Identifier
Values
([2],3) => [2] => 1
([1,1],3) => [1,1] => 1
([3,1],3) => [2,1] => 3
([2,1,1],3) => [1,1,1] => 1
([4,2],3) => [2,2] => 3
([3,1,1],3) => [2,1,1] => 6
([2,2,1,1],3) => [1,1,1,1] => 1
([5,3,1],3) => [2,2,1] => 15
([4,2,1,1],3) => [2,1,1,1] => 10
([3,2,2,1,1],3) => [1,1,1,1,1] => 1
([6,4,2],3) => [2,2,2] => 15
([5,3,1,1],3) => [2,2,1,1] => 45
([4,2,2,1,1],3) => [2,1,1,1,1] => 15
([3,3,2,2,1,1],3) => [1,1,1,1,1,1] => 1
([2],4) => [2] => 1
([1,1],4) => [1,1] => 1
([3],4) => [3] => 2
([2,1],4) => [2,1] => 3
([1,1,1],4) => [1,1,1] => 1
([4,1],4) => [3,1] => 8
([2,2],4) => [2,2] => 3
([3,1,1],4) => [2,1,1] => 6
([2,1,1,1],4) => [1,1,1,1] => 1
([5,2],4) => [3,2] => 20
([4,1,1],4) => [3,1,1] => 20
([3,2,1],4) => [2,2,1] => 15
([3,1,1,1],4) => [2,1,1,1] => 10
([2,2,1,1,1],4) => [1,1,1,1,1] => 1
([6,3],4) => [3,3] => 40
([5,2,1],4) => [3,2,1] => 120
([4,1,1,1],4) => [3,1,1,1] => 40
([4,2,2],4) => [2,2,2] => 15
([3,3,1,1],4) => [2,2,1,1] => 45
([3,2,1,1,1],4) => [2,1,1,1,1] => 15
([2,2,2,1,1,1],4) => [1,1,1,1,1,1] => 1
([2],5) => [2] => 1
([1,1],5) => [1,1] => 1
([3],5) => [3] => 2
([2,1],5) => [2,1] => 3
([1,1,1],5) => [1,1,1] => 1
([4],5) => [4] => 6
([3,1],5) => [3,1] => 8
([2,2],5) => [2,2] => 3
([2,1,1],5) => [2,1,1] => 6
([1,1,1,1],5) => [1,1,1,1] => 1
([5,1],5) => [4,1] => 30
([3,2],5) => [3,2] => 20
([4,1,1],5) => [3,1,1] => 20
([2,2,1],5) => [2,2,1] => 15
([3,1,1,1],5) => [2,1,1,1] => 10
([2,1,1,1,1],5) => [1,1,1,1,1] => 1
([6,2],5) => [4,2] => 90
([5,1,1],5) => [4,1,1] => 90
([3,3],5) => [3,3] => 40
([4,2,1],5) => [3,2,1] => 120
([4,1,1,1],5) => [3,1,1,1] => 40
([2,2,2],5) => [2,2,2] => 15
([3,2,1,1],5) => [2,2,1,1] => 45
([3,1,1,1,1],5) => [2,1,1,1,1] => 15
([2,2,1,1,1,1],5) => [1,1,1,1,1,1] => 1
([2],6) => [2] => 1
([1,1],6) => [1,1] => 1
([3],6) => [3] => 2
([2,1],6) => [2,1] => 3
([1,1,1],6) => [1,1,1] => 1
([4],6) => [4] => 6
([3,1],6) => [3,1] => 8
([2,2],6) => [2,2] => 3
([2,1,1],6) => [2,1,1] => 6
([1,1,1,1],6) => [1,1,1,1] => 1
([5],6) => [5] => 24
([4,1],6) => [4,1] => 30
([3,2],6) => [3,2] => 20
([3,1,1],6) => [3,1,1] => 20
([2,2,1],6) => [2,2,1] => 15
([2,1,1,1],6) => [2,1,1,1] => 10
([1,1,1,1,1],6) => [1,1,1,1,1] => 1
([6,1],6) => [5,1] => 144
([4,2],6) => [4,2] => 90
([5,1,1],6) => [4,1,1] => 90
([3,3],6) => [3,3] => 40
([3,2,1],6) => [3,2,1] => 120
([4,1,1,1],6) => [3,1,1,1] => 40
([2,2,2],6) => [2,2,2] => 15
([2,2,1,1],6) => [2,2,1,1] => 45
([3,1,1,1,1],6) => [2,1,1,1,1] => 15
([2,1,1,1,1,1],6) => [1,1,1,1,1,1] => 1
([7,2],6) => [5,2] => 504
([6,1,1],6) => [5,1,1] => 504
([4,3],6) => [4,3] => 420
([5,2,1],6) => [4,2,1] => 630
([5,1,1,1],6) => [4,1,1,1] => 210
([3,3,1],6) => [3,3,1] => 280
([3,2,2],6) => [3,2,2] => 210
([4,2,1,1],6) => [3,2,1,1] => 420
([4,1,1,1,1],6) => [3,1,1,1,1] => 70
([2,2,2,1],6) => [2,2,2,1] => 105
([3,2,1,1,1],6) => [2,2,1,1,1] => 105
([3,1,1,1,1,1],6) => [2,1,1,1,1,1] => 21
([2,2,1,1,1,1,1],6) => [1,1,1,1,1,1,1] => 1
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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.
Map
to bounded partition
Description
The (k-1)-bounded partition of a k-core.
Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].