6,2,3,4,1,7,8] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[4,5,3,2,6,1,7,8] => [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St001712
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 0
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 0
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 0
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 0
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 0
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 0
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 1
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 0
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 0
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 0
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0
[1,5,2,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[1,5,4,2,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[2,1,4,3,6,5,8,7,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[3,4,1,2,6,5,8,7,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[4,3,2,1,6,5,8,7,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[6,5,4,3,2,1,8,7,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[8,9,5,6,3,4,10,1,2,7] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[4,3,2,1,8,7,6,5,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[8,7,6,5,4,3,2,1,10,9] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[9,7,6,5,4,3,2,10,1,8] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[10,8,9,6,7,4,5,2,3,1] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[6,7,8,9,10,1,2,3,4,5] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[7,3,2,8,9,10,1,4,5,6] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[8,5,4,3,2,9,10,1,6,7] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[6,5,4,3,2,1,10,9,8,7] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[9,7,5,10,3,8,2,6,1,4] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[4,3,2,1,10,9,8,7,6,5] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[2,1,10,9,8,7,6,5,4,3] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[9,10,8,7,6,5,4,3,1,2] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[10,9,8,7,6,5,4,3,2,1] => [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 4
[9,1,2,3,4,8,5,6,7] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[8,1,2,3,9,4,5,6,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[5,1,2,3,9,8,4,6,7] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[9,1,2,3,6,8,4,5,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[4,1,2,9,3,8,5,6,7] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[9,1,2,8,3,4,5,6,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[7,1,2,8,3,4,5,9,6] => [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> ? = 1
[4,1,2,8,9,3,5,6,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[8,1,2,5,9,3,4,6,7] => [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> ? = 1
[8,1,2,9,6,3,4,5,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[4,1,2,5,9,8,3,6,7] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[9,1,2,8,6,7,3,4,5] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[8,1,7,2,3,4,5,9,6] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[3,1,9,8,2,4,5,6,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[7,1,4,9,2,3,5,6,8] => [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> ? = 1
[9,1,4,8,2,3,5,6,7] => [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> ? = 1
[9,1,6,5,2,3,4,7,8] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[3,1,4,8,9,2,5,6,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[8,1,9,5,6,2,3,4,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[3,1,4,9,8,7,2,5,6] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[3,1,9,8,6,7,2,4,5] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[9,1,4,8,6,7,2,3,5] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[9,1,8,5,6,7,2,3,4] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[2,9,1,3,4,8,5,6,7] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[2,8,1,3,9,4,5,6,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[2,9,1,3,8,7,4,5,6] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[2,9,1,8,3,4,5,6,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[2,7,1,6,3,4,8,9,5] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[2,4,1,8,9,3,5,6,7] => [6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9]]
=> ? = 1
[2,9,1,8,6,7,3,4,5] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[9,4,1,2,3,5,6,7,8] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
[7,4,1,2,3,5,8,9,6] => [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 1
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau T is an entry i such that i+1 appears in a higher row than i in English notation.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 17%
Values
[1,2] => [1,2] => [2] => ([],2)
=> ? = 0
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 0
[3,2,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 0
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 0
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> 0
[2,4,3,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[4,1,3,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,4,5,3,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,5,2,3,4] => [1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,2,4,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,3,2,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,5,4,2,3] => [1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,5,4,3,2] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,3,4,1,5] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> 0
[2,3,5,4,1] => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,1,3,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[2,4,3,1,5] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,3,5,1] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,5,1,4,3] => [4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,5,3,1,4] => [3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,5,3,4,1] => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,5,4,3,1] => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,1,4,2,5] => [4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[3,1,5,4,2] => [4,5,2,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[3,2,1,4,5] => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,2,4,1,5] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,4,2,1,5] => [4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,4,5,2,1] => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,5,1,2,4] => [3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,5,1,4,2] => [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,5,2,4,1] => [4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,5,4,1,2] => [4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,1,5,2,3] => [4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,2,5,1,3] => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,2,5,3,1] => [2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[4,3,1,2,5] => [4,2,3,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[4,3,2,1,5] => [3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,3,2,5,1] => [3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,3,5,1,2] => [4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,1,3,2] => [4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,2,1,3] => [4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,3,1,2] => [3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[4,5,3,2,1] => [3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[5,1,4,3,2] => [4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,2,4,1,3] => [2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[5,2,4,3,1] => [2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,3,1,4,2] => [4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[5,3,2,1,4] => [3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,3,2,4,1] => [3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,3,4,2,1] => [4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,4,1,2,3] => [4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[5,4,2,3,1] => [4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000023
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,4,5,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,2,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,5,3,2,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,5,3,4,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,5,4,2,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,4,3,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,4,6,5,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,4,6,7,5] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,4,7,6,5] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,5,4,6,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,5,4,7,6] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,3,5,6,4,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,5,6,7,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,5,7,4,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,5,7,6,4] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,6,4,5,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,6,4,7,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,6,5,4,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,6,5,7,4] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,6,7,4,5] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,3,6,7,5,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,7,4,5,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,7,4,6,5] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,7,5,4,6] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,7,5,6,4] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,7,6,4,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,7,6,5,4] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,4,3,5,6,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,4,3,5,7,6] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,4,3,6,5,7] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,4,3,6,7,5] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,4,3,7,5,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,4,3,7,6,5] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,4,5,3,6,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,4,5,3,7,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,4,5,6,3,7] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,5,6,7,3] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,5,7,3,6] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,5,7,6,3] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,6,3,5,7] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,6,3,7,5] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,6,5,3,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,4,6,5,7,3] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,6,7,3,5] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,4,6,7,5,3] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,7,3,5,6] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,7,3,6,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,7,5,3,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,7,5,6,3] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,4,7,6,3,5] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,7,6,5,3] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,5,3,4,6,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,5,3,4,7,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000779
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,4,5,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,2,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,5,3,2,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,5,3,4,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,5,4,2,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,5,4,3,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,4,6,5,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,4,6,7,5] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,4,7,6,5] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,5,4,6,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,5,4,7,6] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,3,5,6,4,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,5,6,7,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,5,7,4,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,5,7,6,4] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,6,4,5,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,6,4,7,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,6,5,4,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,6,5,7,4] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,6,7,4,5] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,3,6,7,5,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,7,4,5,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,7,4,6,5] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,7,5,4,6] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,3,7,5,6,4] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,3,7,6,4,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,3,7,6,5,4] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,4,3,5,6,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0
[1,2,4,3,5,7,6] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,4,3,6,5,7] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,4,3,6,7,5] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,4,3,7,5,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,4,3,7,6,5] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[1,2,4,5,3,6,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,4,5,3,7,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,4,5,6,3,7] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,5,6,7,3] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,5,7,3,6] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,5,7,6,3] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,6,3,5,7] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,6,3,7,5] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,6,5,3,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,4,6,5,7,3] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,6,7,3,5] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,4,6,7,5,3] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,7,3,5,6] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,7,3,6,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,7,5,3,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,2,4,7,5,6,3] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,4,7,6,3,5] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,4,7,6,5,3] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[1,2,5,3,4,6,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0
[1,2,5,3,4,7,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
Description
The tier of a permutation.
This is the number of elements i such that [i+1,k,i] is an occurrence of the pattern [2,3,1]. For example, [3,5,6,1,2,4] has tier 2, with witnesses [3,5,2] (or [3,6,2]) and [5,6,4].
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
Matching statistic: St000872
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000872: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000872: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[3,2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[1,3,4,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[1,4,2,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[1,4,3,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[2,3,1,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[3,1,2,4] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[3,2,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[4,1,3,2] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[4,2,3,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,2,4,5,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,2,5,3,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,2,5,4,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[1,3,4,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,3,4,5,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,3,5,2,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,4,2,3,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,4,2,5,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,4,3,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,4,5,2,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[1,4,5,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,5,2,3,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,5,2,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,5,3,2,4] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[1,5,3,4,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[1,5,4,2,3] => [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[1,5,4,3,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[2,1,3,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 0
[1,2,3,4,6,5,7] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 0
[1,2,3,4,6,7,5] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,3,4,7,6,5] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 0
[1,2,3,5,4,6,7] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 0
[1,2,3,5,4,7,6] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 1
[1,2,3,5,6,4,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,3,5,6,7,4] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,3,5,7,4,6] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,3,5,7,6,4] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,3,6,4,5,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,3,6,4,7,5] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,3,6,5,4,7] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 0
[1,2,3,6,5,7,4] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,3,6,7,4,5] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 1
[1,2,3,6,7,5,4] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,3,7,4,5,6] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,3,7,4,6,5] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,3,7,5,4,6] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,3,7,5,6,4] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 0
[1,2,3,7,6,4,5] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,3,7,6,5,4] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 1
[1,2,4,3,5,6,7] => [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 0
[1,2,4,3,5,7,6] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 1
[1,2,4,3,6,5,7] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 1
[1,2,4,3,6,7,5] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1
[1,2,4,3,7,5,6] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1
[1,2,4,3,7,6,5] => [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? = 1
[1,2,4,5,3,6,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,4,5,3,7,6] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1
[1,2,4,5,6,3,7] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,4,5,6,7,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0
[1,2,4,5,7,3,6] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0
[1,2,4,5,7,6,3] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,4,6,3,5,7] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,4,6,3,7,5] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0
[1,2,4,6,5,3,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,4,6,5,7,3] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,4,6,7,3,5] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1
[1,2,4,6,7,5,3] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0
[1,2,4,7,3,5,6] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0
[1,2,4,7,3,6,5] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,4,7,5,3,6] => [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 0
[1,2,4,7,5,6,3] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,4,7,6,3,5] => [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 0
[1,2,4,7,6,5,3] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1
[1,2,5,3,4,6,7] => [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 0
[1,2,5,3,4,7,6] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 1
Description
The number of very big descents of a permutation.
A very big descent of a permutation π is an index i such that πi−πi+1>2.
For the number of descents, see [[St000021]] and for the number of big descents, see [[St000647]]. General r-descents were for example be studied in [1, Section 2].
Matching statistic: St000099
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1 = 0 + 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 0 + 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 0 + 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1 = 0 + 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 0 + 1
[1,3,2,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 0 + 1
[1,3,4,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 0 + 1
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 0 + 1
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 1 + 1
[2,3,1,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 0 + 1
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 1 + 1
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 0 + 1
[4,3,2,1] => [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 1 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1 = 0 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 0 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 0 + 1
[1,2,4,5,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,2,5,3,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 0 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 0 + 1
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,3,5,2,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,4,2,3,5] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,4,2,5,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,4,5,2,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[1,4,5,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 0 + 1
[1,5,4,2,3] => [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 0 + 1
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 0 + 1
[1,2,3,4,5,7,6] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0 + 1
[1,2,3,4,6,5,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0 + 1
[1,2,3,4,6,7,5] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,4,7,5,6] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,4,7,6,5] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0 + 1
[1,2,3,5,4,6,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0 + 1
[1,2,3,5,4,7,6] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1 + 1
[1,2,3,5,6,4,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,5,6,7,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,3,5,7,4,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,3,5,7,6,4] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,6,4,5,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,6,4,7,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,3,6,5,4,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0 + 1
[1,2,3,6,5,7,4] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,6,7,4,5] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1 + 1
[1,2,3,6,7,5,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,3,7,4,5,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,3,7,4,6,5] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,7,5,4,6] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,3,7,5,6,4] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0 + 1
[1,2,3,7,6,4,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,3,7,6,5,4] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1 + 1
[1,2,4,3,5,6,7] => [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 0 + 1
[1,2,4,3,5,7,6] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1 + 1
[1,2,4,3,6,5,7] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1 + 1
[1,2,4,3,6,7,5] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1 + 1
[1,2,4,3,7,5,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1 + 1
[1,2,4,3,7,6,5] => [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1 + 1
[1,2,4,5,3,6,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,4,5,3,7,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1 + 1
[1,2,4,5,6,3,7] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,4,5,6,7,3] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0 + 1
[1,2,4,5,7,3,6] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0 + 1
[1,2,4,5,7,6,3] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,4,6,3,5,7] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,4,6,3,7,5] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0 + 1
[1,2,4,6,5,3,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,4,6,5,7,3] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,4,6,7,3,5] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1 + 1
[1,2,4,6,7,5,3] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0 + 1
[1,2,4,7,3,5,6] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0 + 1
[1,2,4,7,3,6,5] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,4,7,5,3,6] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0 + 1
[1,2,4,7,5,6,3] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,4,7,6,3,5] => [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 0 + 1
[1,2,4,7,6,5,3] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1 + 1
[1,2,5,3,4,6,7] => [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 0 + 1
[1,2,5,3,4,7,6] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1 + 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001964The interval resolution global dimension of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.
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