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Identifier
Values
=>
Cc0002;cc-rep
[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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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