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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000182
Mp00108: Permutations —cycle type⟶ Integer partitions
St000182: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000182: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,2] => [1,1]
=> 1
[2,1] => [2]
=> 1
[1,2,3] => [1,1,1]
=> 1
[1,3,2] => [2,1]
=> 3
[2,1,3] => [2,1]
=> 3
[2,3,1] => [3]
=> 2
[3,1,2] => [3]
=> 2
[3,2,1] => [2,1]
=> 3
[1,2,3,4] => [1,1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> 6
[1,3,2,4] => [2,1,1]
=> 6
[1,3,4,2] => [3,1]
=> 8
[1,4,2,3] => [3,1]
=> 8
[1,4,3,2] => [2,1,1]
=> 6
[2,1,3,4] => [2,1,1]
=> 6
[2,1,4,3] => [2,2]
=> 3
[2,3,1,4] => [3,1]
=> 8
[2,3,4,1] => [4]
=> 6
[2,4,1,3] => [4]
=> 6
[2,4,3,1] => [3,1]
=> 8
[3,1,2,4] => [3,1]
=> 8
[3,1,4,2] => [4]
=> 6
[3,2,1,4] => [2,1,1]
=> 6
[3,2,4,1] => [3,1]
=> 8
[3,4,1,2] => [2,2]
=> 3
[3,4,2,1] => [4]
=> 6
[4,1,2,3] => [4]
=> 6
[4,1,3,2] => [3,1]
=> 8
[4,2,1,3] => [3,1]
=> 8
[4,2,3,1] => [2,1,1]
=> 6
[4,3,1,2] => [4]
=> 6
[4,3,2,1] => [2,2]
=> 3
[1,2,3,4,5] => [1,1,1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> 10
[1,2,4,3,5] => [2,1,1,1]
=> 10
[1,2,4,5,3] => [3,1,1]
=> 20
[1,2,5,3,4] => [3,1,1]
=> 20
[1,2,5,4,3] => [2,1,1,1]
=> 10
[1,3,2,4,5] => [2,1,1,1]
=> 10
[1,3,2,5,4] => [2,2,1]
=> 15
[1,3,4,2,5] => [3,1,1]
=> 20
[1,3,4,5,2] => [4,1]
=> 30
[1,3,5,2,4] => [4,1]
=> 30
[1,3,5,4,2] => [3,1,1]
=> 20
[1,4,2,3,5] => [3,1,1]
=> 20
[1,4,2,5,3] => [4,1]
=> 30
[1,4,3,2,5] => [2,1,1,1]
=> 10
[1,4,3,5,2] => [3,1,1]
=> 20
[1,4,5,2,3] => [2,2,1]
=> 15
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 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000690: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 96%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000690: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 96%
Values
[1] => [1] => [.,.]
=> [1] => ? = 1
[1,2] => [1,2] => [.,[.,.]]
=> [2,1] => 1
[2,1] => [2,1] => [[.,.],.]
=> [1,2] => 1
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 2
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 3
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 3
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 3
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 1
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 6
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 6
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 6
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 6
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 6
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 6
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 8
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 8
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 3
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 8
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 3
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 8
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 8
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 8
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 6
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 6
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 3
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 6
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 8
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 8
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 6
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 6
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 6
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 24
[1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,4,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,5,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,6,4,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,6,4,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,6,7,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,7,4,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,7,4,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,7,5,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,7,5,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,7,6,4,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,3,7,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,5,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,6,3,5,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,6,3,7,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,6,7,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,7,3,5,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,7,3,6,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,7,5,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,7,5,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,7,6,3,5] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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[1,2,5,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
=> [2,3,4,5,6,7,1] => ? ∊ 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0,840,840,840,840,840,840,840,840,840,840,840,840,840,840,840,840,840,840,840}
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]].
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