- FindStat

Identifier

St000474: Integer partitions ⟶ ℤ

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