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Your data matches 70 different statistics following compositions of up to 3 maps.
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Matching statistic: St001214
St001214: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 2
[3,3]
=> 3
[3,2,1]
=> 3
[3,1,1,1]
=> 2
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 2
[4,3]
=> 3
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 4
[3,2,2]
=> 4
[3,2,1,1]
=> 3
[3,1,1,1,1]
=> 2
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 2
[5,3]
=> 3
[5,2,1]
=> 3
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
Matching statistic: St000703
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 86%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 86%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,3,2] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,4,2,3] => 2
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,2,3,5] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,4,2,3,5] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,5,2,4,3] => 2
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,2,3,4,5,6] => 0
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,3,2,4,5,6] => 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,4,2,5,3,6] => 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,4,2,3,5,6] => 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,4,2,3,5,6] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,5,2,6,3,4] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,5,2,3,4,6] => 3
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,5,2,4,3,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,6,2,5,3,4] => 3
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,3,2,4,5,6,7] => 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,3,2,4,5,6,7] => 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,4,2,5,3,6,7] => 2
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,4,2,3,5,6,7] => 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,4,2,3,5,6,7] => 2
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,5,2,3,6,4,7] => ? ∊ {2,3,3,3,3,3,4,4}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,5,2,6,3,4,7] => ? ∊ {2,3,3,3,3,3,4,4}
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,5,2,3,4,6,7] => ? ∊ {2,3,3,3,3,3,4,4}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,5,2,4,3,6,7] => ? ∊ {2,3,3,3,3,3,4,4}
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,6,2,4,3,7,5] => ? ∊ {2,3,3,3,3,3,4,4}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,6,2,4,3,5,7] => ? ∊ {2,3,3,3,3,3,4,4}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,6,2,5,3,4,7] => ? ∊ {2,3,3,3,3,3,4,4}
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,7,2,6,3,5,4] => ? ∊ {2,3,3,3,3,3,4,4}
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,3,2,4,5,6,7,8] => 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,4,2,5,3,6,7,8] => 2
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,4,2,3,5,6,7,8] => 2
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,4,2,3,5,6,7,8] => 2
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => 3
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,5,2,3,6,4,7,8] => ? ∊ {4,4}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,5,2,6,3,4,7,8] => 3
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,5,2,3,4,6,7,8] => 3
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,5,2,4,3,6,7,8] => 2
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,6,2,7,3,4,5,8] => 4
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,6,2,3,4,7,5,8] => 4
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,6,2,4,3,7,5,8] => 3
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,7,2,5,3,4,8,6] => ? ∊ {4,4}
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,3,2,4,5,6,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,3,2,4,5,6,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,4,2,5,3,6,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,4,2,3,5,6,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,4,2,3,5,6,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,5,2,6,3,7,4,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,5,2,3,6,4,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,5,2,6,3,4,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,5,2,3,4,6,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,5,2,4,3,6,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,6,2,3,7,4,8,5,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,6,2,7,3,4,5,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [1,6,2,3,4,7,5,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [1,6,2,4,3,7,5,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [1,6,2,4,3,5,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [1,6,2,5,3,4,7,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,7,2,8,3,9,4,5,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [1,7,2,4,3,8,5,6,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [1,7,2,4,3,5,8,6,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,7,2,8,3,5,4,6,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,7,2,5,3,4,8,6,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [1,7,2,5,3,4,6,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [1,7,2,6,3,5,4,8,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,8,2,6,3,9,4,5,7] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [1,8,2,6,3,4,5,9,7] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,9,1,7] => [1,8,2,6,3,5,4,7,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => [1,8,2,7,3,6,4,5,9] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [1,9,2,8,3,7,4,6,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [3,4,1,2,5,6,7,8,9,10] => [1,3,2,4,5,6,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [3,2,1,4,5,6,7,8,9,10] => [1,3,2,4,5,6,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9,10] => [1,4,2,5,3,6,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[7,2,1]
=> [[1,3,6,7,8,9,10],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9,10] => [1,4,2,3,5,6,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9,10] => [1,4,2,3,5,6,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,5,2,6,3,7,4,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [1,5,2,3,6,4,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9,10] => [1,5,2,6,3,4,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[6,2,1,1]
=> [[1,4,7,8,9,10],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9,10] => [1,5,2,3,4,6,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9,10] => [1,5,2,4,3,6,7,8,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [6,7,8,9,10,1,2,3,4,5] => [1,6,2,7,3,8,4,9,5,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [1,6,2,3,7,4,8,5,9,10] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
Description
The number of deficiencies of a permutation.
This is defined as
dec(σ)=#{i:σ(i)<i}.
The number of exceedances is [[St000155]].
Matching statistic: St001232
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 71%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 71%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4]
=> []
=> []
=> []
=> ? ∊ {1,2}
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2}
[5]
=> []
=> []
=> []
=> ? ∊ {2,2,2}
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2}
[6]
=> []
=> []
=> []
=> ? ∊ {2,2,3,3,3}
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3}
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3}
[1,1,1,1,1,1]
=> [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]
=> ? ∊ {2,2,3,3,3}
[7]
=> []
=> []
=> []
=> ? ∊ {2,2,3,3,3,3,4,4}
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4}
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4}
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4}
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4}
[2,1,1,1,1,1]
=> [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]
=> ? ∊ {2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4}
[8]
=> []
=> []
=> []
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[3,1,1,1,1,1]
=> [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]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,3,4,4,4,4,4,4,4,5}
[9]
=> []
=> []
=> []
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[7,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,1,1,1,1,1]
=> [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]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[9,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[8,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[7,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000670
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 71%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 71%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,3,2] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,4,2,3] => 2
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,2,3,5] => 2
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,4,2,3,5] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,5,2,4,3] => 2
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,2,3,4,5,6] => 0
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [1,3,2,4,5,6] => 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,4,2,5,3,6] => 3
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [1,4,2,3,5,6] => 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,4,2,3,5,6] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,5,2,6,3,4] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [1,5,2,3,4,6] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,5,2,4,3,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,6,2,5,3,4] => 3
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,3,2,4,5,6,7] => 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [1,3,2,4,5,6,7] => 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,4,2,5,3,6,7] => 3
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,4,2,3,5,6,7] => 2
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [1,4,2,3,5,6,7] => 2
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [1,5,2,3,6,4,7] => ? ∊ {2,2,3,3,3,3,4,4}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [1,5,2,6,3,4,7] => ? ∊ {2,2,3,3,3,3,4,4}
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,5,2,3,4,6,7] => ? ∊ {2,2,3,3,3,3,4,4}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [1,5,2,4,3,6,7] => ? ∊ {2,2,3,3,3,3,4,4}
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,6,2,4,3,7,5] => ? ∊ {2,2,3,3,3,3,4,4}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [1,6,2,4,3,5,7] => ? ∊ {2,2,3,3,3,3,4,4}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [1,6,2,5,3,4,7] => ? ∊ {2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,7,2,6,3,5,4] => ? ∊ {2,2,3,3,3,3,4,4}
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,3,2,4,5,6,7,8] => 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [1,3,2,4,5,6,7,8] => 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,4,2,5,3,6,7,8] => 3
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,4,2,3,5,6,7,8] => 2
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [1,4,2,3,5,6,7,8] => 2
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,5,2,6,3,7,4,8] => 4
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,5,2,3,6,4,7,8] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [1,5,2,6,3,4,7,8] => 3
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [1,5,2,3,4,6,7,8] => 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,5,2,4,3,6,7,8] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,6,2,7,3,4,5,8] => 3
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [1,6,2,3,4,7,5,8] => 3
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [1,6,2,4,3,7,5,8] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,6,2,4,3,5,7,8] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,6,2,5,3,4,7,8] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,7,2,8,3,5,4,6] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [1,7,2,5,3,4,8,6] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [1,7,2,5,3,4,6,8] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [1,7,2,6,3,5,4,8] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => ? ∊ {2,3,3,4,4,4,4,4,4,5}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [1,3,2,4,5,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [1,3,2,4,5,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,4,2,5,3,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,4,2,3,5,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [1,4,2,3,5,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,5,2,6,3,7,4,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,5,2,3,6,4,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [1,5,2,6,3,4,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [1,5,2,3,4,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [1,5,2,4,3,6,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [1,6,2,3,7,4,8,5,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [1,6,2,7,3,4,5,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [1,6,2,3,4,7,5,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [1,6,2,4,3,7,5,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [1,6,2,4,3,5,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [1,6,2,5,3,4,7,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,7,2,8,3,9,4,5,6] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [1,7,2,4,3,8,5,6,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [1,7,2,4,3,5,8,6,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [1,7,2,8,3,5,4,6,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [1,7,2,5,3,4,8,6,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [1,7,2,5,3,4,6,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [1,7,2,6,3,5,4,8,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,8,2,6,3,9,4,5,7] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => [1,8,2,6,3,4,5,9,7] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [8,6,5,4,3,2,9,1,7] => [1,8,2,6,3,5,4,7,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => [1,8,2,7,3,6,4,5,9] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [1,9,2,8,3,7,4,6,5] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6}
Description
The reversal length of a permutation.
A reversal in a permutation π=[π1,…,πn] is a reversal of a subsequence of the form reversali,j(π)=[π1,…,πi−1,πj,πj−1,…,πi+1,πi,πj+1,…,πn] for 1≤i<j≤n.
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.
Matching statistic: St001557
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 57%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 57%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,5,6] => ? ∊ {0,2}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {0,2}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,6,7] => ? ∊ {1,2,2,3}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => ? ∊ {1,2,2,3}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => ? ∊ {1,2,2,3}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? ∊ {1,2,2,3}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,5,4,3,2,1,7,8] => ? ∊ {2,2,2,2,3,3,4,4}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,5,6] => ? ∊ {2,2,2,2,3,3,4,4}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1,4,6] => ? ∊ {2,2,2,2,3,3,4,4}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1,3,4] => ? ∊ {2,2,2,2,3,3,4,4}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => ? ∊ {2,2,2,2,3,3,4,4}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,1,6] => ? ∊ {2,2,2,2,3,3,4,4}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => ? ∊ {2,2,2,2,3,3,4,4}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1,8] => ? ∊ {2,2,2,2,3,3,4,4}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,6,5,4,3,2,1,8,9] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [8,5,4,3,2,1,6,7] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1,4,5] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,2,1,5,6] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,1,4] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,1,6] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,5,4,3,2,1,7] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1,9] => ? ∊ {0,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [8,7,6,5,4,3,2,1,9,10] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [9,6,5,4,3,2,1,7,8] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [7,5,4,3,2,1,6,8] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [8,7,4,3,2,1,5,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,1,6,7] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,1,4,5] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,1,3,4] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,1,5,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,5,2,1,3,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,3,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,6,4,1,2,3] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,2,3,5,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,4] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,1,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,3,4,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,1,6] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,1,5] => ? ∊ {0,0,1,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
Description
The number of inversions of the second entry of a permutation.
This is, for a permutation π of length n,
#{2<k≤n∣π(2)>π(k)}.
The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001713
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 43%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 43%
Values
[1]
=> []
=> []
=> ?
=> ? = 0
[2]
=> []
=> []
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {0,0}
[3]
=> []
=> []
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[4]
=> []
=> []
=> ?
=> ? ∊ {1,1}
[3,1]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1}
[2,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[5]
=> []
=> []
=> ?
=> ? ∊ {1,1,2}
[4,1]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1,2}
[3,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2}
[6]
=> []
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3}
[5,1]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1,2,2,3,3}
[4,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3}
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3}
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3}
[7]
=> []
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[6,1]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[5,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4}
[8]
=> []
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[7,1]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[6,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[2,1,1,1,1,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,1,1,1,1,1,1],[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5}
[9]
=> []
=> []
=> ?
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,1]
=> [1]
=> [[1]]
=> [[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[7,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[6,3]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[6,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
[5,4]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2
[8,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,1],[1]]
=> 0
[7,3]
=> [3]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 3
[7,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2
[7,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 0
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
Matching statistic: St001514
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 57%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 57%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 2 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,3,3} + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {0,0,1,2,3,3} + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,1,2,3,3} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ? ∊ {0,0,1,2,3,3} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,1,2,3,3} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,2,3,3} + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,4,4} + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6} + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Matching statistic: St001498
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 71%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 71%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[2]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0}
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0}
[3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1}
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1}
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2}
[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]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,4,4,4,4,4,4,5}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 3
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 5
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
[4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 4
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 3
[3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 3
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> 2
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001435
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[1]
=> [1,0]
=> [1,0]
=> [[1],[]]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,1,1,2,2}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {0,1,1,2,2}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {0,1,1,2,2}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {0,1,1,2,2}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {0,1,1,2,2}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> ? ∊ {0,2,2,2,2,3,3,3}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? ∊ {0,2,2,2,2,3,3,3}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {0,2,2,2,2,3,3,3}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? ∊ {0,2,2,2,2,3,3,3}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {0,2,2,2,2,3,3,3}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? ∊ {0,2,2,2,2,3,3,3}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ? ∊ {0,2,2,2,2,3,3,3}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ? ∊ {0,2,2,2,2,3,3,3}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4],[]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[5,1,1]
=> [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]
=> [[4,4,4,4],[3]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1,1,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[4,3,3,3],[2]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2,2],[1,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[4,4,3,3],[2,2]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[5,5,5,5],[]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1,1],[]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1,1]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[5,5,5,2],[1,1,1]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 2
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [[5,5,2],[2,1]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[5,4,4,4],[3]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2],[1]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3,2],[2]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2,2],[1,1,1]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2,2],[1,1,1]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[4,4,3,3,3],[2,2]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[5,5,3,3],[2,2]]
=> ? ∊ {0,0,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[5,5,5,5,5],[]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[5,5,5,5,1],[]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1,1],[]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 0
Description
The number of missing boxes in the first row.
Matching statistic: St001438
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 57%
Values
[1]
=> [1,0]
=> [1,0]
=> [[1],[]]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {0,1,1,2,2}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {0,1,1,2,2}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {0,1,1,2,2}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {0,1,1,2,2}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {0,1,1,2,2}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> ? ∊ {0,1,2,2,2,3,3,3}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ? ∊ {0,1,2,2,2,3,3,3}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {0,1,2,2,2,3,3,3}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ? ∊ {0,1,2,2,2,3,3,3}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {0,1,2,2,2,3,3,3}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> ? ∊ {0,1,2,2,2,3,3,3}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> ? ∊ {0,1,2,2,2,3,3,3}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ? ∊ {0,1,2,2,2,3,3,3}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4],[]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[5,1,1]
=> [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]
=> [[4,4,4,4],[3]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1,1,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[4,3,3,3],[2]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [[3,2,2,2],[1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2,2],[1,1]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[4,4,3,3],[2,2]]
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[5,5,5,5],[]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1,1],[]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1,1]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[5,5,5,2],[1,1,1]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [[5,5,2],[2,1]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[5,4,4,4],[3]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [[4,2,2],[1]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3,2],[2]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2,2],[1,1,1]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 3
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [[5,2],[1]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2,2],[1,1,1]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[4,4,3,3,3],[2,2]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[5,5,3,3],[2,2]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,3,4,4,4,4,4,4,4,5}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[5,5,5,5,5],[]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[5,5,5,5,1],[]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1,1],[]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2]]
=> ? ∊ {0,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,6}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 0
Description
The number of missing boxes of a skew partition.
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001487The number of inner corners of a skew partition. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001569The maximal modular displacement of a permutation. St001330The hat guessing number of a graph. St000648The number of 2-excedences of a permutation. St000732The number of double deficiencies of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001556The number of inversions of the third entry of a permutation. St001822The number of alignments of a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000031The number of cycles in the cycle decomposition of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000527The width of the poset. St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St000664The number of right ropes of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001520The number of strict 3-descents. St001811The Castelnuovo-Mumford regularity of a permutation. St001846The number of elements which do not have a complement in the lattice. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000711The number of big exceedences of a permutation. St000899The maximal number of repetitions of an integer composition. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001889The size of the connectivity set of a signed permutation. St000117The number of centered tunnels of a Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000765The number of weak records in an integer composition. St001267The length of the Lyndon factorization of the binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001530The depth of a Dyck path.
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