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Identifier
Values
=>
Cc0002;cc-rep
[]=>1 [1]=>1 [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 [7]=>720 [6,1]=>840 [5,2]=>504 [5,1,1]=>504 [4,3]=>420 [4,2,1]=>630 [4,1,1,1]=>210 [3,3,1]=>280 [3,2,2]=>210 [3,2,1,1]=>420 [3,1,1,1,1]=>70 [2,2,2,1]=>105 [2,2,1,1,1]=>105 [2,1,1,1,1,1]=>21 [1,1,1,1,1,1,1]=>1 [8]=>5040 [7,1]=>5760 [6,2]=>3360 [6,1,1]=>3360 [5,3]=>2688 [5,2,1]=>4032 [5,1,1,1]=>1344 [4,4]=>1260 [4,3,1]=>3360 [4,2,2]=>1260 [4,2,1,1]=>2520 [4,1,1,1,1]=>420 [3,3,2]=>1120 [3,3,1,1]=>1120 [3,2,2,1]=>1680 [3,2,1,1,1]=>1120 [3,1,1,1,1,1]=>112 [2,2,2,2]=>105 [2,2,2,1,1]=>420 [2,2,1,1,1,1]=>210 [2,1,1,1,1,1,1]=>28 [1,1,1,1,1,1,1,1]=>1 [9]=>40320 [8,1]=>45360 [7,2]=>25920 [7,1,1]=>25920 [6,3]=>20160 [6,2,1]=>30240 [6,1,1,1]=>10080 [5,4]=>18144 [5,3,1]=>24192 [5,2,2]=>9072 [5,2,1,1]=>18144 [5,1,1,1,1]=>3024 [4,4,1]=>11340 [4,3,2]=>15120 [4,3,1,1]=>15120 [4,2,2,1]=>11340 [4,2,1,1,1]=>7560 [4,1,1,1,1,1]=>756 [3,3,3]=>2240 [3,3,2,1]=>10080 [3,3,1,1,1]=>3360 [3,2,2,2]=>2520 [3,2,2,1,1]=>7560 [3,2,1,1,1,1]=>2520 [3,1,1,1,1,1,1]=>168 [2,2,2,2,1]=>945 [2,2,2,1,1,1]=>1260 [2,2,1,1,1,1,1]=>378 [2,1,1,1,1,1,1,1]=>36 [1,1,1,1,1,1,1,1,1]=>1 [10]=>362880 [9,1]=>403200 [8,2]=>226800 [8,1,1]=>226800 [7,3]=>172800 [7,2,1]=>259200 [7,1,1,1]=>86400 [6,4]=>151200 [6,3,1]=>201600 [6,2,2]=>75600 [6,2,1,1]=>151200 [6,1,1,1,1]=>25200 [5,5]=>72576 [5,4,1]=>181440 [5,3,2]=>120960 [5,3,1,1]=>120960 [5,2,2,1]=>90720 [5,2,1,1,1]=>60480 [5,1,1,1,1,1]=>6048 [4,4,2]=>56700 [4,4,1,1]=>56700 [4,3,3]=>50400 [4,3,2,1]=>151200 [4,3,1,1,1]=>50400 [4,2,2,2]=>18900 [4,2,2,1,1]=>56700 [4,2,1,1,1,1]=>18900 [4,1,1,1,1,1,1]=>1260 [3,3,3,1]=>22400 [3,3,2,2]=>25200 [3,3,2,1,1]=>50400 [3,3,1,1,1,1]=>8400 [3,2,2,2,1]=>25200 [3,2,2,1,1,1]=>25200 [3,2,1,1,1,1,1]=>5040 [3,1,1,1,1,1,1,1]=>240 [2,2,2,2,2]=>945 [2,2,2,2,1,1]=>4725 [2,2,2,1,1,1,1]=>3150 [2,2,1,1,1,1,1,1]=>630 [2,1,1,1,1,1,1,1,1]=>45 [1,1,1,1,1,1,1,1,1,1]=>1 [11]=>3628800 [10,1]=>3991680 [9,1,1]=>2217600 [8,3]=>1663200 [7,4]=>1425600 [6,5]=>1330560 [6,4,1]=>1663200 [6,1,1,1,1,1]=>55440 [5,5,1]=>798336 [5,4,2]=>997920 [5,4,1,1]=>997920 [5,3,3]=>443520 [5,3,2,1]=>1330560 [5,3,1,1,1]=>443520 [5,2,2,2]=>166320 [5,2,2,1,1]=>498960 [5,2,1,1,1,1]=>166320 [4,4,3]=>415800 [4,4,2,1]=>623700 [4,4,1,1,1]=>207900 [4,3,3,1]=>554400 [4,3,2,2]=>415800 [4,3,2,1,1]=>831600 [4,2,2,2,1]=>207900 [3,3,3,2]=>123200 [3,3,3,1,1]=>123200 [3,3,2,2,1]=>277200 [3,2,2,2,2]=>34650 [2,2,2,2,2,1]=>10395 [2,1,1,1,1,1,1,1,1,1]=>55 [1,1,1,1,1,1,1,1,1,1,1]=>1 [12]=>39916800 [11,1]=>43545600 [10,1,1]=>23950080 [9,3]=>17740800 [7,5]=>13685760 [7,4,1]=>17107200 [6,6]=>6652800 [6,4,2]=>9979200 [5,5,2]=>4790016 [5,4,3]=>7983360 [5,4,2,1]=>11975040 [5,4,1,1,1]=>3991680 [5,3,3,1]=>5322240 [5,3,2,2]=>3991680 [5,3,2,1,1]=>7983360 [5,2,2,2,1]=>1995840 [5,2,2,1,1,1]=>1995840 [4,4,4]=>1247400 [4,4,3,1]=>4989600 [4,4,2,2]=>1871100 [4,4,2,1,1]=>3742200 [4,3,3,2]=>3326400 [4,3,3,1,1]=>3326400 [4,3,2,2,1]=>4989600 [3,3,3,3]=>246400 [3,3,3,2,1]=>1478400 [3,3,2,2,2]=>554400 [3,3,2,2,1,1]=>1663200 [3,2,2,2,2,1]=>415800 [2,2,2,2,2,2]=>10395 [1,1,1,1,1,1,1,1,1,1,1,1]=>1 [13]=>479001600 [12,1]=>518918400 [10,3]=>207567360 [8,5]=>155675520 [7,6]=>148262400 [7,5,1]=>177914880 [7,4,2]=>111196800 [6,6,1]=>86486400 [6,4,2,1]=>129729600 [5,5,3]=>41513472 [5,4,4]=>38918880 [5,4,3,1]=>103783680 [5,4,2,2]=>38918880 [5,4,2,1,1]=>77837760 [5,4,1,1,1,1]=>12972960 [5,3,3,2]=>34594560 [5,3,3,1,1]=>34594560 [5,3,2,2,1]=>51891840 [5,3,2,1,1,1]=>34594560 [4,4,4,1]=>16216200 [4,4,3,2]=>32432400 [4,4,3,1,1]=>32432400 [4,4,2,2,1]=>24324300 [4,3,3,3]=>9609600 [4,3,3,2,1]=>43243200 [3,3,3,3,1]=>3203200 [3,3,3,2,2]=>4804800 [3,3,2,2,2,1]=>7207200 [3,2,2,2,2,2]=>540540 [2,2,2,2,2,2,1]=>135135 [1,1,1,1,1,1,1,1,1,1,1,1,1]=>1 [9,5]=>1937295360 [7,7]=>889574400 [7,5,2]=>1245404160 [7,4,3]=>1037836800 [6,6,2]=>605404800 [6,4,4]=>454053600 [6,2,2,2,2]=>37837800 [5,5,4]=>435891456 [5,5,1,1,1,1]=>72648576 [5,4,3,2]=>726485760 [5,4,3,1,1]=>726485760 [5,4,2,2,1]=>544864320 [5,4,2,1,1,1]=>363242880 [5,3,3,3]=>107627520 [5,3,3,2,1]=>484323840 [5,2,2,2,2,1]=>45405360 [4,4,4,2]=>113513400 [4,4,3,3]=>151351200 [4,4,3,2,1]=>454053600 [4,3,2,2,2,1]=>151351200 [3,3,3,3,2]=>22422400 [3,3,3,3,1,1]=>22422400 [3,3,2,2,2,2]=>12612600 [2,2,2,2,2,2,2]=>135135 [1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1 [6,5,1,1,1,1]=>1816214400 [6,3,3,3]=>1345344000 [6,2,2,2,2,1]=>567567000 [5,5,5]=>1743565824 [5,3,2,2,2,1]=>1816214400 [4,4,4,3]=>1135134000 [4,4,4,1,1,1]=>567567000 [4,3,3,3,2]=>1009008000 [3,3,3,3,3]=>44844800 [3,3,3,3,2,1]=>336336000 [3,3,3,2,2,2]=>168168000 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1 [3,3,3,3,2,2]=>1345344000 [2,2,2,2,2,2,2,2]=>2027025 [2,2,2,2,2,2,2,2,2]=>34459425
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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.
References
[1] Section 1.3 p24 Kerber, A. Algebraic combinatorics via finite group actions MathSciNet:1115208
Code
def statistic(p):
    return p.conjugacy_class_size()

def statistic_alternative(la):
    la = list(la)
    return factorial(sum(la))/prod(la)/prod(factorial(la.count(j)) for j in [1..la[0]+1])
Created
May 03, 2014 at 21:11 by Lahiru Kariyawasam
Updated
May 25, 2023 at 14:21 by Martin Rubey