Identifier
Identifier
Values
[1] generating graphics... => -1
[2] generating graphics... => 2
[1,1] generating graphics... => -2
[3] generating graphics... => 3
[2,1] generating graphics... => 0
[1,1,1] generating graphics... => -3
[4] generating graphics... => 4
[3,1] generating graphics... => 0
[2,2] generating graphics... => 2
[2,1,1] generating graphics... => -2
[1,1,1,1] generating graphics... => -4
[5] generating graphics... => 5
[4,1] generating graphics... => 0
[3,2] generating graphics... => 3
[3,1,1] generating graphics... => -1
[2,2,1] generating graphics... => 1
[2,1,1,1] generating graphics... => -3
[1,1,1,1,1] generating graphics... => -5
[6] generating graphics... => 6
[5,1] generating graphics... => 0
[4,2] generating graphics... => 4
[4,1,1] generating graphics... => -1
[3,3] generating graphics... => 3
[3,2,1] generating graphics... => 1
[3,1,1,1] generating graphics... => -3
[2,2,2] generating graphics... => 2
[2,2,1,1] generating graphics... => -2
[2,1,1,1,1] generating graphics... => -4
[1,1,1,1,1,1] generating graphics... => -6
[7] generating graphics... => 7
[6,1] generating graphics... => 0
[5,2] generating graphics... => 5
[5,1,1] generating graphics... => -1
[4,3] generating graphics... => 4
[4,2,1] generating graphics... => 1
[4,1,1,1] generating graphics... => -2
[3,3,1] generating graphics... => 1
[3,2,2] generating graphics... => 3
[3,2,1,1] generating graphics... => -1
[3,1,1,1,1] generating graphics... => -4
[2,2,2,1] generating graphics... => 2
[2,2,1,1,1] generating graphics... => -3
[2,1,1,1,1,1] generating graphics... => -5
[1,1,1,1,1,1,1] generating graphics... => -7
[8] generating graphics... => 8
[7,1] generating graphics... => 0
[6,2] generating graphics... => 6
[6,1,1] generating graphics... => -1
[5,3] generating graphics... => 5
[5,2,1] generating graphics... => 1
[5,1,1,1] generating graphics... => -2
[4,4] generating graphics... => 4
[4,3,1] generating graphics... => 1
[4,2,2] generating graphics... => 4
[4,2,1,1] generating graphics... => -1
[4,1,1,1,1] generating graphics... => -4
[3,3,2] generating graphics... => 3
[3,3,1,1] generating graphics... => 0
[3,2,2,1] generating graphics... => 2
[3,2,1,1,1] generating graphics... => -3
[3,1,1,1,1,1] generating graphics... => -5
[2,2,2,2] generating graphics... => 2
[2,2,2,1,1] generating graphics... => -2
[2,2,1,1,1,1] generating graphics... => -4
[2,1,1,1,1,1,1] generating graphics... => -6
[1,1,1,1,1,1,1,1] generating graphics... => -8
[9] generating graphics... => 9
[8,1] generating graphics... => 0
[7,2] generating graphics... => 7
[7,1,1] generating graphics... => -1
[6,3] generating graphics... => 6
[6,2,1] generating graphics... => 1
[6,1,1,1] generating graphics... => -2
[5,4] generating graphics... => 5
[5,3,1] generating graphics... => 1
[5,2,2] generating graphics... => 5
[5,2,1,1] generating graphics... => -1
[5,1,1,1,1] generating graphics... => -3
[4,4,1] generating graphics... => 1
[4,3,2] generating graphics... => 4
[4,3,1,1] generating graphics... => 0
[4,2,2,1] generating graphics... => 2
[4,2,1,1,1] generating graphics... => -2
[4,1,1,1,1,1] generating graphics... => -5
[3,3,3] generating graphics... => 3
[3,3,2,1] generating graphics... => 2
[3,3,1,1,1] generating graphics... => -3
[3,2,2,2] generating graphics... => 3
[3,2,2,1,1] generating graphics... => -1
[3,2,1,1,1,1] generating graphics... => -4
[3,1,1,1,1,1,1] generating graphics... => -6
[2,2,2,2,1] generating graphics... => 3
[2,2,2,1,1,1] generating graphics... => -3
[2,2,1,1,1,1,1] generating graphics... => -5
[2,1,1,1,1,1,1,1] generating graphics... => -7
[1,1,1,1,1,1,1,1,1] generating graphics... => -9
[10] generating graphics... => 10
[9,1] generating graphics... => 0
[8,2] generating graphics... => 8
[8,1,1] generating graphics... => -1
[7,3] generating graphics... => 7
[7,2,1] generating graphics... => 1
[7,1,1,1] generating graphics... => -2
[6,4] generating graphics... => 6
[6,3,1] generating graphics... => 1
[6,2,2] generating graphics... => 6
[6,2,1,1] generating graphics... => -1
[6,1,1,1,1] generating graphics... => -3
[5,5] generating graphics... => 5
[5,4,1] generating graphics... => 1
[5,3,2] generating graphics... => 5
[5,3,1,1] generating graphics... => 0
[5,2,2,1] generating graphics... => 2
[5,2,1,1,1] generating graphics... => -2
[5,1,1,1,1,1] generating graphics... => -5
[4,4,2] generating graphics... => 4
[4,4,1,1] generating graphics... => 0
[4,3,3] generating graphics... => 4
[4,3,2,1] generating graphics... => 2
[4,3,1,1,1] generating graphics... => -2
[4,2,2,2] generating graphics... => 4
[4,2,2,1,1] generating graphics... => -1
[4,2,1,1,1,1] generating graphics... => -4
[4,1,1,1,1,1,1] generating graphics... => -6
[3,3,3,1] generating graphics... => 2
[3,3,2,2] generating graphics... => 3
[3,3,2,1,1] generating graphics... => 0
[3,3,1,1,1,1] generating graphics... => -4
[3,2,2,2,1] generating graphics... => 3
[3,2,2,1,1,1] generating graphics... => -3
[3,2,1,1,1,1,1] generating graphics... => -5
[3,1,1,1,1,1,1,1] generating graphics... => -7
[2,2,2,2,2] generating graphics... => 2
[2,2,2,2,1,1] generating graphics... => -2
[2,2,2,1,1,1,1] generating graphics... => -4
[2,2,1,1,1,1,1,1] generating graphics... => -6
[2,1,1,1,1,1,1,1,1] generating graphics... => -8
[1,1,1,1,1,1,1,1,1,1] generating graphics... => -10
[11] generating graphics... => 11
[10,1] generating graphics... => 0
[9,2] generating graphics... => 9
[9,1,1] generating graphics... => -1
[8,3] generating graphics... => 8
[8,2,1] generating graphics... => 1
[8,1,1,1] generating graphics... => -2
[7,4] generating graphics... => 7
[7,3,1] generating graphics... => 1
[7,2,2] generating graphics... => 7
[7,2,1,1] generating graphics... => -1
[7,1,1,1,1] generating graphics... => -3
[6,5] generating graphics... => 6
[6,4,1] generating graphics... => 1
[6,3,2] generating graphics... => 6
[6,3,1,1] generating graphics... => 0
[6,2,2,1] generating graphics... => 2
[6,2,1,1,1] generating graphics... => -2
[6,1,1,1,1,1] generating graphics... => -4
[5,5,1] generating graphics... => 1
[5,4,2] generating graphics... => 5
[5,4,1,1] generating graphics... => 0
[5,3,3] generating graphics... => 5
[5,3,2,1] generating graphics... => 2
[5,3,1,1,1] generating graphics... => -2
[5,2,2,2] generating graphics... => 5
[5,2,2,1,1] generating graphics... => -1
[5,2,1,1,1,1] generating graphics... => -3
[5,1,1,1,1,1,1] generating graphics... => -6
[4,4,3] generating graphics... => 4
[4,4,2,1] generating graphics... => 2
[4,4,1,1,1] generating graphics... => -1
[4,3,3,1] generating graphics... => 2
[4,3,2,2] generating graphics... => 4
[4,3,2,1,1] generating graphics... => 0
[4,3,1,1,1,1] generating graphics... => -4
[4,2,2,2,1] generating graphics... => 3
[4,2,2,1,1,1] generating graphics... => -2
[4,2,1,1,1,1,1] generating graphics... => -5
[4,1,1,1,1,1,1,1] generating graphics... => -7
[3,3,3,2] generating graphics... => 3
[3,3,3,1,1] generating graphics... => 1
[3,3,2,2,1] generating graphics... => 3
[3,3,2,1,1,1] generating graphics... => -3
[3,3,1,1,1,1,1] generating graphics... => -5
[3,2,2,2,2] generating graphics... => 3
[3,2,2,2,1,1] generating graphics... => -1
[3,2,2,1,1,1,1] generating graphics... => -4
[3,2,1,1,1,1,1,1] generating graphics... => -6
[3,1,1,1,1,1,1,1,1] generating graphics... => -8
[2,2,2,2,2,1] generating graphics... => 4
[2,2,2,2,1,1,1] generating graphics... => -3
[2,2,2,1,1,1,1,1] generating graphics... => -5
[2,2,1,1,1,1,1,1,1] generating graphics... => -7
[2,1,1,1,1,1,1,1,1,1] generating graphics... => -9
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => -11
[12] generating graphics... => 12
[11,1] generating graphics... => 0
[10,2] generating graphics... => 10
[10,1,1] generating graphics... => -1
[9,3] generating graphics... => 9
[9,2,1] generating graphics... => 1
[9,1,1,1] generating graphics... => -2
[8,4] generating graphics... => 8
[8,3,1] generating graphics... => 1
[8,2,2] generating graphics... => 8
[8,2,1,1] generating graphics... => -1
[8,1,1,1,1] generating graphics... => -3
[7,5] generating graphics... => 7
[7,4,1] generating graphics... => 1
[7,3,2] generating graphics... => 7
[7,3,1,1] generating graphics... => 0
[7,2,2,1] generating graphics... => 2
[7,2,1,1,1] generating graphics... => -2
[7,1,1,1,1,1] generating graphics... => -4
[6,6] generating graphics... => 6
[6,5,1] generating graphics... => 1
[6,4,2] generating graphics... => 6
[6,4,1,1] generating graphics... => 0
[6,3,3] generating graphics... => 6
[6,3,2,1] generating graphics... => 2
[6,3,1,1,1] generating graphics... => -2
[6,2,2,2] generating graphics... => 6
[6,2,2,1,1] generating graphics... => -1
[6,2,1,1,1,1] generating graphics... => -3
[6,1,1,1,1,1,1] generating graphics... => -6
[5,5,2] generating graphics... => 5
[5,5,1,1] generating graphics... => 0
[5,4,3] generating graphics... => 5
[5,4,2,1] generating graphics... => 2
[5,4,1,1,1] generating graphics... => -1
[5,3,3,1] generating graphics... => 2
[5,3,2,2] generating graphics... => 5
[5,3,2,1,1] generating graphics... => 0
[5,3,1,1,1,1] generating graphics... => -3
[5,2,2,2,1] generating graphics... => 3
[5,2,2,1,1,1] generating graphics... => -2
[5,2,1,1,1,1,1] generating graphics... => -5
[5,1,1,1,1,1,1,1] generating graphics... => -7
[4,4,4] generating graphics... => 4
[4,4,3,1] generating graphics... => 2
[4,4,2,2] generating graphics... => 4
[4,4,2,1,1] generating graphics... => 0
[4,4,1,1,1,1] generating graphics... => -4
[4,3,3,2] generating graphics... => 4
[4,3,3,1,1] generating graphics... => 1
[4,3,2,2,1] generating graphics... => 3
[4,3,2,1,1,1] generating graphics... => -2
[4,3,1,1,1,1,1] generating graphics... => -5
[4,2,2,2,2] generating graphics... => 4
[4,2,2,2,1,1] generating graphics... => -1
[4,2,2,1,1,1,1] generating graphics... => -4
[4,2,1,1,1,1,1,1] generating graphics... => -6
[4,1,1,1,1,1,1,1,1] generating graphics... => -8
[3,3,3,3] generating graphics... => 3
[3,3,3,2,1] generating graphics... => 3
[3,3,3,1,1,1] generating graphics... => -3
[3,3,2,2,2] generating graphics... => 3
[3,3,2,2,1,1] generating graphics... => 0
[3,3,2,1,1,1,1] generating graphics... => -4
[3,3,1,1,1,1,1,1] generating graphics... => -6
[3,2,2,2,2,1] generating graphics... => 4
[3,2,2,2,1,1,1] generating graphics... => -3
[3,2,2,1,1,1,1,1] generating graphics... => -5
[3,2,1,1,1,1,1,1,1] generating graphics... => -7
[3,1,1,1,1,1,1,1,1,1] generating graphics... => -9
[2,2,2,2,2,2] generating graphics... => 2
[2,2,2,2,2,1,1] generating graphics... => -2
[2,2,2,2,1,1,1,1] generating graphics... => -4
[2,2,2,1,1,1,1,1,1] generating graphics... => -6
[2,2,1,1,1,1,1,1,1,1] generating graphics... => -8
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => -10
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => -12
click to show generating function       
Description
Dyson's crank of a partition.
Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 (St000475The number of parts equal to 1 in a partition.), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ (St000473The number of parts of a partition that are strictly bigger than the number of ones.). Dyson's crank is then defined as
$$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
References
[1] Andrews, G. E., Garvan, F. G. Dyson's crank of a partition MathSciNet:0929094
Code
def statistic(L):
    ones = list(L).count(1)
    if ones == 0:
        return L[0]
    else:
        return sum(1 for part in L if part > ones) - ones
Created
Apr 19, 2016 at 10:47 by Christian Stump
Updated
Oct 29, 2017 at 21:22 by Martin Rubey