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Statistic identifier: St000474

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Collection: Integer partitions

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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 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]).  Dyson's crank is then defined as
$$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda)       & o(\lambda) > 0. \end{cases}$$

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References: [1]   Andrews, G. E., Garvan, F. G. Dyson's crank of a partition [[MathSciNet:0929094]]

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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

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Statistic values:

[1]                       => -1
[2]                       => 2
[1,1]                     => -2
[3]                       => 3
[2,1]                     => 0
[1,1,1]                   => -3
[4]                       => 4
[3,1]                     => 0
[2,2]                     => 2
[2,1,1]                   => -2
[1,1,1,1]                 => -4
[5]                       => 5
[4,1]                     => 0
[3,2]                     => 3
[3,1,1]                   => -1
[2,2,1]                   => 1
[2,1,1,1]                 => -3
[1,1,1,1,1]               => -5
[6]                       => 6
[5,1]                     => 0
[4,2]                     => 4
[4,1,1]                   => -1
[3,3]                     => 3
[3,2,1]                   => 1
[3,1,1,1]                 => -3
[2,2,2]                   => 2
[2,2,1,1]                 => -2
[2,1,1,1,1]               => -4
[1,1,1,1,1,1]             => -6
[7]                       => 7
[6,1]                     => 0
[5,2]                     => 5
[5,1,1]                   => -1
[4,3]                     => 4
[4,2,1]                   => 1
[4,1,1,1]                 => -2
[3,3,1]                   => 1
[3,2,2]                   => 3
[3,2,1,1]                 => -1
[3,1,1,1,1]               => -4
[2,2,2,1]                 => 2
[2,2,1,1,1]               => -3
[2,1,1,1,1,1]             => -5
[1,1,1,1,1,1,1]           => -7
[8]                       => 8
[7,1]                     => 0
[6,2]                     => 6
[6,1,1]                   => -1
[5,3]                     => 5
[5,2,1]                   => 1
[5,1,1,1]                 => -2
[4,4]                     => 4
[4,3,1]                   => 1
[4,2,2]                   => 4
[4,2,1,1]                 => -1
[4,1,1,1,1]               => -4
[3,3,2]                   => 3
[3,3,1,1]                 => 0
[3,2,2,1]                 => 2
[3,2,1,1,1]               => -3
[3,1,1,1,1,1]             => -5
[2,2,2,2]                 => 2
[2,2,2,1,1]               => -2
[2,2,1,1,1,1]             => -4
[2,1,1,1,1,1,1]           => -6
[1,1,1,1,1,1,1,1]         => -8
[9]                       => 9
[8,1]                     => 0
[7,2]                     => 7
[7,1,1]                   => -1
[6,3]                     => 6
[6,2,1]                   => 1
[6,1,1,1]                 => -2
[5,4]                     => 5
[5,3,1]                   => 1
[5,2,2]                   => 5
[5,2,1,1]                 => -1
[5,1,1,1,1]               => -3
[4,4,1]                   => 1
[4,3,2]                   => 4
[4,3,1,1]                 => 0
[4,2,2,1]                 => 2
[4,2,1,1,1]               => -2
[4,1,1,1,1,1]             => -5
[3,3,3]                   => 3
[3,3,2,1]                 => 2
[3,3,1,1,1]               => -3
[3,2,2,2]                 => 3
[3,2,2,1,1]               => -1
[3,2,1,1,1,1]             => -4
[3,1,1,1,1,1,1]           => -6
[2,2,2,2,1]               => 3
[2,2,2,1,1,1]             => -3
[2,2,1,1,1,1,1]           => -5
[2,1,1,1,1,1,1,1]         => -7
[1,1,1,1,1,1,1,1,1]       => -9
[10]                      => 10
[9,1]                     => 0
[8,2]                     => 8
[8,1,1]                   => -1
[7,3]                     => 7
[7,2,1]                   => 1
[7,1,1,1]                 => -2
[6,4]                     => 6
[6,3,1]                   => 1
[6,2,2]                   => 6
[6,2,1,1]                 => -1
[6,1,1,1,1]               => -3
[5,5]                     => 5
[5,4,1]                   => 1
[5,3,2]                   => 5
[5,3,1,1]                 => 0
[5,2,2,1]                 => 2
[5,2,1,1,1]               => -2
[5,1,1,1,1,1]             => -5
[4,4,2]                   => 4
[4,4,1,1]                 => 0
[4,3,3]                   => 4
[4,3,2,1]                 => 2
[4,3,1,1,1]               => -2
[4,2,2,2]                 => 4
[4,2,2,1,1]               => -1
[4,2,1,1,1,1]             => -4
[4,1,1,1,1,1,1]           => -6
[3,3,3,1]                 => 2
[3,3,2,2]                 => 3
[3,3,2,1,1]               => 0
[3,3,1,1,1,1]             => -4
[3,2,2,2,1]               => 3
[3,2,2,1,1,1]             => -3
[3,2,1,1,1,1,1]           => -5
[3,1,1,1,1,1,1,1]         => -7
[2,2,2,2,2]               => 2
[2,2,2,2,1,1]             => -2
[2,2,2,1,1,1,1]           => -4
[2,2,1,1,1,1,1,1]         => -6
[2,1,1,1,1,1,1,1,1]       => -8
[1,1,1,1,1,1,1,1,1,1]     => -10
[11]                      => 11
[10,1]                    => 0
[9,2]                     => 9
[9,1,1]                   => -1
[8,3]                     => 8
[8,2,1]                   => 1
[8,1,1,1]                 => -2
[7,4]                     => 7
[7,3,1]                   => 1
[7,2,2]                   => 7
[7,2,1,1]                 => -1
[7,1,1,1,1]               => -3
[6,5]                     => 6
[6,4,1]                   => 1
[6,3,2]                   => 6
[6,3,1,1]                 => 0
[6,2,2,1]                 => 2
[6,2,1,1,1]               => -2
[6,1,1,1,1,1]             => -4
[5,5,1]                   => 1
[5,4,2]                   => 5
[5,4,1,1]                 => 0
[5,3,3]                   => 5
[5,3,2,1]                 => 2
[5,3,1,1,1]               => -2
[5,2,2,2]                 => 5
[5,2,2,1,1]               => -1
[5,2,1,1,1,1]             => -3
[5,1,1,1,1,1,1]           => -6
[4,4,3]                   => 4
[4,4,2,1]                 => 2
[4,4,1,1,1]               => -1
[4,3,3,1]                 => 2
[4,3,2,2]                 => 4
[4,3,2,1,1]               => 0
[4,3,1,1,1,1]             => -4
[4,2,2,2,1]               => 3
[4,2,2,1,1,1]             => -2
[4,2,1,1,1,1,1]           => -5
[4,1,1,1,1,1,1,1]         => -7
[3,3,3,2]                 => 3
[3,3,3,1,1]               => 1
[3,3,2,2,1]               => 3
[3,3,2,1,1,1]             => -3
[3,3,1,1,1,1,1]           => -5
[3,2,2,2,2]               => 3
[3,2,2,2,1,1]             => -1
[3,2,2,1,1,1,1]           => -4
[3,2,1,1,1,1,1,1]         => -6
[3,1,1,1,1,1,1,1,1]       => -8
[2,2,2,2,2,1]             => 4
[2,2,2,2,1,1,1]           => -3
[2,2,2,1,1,1,1,1]         => -5
[2,2,1,1,1,1,1,1,1]       => -7
[2,1,1,1,1,1,1,1,1,1]     => -9
[1,1,1,1,1,1,1,1,1,1,1]   => -11
[12]                      => 12
[11,1]                    => 0
[10,2]                    => 10
[10,1,1]                  => -1
[9,3]                     => 9
[9,2,1]                   => 1
[9,1,1,1]                 => -2
[8,4]                     => 8
[8,3,1]                   => 1
[8,2,2]                   => 8
[8,2,1,1]                 => -1
[8,1,1,1,1]               => -3
[7,5]                     => 7
[7,4,1]                   => 1
[7,3,2]                   => 7
[7,3,1,1]                 => 0
[7,2,2,1]                 => 2
[7,2,1,1,1]               => -2
[7,1,1,1,1,1]             => -4
[6,6]                     => 6
[6,5,1]                   => 1
[6,4,2]                   => 6
[6,4,1,1]                 => 0
[6,3,3]                   => 6
[6,3,2,1]                 => 2
[6,3,1,1,1]               => -2
[6,2,2,2]                 => 6
[6,2,2,1,1]               => -1
[6,2,1,1,1,1]             => -3
[6,1,1,1,1,1,1]           => -6
[5,5,2]                   => 5
[5,5,1,1]                 => 0
[5,4,3]                   => 5
[5,4,2,1]                 => 2
[5,4,1,1,1]               => -1
[5,3,3,1]                 => 2
[5,3,2,2]                 => 5
[5,3,2,1,1]               => 0
[5,3,1,1,1,1]             => -3
[5,2,2,2,1]               => 3
[5,2,2,1,1,1]             => -2
[5,2,1,1,1,1,1]           => -5
[5,1,1,1,1,1,1,1]         => -7
[4,4,4]                   => 4
[4,4,3,1]                 => 2
[4,4,2,2]                 => 4
[4,4,2,1,1]               => 0
[4,4,1,1,1,1]             => -4
[4,3,3,2]                 => 4
[4,3,3,1,1]               => 1
[4,3,2,2,1]               => 3
[4,3,2,1,1,1]             => -2
[4,3,1,1,1,1,1]           => -5
[4,2,2,2,2]               => 4
[4,2,2,2,1,1]             => -1
[4,2,2,1,1,1,1]           => -4
[4,2,1,1,1,1,1,1]         => -6
[4,1,1,1,1,1,1,1,1]       => -8
[3,3,3,3]                 => 3
[3,3,3,2,1]               => 3
[3,3,3,1,1,1]             => -3
[3,3,2,2,2]               => 3
[3,3,2,2,1,1]             => 0
[3,3,2,1,1,1,1]           => -4
[3,3,1,1,1,1,1,1]         => -6
[3,2,2,2,2,1]             => 4
[3,2,2,2,1,1,1]           => -3
[3,2,2,1,1,1,1,1]         => -5
[3,2,1,1,1,1,1,1,1]       => -7
[3,1,1,1,1,1,1,1,1,1]     => -9
[2,2,2,2,2,2]             => 2
[2,2,2,2,2,1,1]           => -2
[2,2,2,2,1,1,1,1]         => -4
[2,2,2,1,1,1,1,1,1]       => -6
[2,2,1,1,1,1,1,1,1,1]     => -8
[2,1,1,1,1,1,1,1,1,1,1]   => -10
[1,1,1,1,1,1,1,1,1,1,1,1] => -12

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Created: Apr 19, 2016 at 10:47 by Christian Stump

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Last Updated: Oct 29, 2017 at 21:22 by Martin Rubey