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Your data matches 475 different statistics following compositions of up to 3 maps.
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Matching statistic: St000182
St000182: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 1
[1,1]
=> 1
[3]
=> 2
[2,1]
=> 3
[1,1,1]
=> 1
[4]
=> 6
[3,1]
=> 8
[2,2]
=> 3
[2,1,1]
=> 6
[1,1,1,1]
=> 1
[5]
=> 24
[4,1]
=> 30
[3,2]
=> 20
[3,1,1]
=> 20
[2,2,1]
=> 15
[2,1,1,1]
=> 10
[1,1,1,1,1]
=> 1
[6]
=> 120
[5,1]
=> 144
[4,2]
=> 90
[4,1,1]
=> 90
[3,3]
=> 40
[3,2,1]
=> 120
[3,1,1,1]
=> 40
[2,2,2]
=> 15
[2,2,1,1]
=> 45
[2,1,1,1,1]
=> 15
[1,1,1,1,1,1]
=> 1
Description
The number of permutations whose cycle type is the given integer partition.
This number is given by
$$\{ \pi \in \mathfrak{S}_n : \text{type}(\pi) = \lambda\} = \frac{n!}{\lambda_1 \cdots \lambda_k \mu_1(\lambda)! \cdots \mu_n(\lambda)!}$$
where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$.
All permutations with the same cycle type form a [[wikipedia:Conjugacy class]].
Matching statistic: St000690
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000690: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000690: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> {{1,2}}
=> [2,1] => 1
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> [1,2] => 1
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 2
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> [3,2,1] => 3
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 6
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> [3,2,4,1] => 8
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => 3
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 6
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 24
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 30
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 20
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 20
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 15
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 10
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 120
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 144
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 90
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => 90
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => 40
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => 120
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => 40
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => 15
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => 45
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => 15
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => 1
Description
The size of the conjugacy class of a permutation.
Two permutations are conjugate if and only if they have the same cycle type, this statistic is then computed as described in [[St000182]].
Matching statistic: St001582
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Values
[2]
=> [1,0,1,0]
=> [3,1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? ∊ {1,6,6,8}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? ∊ {1,6,6,8}
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? ∊ {1,6,6,8}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ? ∊ {1,6,6,8}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {1,10,15,20,20,24,30}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {1,10,15,20,20,24,30}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? ∊ {1,10,15,20,20,24,30}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? ∊ {1,10,15,20,20,24,30}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? ∊ {1,10,15,20,20,24,30}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? ∊ {1,10,15,20,20,24,30}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? ∊ {1,10,15,20,20,24,30}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001583
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Values
[2]
=> [1,0,1,0]
=> [3,1,2] => [2,3,1] => 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,3,1] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [2,4,3,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? ∊ {1,6,6,8}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,5,3,4,1] => ? ∊ {1,6,6,8}
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,4,2,5,1] => ? ∊ {1,6,6,8}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [2,3,5,4,1] => ? ∊ {1,6,6,8}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => ? ∊ {1,10,15,20,20,24,30}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [2,3,6,4,5,1] => ? ∊ {1,10,15,20,20,24,30}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,2,3,1,4] => ? ∊ {1,10,15,20,20,24,30}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [2,4,5,3,6,1] => ? ∊ {1,10,15,20,20,24,30}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,5,4,1,2] => ? ∊ {1,10,15,20,20,24,30}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [3,4,5,2,6,1] => ? ∊ {1,10,15,20,20,24,30}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [2,3,4,6,1,5] => ? ∊ {1,10,15,20,20,24,30}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [2,3,4,7,5,6,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [2,6,3,4,1,5] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [2,3,5,6,4,7,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,4,1,5,3] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [4,6,5,2,3,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [2,4,5,6,3,7,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [3,4,6,5,1,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [3,4,5,7,2,6,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001722
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Values
[2]
=> [1,0,1,0]
=> 1010 => 1001 => 1
[1,1]
=> [1,1,0,0]
=> 1100 => 1011 => 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => 011001 => 1
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => 010001 => 3
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 010011 => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 10011001 => ? ∊ {3,6,6,8}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 10111001 => ? ∊ {3,6,6,8}
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => 010111 => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 10110001 => ? ∊ {3,6,6,8}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 10110011 => ? ∊ {3,6,6,8}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0110011001 => ? ∊ {1,10,15,20,20,24,30}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0100011001 => ? ∊ {1,10,15,20,20,24,30}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 10100001 => ? ∊ {1,10,15,20,20,24,30}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0100111001 => ? ∊ {1,10,15,20,20,24,30}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 10110111 => ? ∊ {1,10,15,20,20,24,30}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0100110001 => ? ∊ {1,10,15,20,20,24,30}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 0100110011 => ? ∊ {1,10,15,20,20,24,30}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 100110011001 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 101110011001 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101111001 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => 101100011001 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 10100111 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0100100001 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => 101100111001 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 10101111 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 0100110111 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => 101100110001 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => 101100110011 => ? ∊ {1,15,15,40,40,45,90,90,120,120,144}
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000089
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000089: Integer compositions ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000089: Integer compositions ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Values
[2]
=> [1,0,1,0]
=> 1010 => [1,1,1,1] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> 1100 => [2,2] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => [1,1,1,1,1,1] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => [1,1,2,2] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => [2,1,1,2] => 2 = 3 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => [1,1,1,1,1,1,1,1] => ? ∊ {3,6,6,8} - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => [1,1,1,1,2,2] => ? ∊ {3,6,6,8} - 1
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => [3,3] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => [1,1,2,1,1,2] => ? ∊ {3,6,6,8} - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => [2,1,1,1,1,2] => ? ∊ {3,6,6,8} - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => [1,1,1,1,1,1,1,1,1,1] => ? ∊ {1,10,15,20,20,24,30} - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => [1,1,1,1,1,1,2,2] => ? ∊ {1,10,15,20,20,24,30} - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => [1,1,3,3] => ? ∊ {1,10,15,20,20,24,30} - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => [1,1,1,1,2,1,1,2] => ? ∊ {1,10,15,20,20,24,30} - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => [3,2,1,2] => ? ∊ {1,10,15,20,20,24,30} - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => [1,1,2,1,1,1,1,2] => ? ∊ {1,10,15,20,20,24,30} - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => [2,1,1,1,1,1,1,2] => ? ∊ {1,10,15,20,20,24,30} - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => [1,1,1,1,1,1,1,1,2,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => [1,1,1,1,3,3] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => [1,1,1,1,1,1,2,1,1,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => [3,1,1,3] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => [1,1,3,2,1,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => [1,1,1,1,2,1,1,1,1,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => [4,4] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => [3,2,1,1,1,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => [1,1,2,1,1,1,1,1,1,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => [2,1,1,1,1,1,1,1,1,2] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
Description
The absolute variation of a composition.
Matching statistic: St000527
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000527: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 21%●distinct values known / distinct values provided: 13%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000527: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 21%●distinct values known / distinct values provided: 13%
Values
[2]
=> 100 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[1,1]
=> 110 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3]
=> 1000 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> 3 = 2 + 1
[2,1]
=> 1010 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 3 + 1
[1,1,1]
=> 1110 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2 = 1 + 1
[4]
=> 10000 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? ∊ {3,6,6,8} + 1
[3,1]
=> 10010 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? ∊ {3,6,6,8} + 1
[2,2]
=> 1100 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? ∊ {3,6,6,8} + 1
[2,1,1]
=> 10110 => 01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? ∊ {3,6,6,8} + 1
[1,1,1,1]
=> 11110 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 2 = 1 + 1
[5]
=> 100000 => 000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? ∊ {10,15,20,20,24,30} + 1
[4,1]
=> 100010 => 000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {10,15,20,20,24,30} + 1
[3,2]
=> 10100 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? ∊ {10,15,20,20,24,30} + 1
[3,1,1]
=> 100110 => 001110 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? ∊ {10,15,20,20,24,30} + 1
[2,2,1]
=> 11010 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? ∊ {10,15,20,20,24,30} + 1
[2,1,1,1]
=> 101110 => 011110 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? ∊ {10,15,20,20,24,30} + 1
[1,1,1,1,1]
=> 111110 => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> 2 = 1 + 1
[6]
=> 1000000 => 0000010 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[5,1]
=> 1000010 => 0000110 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[4,2]
=> 100100 => 100010 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[4,1,1]
=> 1000110 => 0001110 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[3,3]
=> 11000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[3,2,1]
=> 101010 => 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[3,1,1,1]
=> 1001110 => 0011110 => ([(0,5),(0,6),(1,14),(2,4),(2,9),(2,18),(3,16),(3,20),(4,3),(4,15),(4,19),(5,2),(5,10),(5,17),(6,1),(6,10),(6,17),(8,12),(9,15),(10,9),(11,8),(12,7),(13,7),(14,11),(15,16),(16,13),(17,14),(17,18),(18,11),(18,19),(19,8),(19,20),(20,12),(20,13)],21)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[2,2,2]
=> 11100 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[2,2,1,1]
=> 110110 => 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[2,1,1,1,1]
=> 1011110 => 0111110 => ([(0,5),(0,6),(1,4),(1,8),(1,9),(2,16),(2,17),(3,2),(3,12),(3,13),(4,3),(4,14),(4,15),(5,10),(5,11),(6,1),(6,10),(6,11),(8,15),(9,14),(10,9),(11,8),(12,16),(13,17),(14,12),(15,13),(16,7),(17,7)],18)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[1,1,1,1,1,1]
=> 1111110 => 1111110 => ([(0,2),(0,7),(1,9),(2,8),(3,4),(3,11),(4,6),(4,10),(5,3),(5,13),(6,1),(6,12),(7,5),(7,8),(8,13),(10,12),(11,10),(12,9),(13,11)],14)
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000538
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000538: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000538: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Values
[2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 2 = 1 + 1
[1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 2 = 1 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 3 = 2 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => 4 = 3 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ? ∊ {1,6,6,8} + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => ? ∊ {1,6,6,8} + 1
[2,2]
=> [1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 4 = 3 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [2,5,7,8,1,3,4,6] => ? ∊ {1,6,6,8} + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => ? ∊ {1,6,6,8} + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => ? ∊ {1,10,15,20,20,24,30} + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [2,4,6,9,10,1,3,5,7,8] => ? ∊ {1,10,15,20,20,24,30} + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [2,6,7,8,1,3,4,5] => ? ∊ {1,10,15,20,20,24,30} + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [2,4,7,9,10,1,3,5,6,8] => ? ∊ {1,10,15,20,20,24,30} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [4,5,7,8,1,2,3,6] => ? ∊ {1,10,15,20,20,24,30} + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,5,7,9,10,1,3,4,6,8] => ? ∊ {1,10,15,20,20,24,30} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [3,5,7,9,10,1,2,4,6,8] => ? ∊ {1,10,15,20,20,24,30} + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> [2,4,6,8,11,12,1,3,5,7,9,10] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [2,4,8,9,10,1,3,5,6,7] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> [2,4,6,9,11,12,1,3,5,7,8,10] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [4,6,7,8,1,2,3,5] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> [2,6,7,9,10,1,3,4,5,8] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> [2,4,7,9,11,12,1,3,5,6,8,10] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [4,5,7,9,10,1,2,3,6,8] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> [2,5,7,9,11,12,1,3,4,6,8,10] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> [3,5,7,9,11,12,1,2,4,6,8,10] => ? ∊ {1,15,15,40,40,45,90,90,120,120,144} + 1
Description
The number of even inversions of a permutation.
An inversion $i < j$ of a permutation is even if $i \equiv j~(\operatorname{mod} 2)$. See [[St000539]] for odd inversions.
Matching statistic: St000596
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000596: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000596: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Values
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> {{1,4},{2,3}}
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> 0 = 1 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? ∊ {3,6,6,8} - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? ∊ {3,6,6,8} - 1
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> ? ∊ {3,6,6,8} - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> ? ∊ {3,6,6,8} - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> {{1,2},{3,8},{4,7},{5,6}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> {{1,2},{3,4},{5,6},{7,8},{9,12},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> {{1,2},{3,4},{5,10},{6,9},{7,8}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> {{1,2},{3,4},{5,6},{7,12},{8,9},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> {{1,8},{2,7},{3,4},{5,6}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> {{1,2},{3,4},{5,12},{6,7},{8,9},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> {{1,10},{2,5},{3,4},{6,7},{8,9}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> {{1,2},{3,12},{4,5},{6,7},{8,9},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> {{1,12},{2,3},{4,5},{6,7},{8,9},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
Description
The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal.
Matching statistic: St000604
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000604: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000604: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 21%●distinct values known / distinct values provided: 20%
Values
[2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> {{1,4},{2,3}}
=> 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> 0 = 1 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> 1 = 2 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? ∊ {3,6,6,8} - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? ∊ {3,6,6,8} - 1
[2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> ? ∊ {3,6,6,8} - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> ? ∊ {3,6,6,8} - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> {{1,2},{3,8},{4,7},{5,6}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> ? ∊ {1,10,15,20,20,24,30} - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> {{1,2},{3,4},{5,6},{7,8},{9,12},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> {{1,2},{3,4},{5,10},{6,9},{7,8}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> {{1,2},{3,4},{5,6},{7,12},{8,9},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> {{1,8},{2,7},{3,4},{5,6}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> {{1,2},{3,4},{5,12},{6,7},{8,9},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> {{1,8},{2,7},{3,6},{4,5}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> {{1,10},{2,5},{3,4},{6,7},{8,9}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> {{1,2},{3,12},{4,5},{6,7},{8,9},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> {{1,12},{2,3},{4,5},{6,7},{8,9},{10,11}}
=> ? ∊ {1,15,15,40,40,45,90,90,120,120,144} - 1
Description
The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal.
The following 465 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001565The number of arithmetic progressions of length 2 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001487The number of inner corners of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000731The number of double exceedences of a permutation. St000842The breadth of a permutation. St000862The number of parts of the shifted shape of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000068The number of minimal elements in a poset. St001490The number of connected components of a skew partition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000210Minimum over maximum difference of elements in cycles. St000232The number of crossings of a set partition. St000234The number of global ascents of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000563The number of overlapping pairs of blocks of a set partition. St000570The Edelman-Greene number of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000633The size of the automorphism group of a poset. St000646The number of big ascents of a permutation. St000654The first descent of a permutation. St000663The number of right floats of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000732The number of double deficiencies of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St001114The number of odd descents of a permutation. St001162The minimum jump of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001388The number of non-attacking neighbors of a permutation. St001399The distinguishing number of a poset. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001625The Möbius invariant of a lattice. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001712The number of natural descents of a standard Young tableau. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001928The number of non-overlapping descents in a permutation. St000039The number of crossings of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000222The number of alignments in the permutation. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000251The number of nonsingleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000295The length of the border of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000317The cycle descent number of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000338The number of pixed points of a permutation. St000354The number of recoils of a permutation. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000422The energy of a graph, if it is integral. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000462The major index minus the number of excedences of a permutation. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000619The number of cyclic descents of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000679The pruning number of an ordered tree. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000779The tier of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St000991The number of right-to-left minima of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001151The number of blocks with odd minimum. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001301The first Betti number of the order complex associated with the poset. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St001513The number of nested exceedences of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001569The maximal modular displacement of a permutation. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001810The number of fixed points of a permutation smaller than its largest moved point. St001841The number of inversions of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001911A descent variant minus the number of inversions. St000004The major index of a permutation. St000037The sign of a permutation. St000080The rank of the poset. St000136The dinv of a parking function. St000155The number of exceedances (also excedences) of a permutation. St000166The depth minus 1 of an ordered tree. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000348The non-inversion sum of a binary word. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000461The rix statistic of a permutation. St000490The intertwining number of a set partition. St000493The los statistic of a set partition. St000495The number of inversions of distance at most 2 of a permutation. St000499The rcb statistic of a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000572The dimension exponent of a set partition. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000638The number of up-down runs of a permutation. St000653The last descent of a permutation. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St000717The number of ordinal summands of a poset. St000747A variant of the major index of a set partition. St000748The major index of the permutation obtained by flattening the set partition. St000794The mak of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000823The number of unsplittable factors of the set partition. St000824The sum of the number of descents and the number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000873The aix statistic of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000956The maximal displacement of a permutation. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St001405The number of bonds in a permutation. St001424The number of distinct squares in a binary word. St001472The permanent of the Coxeter matrix of the poset. St001497The position of the largest weak excedence of a permutation. St001517The length of a longest pair of twins in a permutation. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001566The length of the longest arithmetic progression in a permutation. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001667The maximal size of a pair of weak twins for a permutation. St001731The factorization defect of a permutation. St001769The reflection length of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St001894The depth of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000116The major index of a semistandard tableau obtained by standardizing. St000135The number of lucky cars of the parking function. St000498The lcs statistic of a set partition. St000503The maximal difference between two elements in a common block. St000519The largest length of a factor maximising the subword complexity. St000539The number of odd inversions of a permutation. St000677The standardized bi-alternating inversion number of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000863The length of the first row of the shifted shape of a permutation. St000906The length of the shortest maximal chain in a poset. St000922The minimal number such that all substrings of this length are unique. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001375The pancake length of a permutation. St001377The major index minus the number of inversions of a permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001536The number of cyclic misalignments of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001821The sorting index of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001927Sparre Andersen's number of positives of a signed permutation. St000030The sum of the descent differences of a permutations. St000044The number of vertices of the unicellular map given by a perfect matching. St000060The greater neighbor of the maximum. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000304The load of a permutation. St000492The rob statistic of a set partition. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000680The Grundy value for Hackendot on posets. St001077The prefix exchange distance of a permutation. St001160The number of proper blocks (or intervals) of a permutations. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001718The number of non-empty open intervals in a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001848The atomic length of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001958The degree of the polynomial interpolating the values of a permutation. St000017The number of inversions of a standard tableau. St000064The number of one-box pattern of a permutation. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000156The Denert index of a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000501The size of the first part in the decomposition of a permutation. St000540The sum of the entries of a parking function minus its length. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000673The number of non-fixed points of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001267The length of the Lyndon factorization of the binary word. St001519The pinnacle sum of a permutation. St001807The lower middle entry of a permutation. St000728The dimension of a set partition. St000795The mad of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001468The smallest fixpoint of a permutation. St001721The degree of a binary word. St000008The major index of the composition. St000230Sum of the minimal elements of the blocks of a set partition. St000305The inverse major index of a permutation. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000543The size of the conjugacy class of a binary word. St000625The sum of the minimal distances to a greater element. St000626The minimal period of a binary word. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001671Haglund's hag of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000564The number of occurrences of the pattern {{1},{2}} in a set partition. St000797The stat`` of a permutation. St001759The Rajchgot index of a permutation. St000165The sum of the entries of a parking function. St001858The number of covering elements of a signed permutation in absolute order. St001865The number of alignments of a signed permutation. St000391The sum of the positions of the ones in a binary word. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001684The reduced word complexity of a permutation. St000520The number of patterns in a permutation. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St000016The number of attacking pairs of a standard tableau. St000530The number of permutations with the same descent word as the given permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001852The size of the conjugacy class of the signed permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St001885The number of binary words with the same proper border set. St000324The shape of the tree associated to a permutation. St000529The number of permutations whose descent word is the given binary word. St001528The number of permutations such that the product with the permutation has the same number of fixed points.
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