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Statistic identifier: St000182
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Collection: Integer partitions
-----------------------------------------------------------------------------
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]].
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References: [1] Section 1.3 p24 Kerber, A. Algebraic combinatorics via finite group actions [[MathSciNet:1115208]]
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Code:
def statistic(p):
return p.conjugacy_class_size()
-----------------------------------------------------------------------------
Statistic values:
[] => 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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Created: May 03, 2014 at 21:11 by Lahiru Kariyawasam
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Last Updated: May 25, 2023 at 14:21 by Martin Rubey