Identifier
Values
([2],3) => [2] => 2
([1,1],3) => [1,1] => 2
([3,1],3) => [2,1] => 2
([2,1,1],3) => [1,1,1] => 6
([4,2],3) => [2,2] => 8
([3,1,1],3) => [2,1,1] => 4
([2,2,1,1],3) => [1,1,1,1] => 24
([5,3,1],3) => [2,2,1] => 8
([4,2,1,1],3) => [2,1,1,1] => 12
([3,2,2,1,1],3) => [1,1,1,1,1] => 120
([6,4,2],3) => [2,2,2] => 48
([5,3,1,1],3) => [2,2,1,1] => 16
([4,2,2,1,1],3) => [2,1,1,1,1] => 48
([3,3,2,2,1,1],3) => [1,1,1,1,1,1] => 720
([2],4) => [2] => 2
([1,1],4) => [1,1] => 2
([3],4) => [3] => 3
([2,1],4) => [2,1] => 2
([1,1,1],4) => [1,1,1] => 6
([4,1],4) => [3,1] => 3
([2,2],4) => [2,2] => 8
([3,1,1],4) => [2,1,1] => 4
([2,1,1,1],4) => [1,1,1,1] => 24
([5,2],4) => [3,2] => 6
([4,1,1],4) => [3,1,1] => 6
([3,2,1],4) => [2,2,1] => 8
([3,1,1,1],4) => [2,1,1,1] => 12
([2,2,1,1,1],4) => [1,1,1,1,1] => 120
([6,3],4) => [3,3] => 18
([5,2,1],4) => [3,2,1] => 6
([4,1,1,1],4) => [3,1,1,1] => 18
([4,2,2],4) => [2,2,2] => 48
([3,3,1,1],4) => [2,2,1,1] => 16
([3,2,1,1,1],4) => [2,1,1,1,1] => 48
([2,2,2,1,1,1],4) => [1,1,1,1,1,1] => 720
([2],5) => [2] => 2
([1,1],5) => [1,1] => 2
([3],5) => [3] => 3
([2,1],5) => [2,1] => 2
([1,1,1],5) => [1,1,1] => 6
([4],5) => [4] => 4
([3,1],5) => [3,1] => 3
([2,2],5) => [2,2] => 8
([2,1,1],5) => [2,1,1] => 4
([1,1,1,1],5) => [1,1,1,1] => 24
([5,1],5) => [4,1] => 4
([3,2],5) => [3,2] => 6
([4,1,1],5) => [3,1,1] => 6
([2,2,1],5) => [2,2,1] => 8
([3,1,1,1],5) => [2,1,1,1] => 12
([2,1,1,1,1],5) => [1,1,1,1,1] => 120
([6,2],5) => [4,2] => 8
([5,1,1],5) => [4,1,1] => 8
([3,3],5) => [3,3] => 18
([4,2,1],5) => [3,2,1] => 6
([4,1,1,1],5) => [3,1,1,1] => 18
([2,2,2],5) => [2,2,2] => 48
([3,2,1,1],5) => [2,2,1,1] => 16
([3,1,1,1,1],5) => [2,1,1,1,1] => 48
([2,2,1,1,1,1],5) => [1,1,1,1,1,1] => 720
([2],6) => [2] => 2
([1,1],6) => [1,1] => 2
([3],6) => [3] => 3
([2,1],6) => [2,1] => 2
([1,1,1],6) => [1,1,1] => 6
([4],6) => [4] => 4
([3,1],6) => [3,1] => 3
([2,2],6) => [2,2] => 8
([2,1,1],6) => [2,1,1] => 4
([1,1,1,1],6) => [1,1,1,1] => 24
([5],6) => [5] => 5
([4,1],6) => [4,1] => 4
([3,2],6) => [3,2] => 6
([3,1,1],6) => [3,1,1] => 6
([2,2,1],6) => [2,2,1] => 8
([2,1,1,1],6) => [2,1,1,1] => 12
([1,1,1,1,1],6) => [1,1,1,1,1] => 120
([6,1],6) => [5,1] => 5
([4,2],6) => [4,2] => 8
([5,1,1],6) => [4,1,1] => 8
([3,3],6) => [3,3] => 18
([3,2,1],6) => [3,2,1] => 6
([4,1,1,1],6) => [3,1,1,1] => 18
([2,2,2],6) => [2,2,2] => 48
([2,2,1,1],6) => [2,2,1,1] => 16
([3,1,1,1,1],6) => [2,1,1,1,1] => 48
([2,1,1,1,1,1],6) => [1,1,1,1,1,1] => 720
([7,2],6) => [5,2] => 10
([6,1,1],6) => [5,1,1] => 10
([4,3],6) => [4,3] => 12
([5,2,1],6) => [4,2,1] => 8
([5,1,1,1],6) => [4,1,1,1] => 24
([3,3,1],6) => [3,3,1] => 18
([3,2,2],6) => [3,2,2] => 24
([4,2,1,1],6) => [3,2,1,1] => 12
([4,1,1,1,1],6) => [3,1,1,1,1] => 72
([2,2,2,1],6) => [2,2,2,1] => 48
([3,2,1,1,1],6) => [2,2,1,1,1] => 48
([3,1,1,1,1,1],6) => [2,1,1,1,1,1] => 240
([2,2,1,1,1,1,1],6) => [1,1,1,1,1,1,1] => 5040
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Description
The size of the centralizer of any permutation of given cycle type.
The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$:
$$C_g = \{h \in G : hgh^{-1} = g\}.$$
Its size thus depends only on the conjugacy class of $g$.
The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is
$$|C| = \Pi j^{a_j} a_j!$$
For example, for any permutation with cycle type $\lambda = (3,2,2,1)$,
$$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$
There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.
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].