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Identifier
Values
=>
Cc0002;cc-rep
[]=>1 [1]=>1 [2]=>2 [1,1]=>2 [3]=>3 [2,1]=>2 [1,1,1]=>6 [4]=>4 [3,1]=>3 [2,2]=>8 [2,1,1]=>4 [1,1,1,1]=>24 [5]=>5 [4,1]=>4 [3,2]=>6 [3,1,1]=>6 [2,2,1]=>8 [2,1,1,1]=>12 [1,1,1,1,1]=>120 [6]=>6 [5,1]=>5 [4,2]=>8 [4,1,1]=>8 [3,3]=>18 [3,2,1]=>6 [3,1,1,1]=>18 [2,2,2]=>48 [2,2,1,1]=>16 [2,1,1,1,1]=>48 [1,1,1,1,1,1]=>720 [7]=>7 [6,1]=>6 [5,2]=>10 [5,1,1]=>10 [4,3]=>12 [4,2,1]=>8 [4,1,1,1]=>24 [3,3,1]=>18 [3,2,2]=>24 [3,2,1,1]=>12 [3,1,1,1,1]=>72 [2,2,2,1]=>48 [2,2,1,1,1]=>48 [2,1,1,1,1,1]=>240 [1,1,1,1,1,1,1]=>5040 [8]=>8 [7,1]=>7 [6,2]=>12 [6,1,1]=>12 [5,3]=>15 [5,2,1]=>10 [5,1,1,1]=>30 [4,4]=>32 [4,3,1]=>12 [4,2,2]=>32 [4,2,1,1]=>16 [4,1,1,1,1]=>96 [3,3,2]=>36 [3,3,1,1]=>36 [3,2,2,1]=>24 [3,2,1,1,1]=>36 [3,1,1,1,1,1]=>360 [2,2,2,2]=>384 [2,2,2,1,1]=>96 [2,2,1,1,1,1]=>192 [2,1,1,1,1,1,1]=>1440 [1,1,1,1,1,1,1,1]=>40320 [9]=>9 [8,1]=>8 [7,2]=>14 [7,1,1]=>14 [6,3]=>18 [6,2,1]=>12 [6,1,1,1]=>36 [5,4]=>20 [5,3,1]=>15 [5,2,2]=>40 [5,2,1,1]=>20 [5,1,1,1,1]=>120 [4,4,1]=>32 [4,3,2]=>24 [4,3,1,1]=>24 [4,2,2,1]=>32 [4,2,1,1,1]=>48 [4,1,1,1,1,1]=>480 [3,3,3]=>162 [3,3,2,1]=>36 [3,3,1,1,1]=>108 [3,2,2,2]=>144 [3,2,2,1,1]=>48 [3,2,1,1,1,1]=>144 [3,1,1,1,1,1,1]=>2160 [2,2,2,2,1]=>384 [2,2,2,1,1,1]=>288 [2,2,1,1,1,1,1]=>960 [2,1,1,1,1,1,1,1]=>10080 [1,1,1,1,1,1,1,1,1]=>362880 [10]=>10 [9,1]=>9 [8,2]=>16 [8,1,1]=>16 [7,3]=>21 [7,2,1]=>14 [7,1,1,1]=>42 [6,4]=>24 [6,3,1]=>18 [6,2,2]=>48 [6,2,1,1]=>24 [6,1,1,1,1]=>144 [5,5]=>50 [5,4,1]=>20 [5,3,2]=>30 [5,3,1,1]=>30 [5,2,2,1]=>40 [5,2,1,1,1]=>60 [5,1,1,1,1,1]=>600 [4,4,2]=>64 [4,4,1,1]=>64 [4,3,3]=>72 [4,3,2,1]=>24 [4,3,1,1,1]=>72 [4,2,2,2]=>192 [4,2,2,1,1]=>64 [4,2,1,1,1,1]=>192 [4,1,1,1,1,1,1]=>2880 [3,3,3,1]=>162 [3,3,2,2]=>144 [3,3,2,1,1]=>72 [3,3,1,1,1,1]=>432 [3,2,2,2,1]=>144 [3,2,2,1,1,1]=>144 [3,2,1,1,1,1,1]=>720 [3,1,1,1,1,1,1,1]=>15120 [2,2,2,2,2]=>3840 [2,2,2,2,1,1]=>768 [2,2,2,1,1,1,1]=>1152 [2,2,1,1,1,1,1,1]=>5760 [2,1,1,1,1,1,1,1,1]=>80640 [1,1,1,1,1,1,1,1,1,1]=>3628800
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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 $1$.
Code
def statistic(p):
    return p.centralizer_size()
Created
May 04, 2014 at 23:41 by Lahiru Kariyawasam
Updated
Oct 29, 2017 at 16:33 by Martin Rubey