Identifier
Identifier
Values
[] generating graphics... => 1
[1] generating graphics... => 1
[2] generating graphics... => 1
[1,1] generating graphics... => 1
[3] generating graphics... => 2
[2,1] generating graphics... => 3
[1,1,1] generating graphics... => 1
[4] generating graphics... => 6
[3,1] generating graphics... => 8
[2,2] generating graphics... => 3
[2,1,1] generating graphics... => 6
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 24
[4,1] generating graphics... => 30
[3,2] generating graphics... => 20
[3,1,1] generating graphics... => 20
[2,2,1] generating graphics... => 15
[2,1,1,1] generating graphics... => 10
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 120
[5,1] generating graphics... => 144
[4,2] generating graphics... => 90
[4,1,1] generating graphics... => 90
[3,3] generating graphics... => 40
[3,2,1] generating graphics... => 120
[3,1,1,1] generating graphics... => 40
[2,2,2] generating graphics... => 15
[2,2,1,1] generating graphics... => 45
[2,1,1,1,1] generating graphics... => 15
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 720
[6,1] generating graphics... => 840
[5,2] generating graphics... => 504
[5,1,1] generating graphics... => 504
[4,3] generating graphics... => 420
[4,2,1] generating graphics... => 630
[4,1,1,1] generating graphics... => 210
[3,3,1] generating graphics... => 280
[3,2,2] generating graphics... => 210
[3,2,1,1] generating graphics... => 420
[3,1,1,1,1] generating graphics... => 70
[2,2,2,1] generating graphics... => 105
[2,2,1,1,1] generating graphics... => 105
[2,1,1,1,1,1] generating graphics... => 21
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 5040
[7,1] generating graphics... => 5760
[6,2] generating graphics... => 3360
[6,1,1] generating graphics... => 3360
[5,3] generating graphics... => 2688
[5,2,1] generating graphics... => 4032
[5,1,1,1] generating graphics... => 1344
[4,4] generating graphics... => 1260
[4,3,1] generating graphics... => 3360
[4,2,2] generating graphics... => 1260
[4,2,1,1] generating graphics... => 2520
[4,1,1,1,1] generating graphics... => 420
[3,3,2] generating graphics... => 1120
[3,3,1,1] generating graphics... => 1120
[3,2,2,1] generating graphics... => 1680
[3,2,1,1,1] generating graphics... => 1120
[3,1,1,1,1,1] generating graphics... => 112
[2,2,2,2] generating graphics... => 105
[2,2,2,1,1] generating graphics... => 420
[2,2,1,1,1,1] generating graphics... => 210
[2,1,1,1,1,1,1] generating graphics... => 28
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 40320
[8,1] generating graphics... => 45360
[7,2] generating graphics... => 25920
[7,1,1] generating graphics... => 25920
[6,3] generating graphics... => 20160
[6,2,1] generating graphics... => 30240
[6,1,1,1] generating graphics... => 10080
[5,4] generating graphics... => 18144
[5,3,1] generating graphics... => 24192
[5,2,2] generating graphics... => 9072
[5,2,1,1] generating graphics... => 18144
[5,1,1,1,1] generating graphics... => 3024
[4,4,1] generating graphics... => 11340
[4,3,2] generating graphics... => 15120
[4,3,1,1] generating graphics... => 15120
[4,2,2,1] generating graphics... => 11340
[4,2,1,1,1] generating graphics... => 7560
[4,1,1,1,1,1] generating graphics... => 756
[3,3,3] generating graphics... => 2240
[3,3,2,1] generating graphics... => 10080
[3,3,1,1,1] generating graphics... => 3360
[3,2,2,2] generating graphics... => 2520
[3,2,2,1,1] generating graphics... => 7560
[3,2,1,1,1,1] generating graphics... => 2520
[3,1,1,1,1,1,1] generating graphics... => 168
[2,2,2,2,1] generating graphics... => 945
[2,2,2,1,1,1] generating graphics... => 1260
[2,2,1,1,1,1,1] generating graphics... => 378
[2,1,1,1,1,1,1,1] generating graphics... => 36
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 362880
[9,1] generating graphics... => 403200
[8,2] generating graphics... => 226800
[8,1,1] generating graphics... => 226800
[7,3] generating graphics... => 172800
[7,2,1] generating graphics... => 259200
[7,1,1,1] generating graphics... => 86400
[6,4] generating graphics... => 151200
[6,3,1] generating graphics... => 201600
[6,2,2] generating graphics... => 75600
[6,2,1,1] generating graphics... => 151200
[6,1,1,1,1] generating graphics... => 25200
[5,5] generating graphics... => 72576
[5,4,1] generating graphics... => 181440
[5,3,2] generating graphics... => 120960
[5,3,1,1] generating graphics... => 120960
[5,2,2,1] generating graphics... => 90720
[5,2,1,1,1] generating graphics... => 60480
[5,1,1,1,1,1] generating graphics... => 6048
[4,4,2] generating graphics... => 56700
[4,4,1,1] generating graphics... => 56700
[4,3,3] generating graphics... => 50400
[4,3,2,1] generating graphics... => 151200
[4,3,1,1,1] generating graphics... => 50400
[4,2,2,2] generating graphics... => 18900
[4,2,2,1,1] generating graphics... => 56700
[4,2,1,1,1,1] generating graphics... => 18900
[4,1,1,1,1,1,1] generating graphics... => 1260
[3,3,3,1] generating graphics... => 22400
[3,3,2,2] generating graphics... => 25200
[3,3,2,1,1] generating graphics... => 50400
[3,3,1,1,1,1] generating graphics... => 8400
[3,2,2,2,1] generating graphics... => 25200
[3,2,2,1,1,1] generating graphics... => 25200
[3,2,1,1,1,1,1] generating graphics... => 5040
[3,1,1,1,1,1,1,1] generating graphics... => 240
[2,2,2,2,2] generating graphics... => 945
[2,2,2,2,1,1] generating graphics... => 4725
[2,2,2,1,1,1,1] generating graphics... => 3150
[2,2,1,1,1,1,1,1] generating graphics... => 630
[2,1,1,1,1,1,1,1,1] generating graphics... => 45
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
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
Oct 29, 2017 at 20:43 by Martin Rubey