Identifier
Values
=>
Cc0002;cc-rep
[]=>0
[1]=>-1
[2]=>1
[1,1]=>-2
[3]=>1
[2,1]=>0
[1,1,1]=>-3
[4]=>1
[3,1]=>0
[2,2]=>2
[2,1,1]=>-2
[1,1,1,1]=>-4
[5]=>1
[4,1]=>0
[3,2]=>2
[3,1,1]=>-1
[2,2,1]=>1
[2,1,1,1]=>-3
[1,1,1,1,1]=>-5
[6]=>1
[5,1]=>0
[4,2]=>2
[4,1,1]=>-1
[3,3]=>2
[3,2,1]=>1
[3,1,1,1]=>-3
[2,2,2]=>3
[2,2,1,1]=>-2
[2,1,1,1,1]=>-4
[1,1,1,1,1,1]=>-6
[7]=>1
[6,1]=>0
[5,2]=>2
[5,1,1]=>-1
[4,3]=>2
[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]=>1
[7,1]=>0
[6,2]=>2
[6,1,1]=>-1
[5,3]=>2
[5,2,1]=>1
[5,1,1,1]=>-2
[4,4]=>2
[4,3,1]=>1
[4,2,2]=>3
[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]=>4
[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]=>1
[8,1]=>0
[7,2]=>2
[7,1,1]=>-1
[6,3]=>2
[6,2,1]=>1
[6,1,1,1]=>-2
[5,4]=>2
[5,3,1]=>1
[5,2,2]=>3
[5,2,1,1]=>-1
[5,1,1,1,1]=>-3
[4,4,1]=>1
[4,3,2]=>3
[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]=>4
[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]=>1
[9,1]=>0
[8,2]=>2
[8,1,1]=>-1
[7,3]=>2
[7,2,1]=>1
[7,1,1,1]=>-2
[6,4]=>2
[6,3,1]=>1
[6,2,2]=>3
[6,2,1,1]=>-1
[6,1,1,1,1]=>-3
[5,5]=>2
[5,4,1]=>1
[5,3,2]=>3
[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]=>3
[4,4,1,1]=>0
[4,3,3]=>3
[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]=>4
[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]=>5
[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
[8,3]=>2
[7,4]=>2
[6,5]=>2
[6,4,1]=>1
[6,1,1,1,1,1]=>-4
[5,5,1]=>1
[5,4,2]=>3
[5,4,1,1]=>0
[5,3,3]=>3
[5,3,2,1]=>2
[5,3,1,1,1]=>-2
[5,2,2,2]=>4
[5,2,2,1,1]=>-1
[5,2,1,1,1,1]=>-3
[4,4,3]=>3
[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
[4,2,2,2,1]=>3
[3,3,3,2]=>4
[3,3,3,1,1]=>1
[3,3,2,2,1]=>3
[3,2,2,2,2]=>5
[2,2,2,2,2,1]=>4
[7,5]=>2
[7,4,1]=>1
[6,6]=>2
[6,4,2]=>3
[5,5,2]=>3
[5,4,3]=>3
[5,4,2,1]=>2
[5,4,1,1,1]=>-1
[5,3,3,1]=>2
[5,3,2,2]=>4
[5,3,2,1,1]=>0
[5,2,2,2,1]=>3
[5,2,2,1,1,1]=>-2
[4,4,4]=>3
[4,4,3,1]=>2
[4,4,2,2]=>4
[4,4,2,1,1]=>0
[4,3,3,2]=>4
[4,3,3,1,1]=>1
[4,3,2,2,1]=>3
[3,3,3,3]=>4
[3,3,3,2,1]=>3
[3,3,2,2,2]=>5
[3,3,2,2,1,1]=>0
[3,2,2,2,2,1]=>4
[2,2,2,2,2,2]=>6
[8,5]=>2
[7,5,1]=>1
[7,4,2]=>3
[5,5,3]=>3
[5,4,4]=>3
[5,4,3,1]=>2
[5,4,2,2]=>4
[5,4,2,1,1]=>0
[5,4,1,1,1,1]=>-3
[5,3,3,2]=>4
[5,3,3,1,1]=>1
[5,3,2,2,1]=>3
[5,3,2,1,1,1]=>-2
[4,4,4,1]=>2
[4,4,3,2]=>4
[4,4,3,1,1]=>1
[4,4,2,2,1]=>3
[4,3,3,3]=>4
[4,3,3,2,1]=>3
[3,3,3,3,1]=>3
[3,3,3,2,2]=>5
[3,3,2,2,2,1]=>4
[9,5]=>2
[8,5,1]=>1
[7,5,2]=>3
[7,4,3]=>3
[6,4,4]=>3
[6,2,2,2,2]=>5
[5,5,4]=>3
[5,5,1,1,1,1]=>-2
[5,4,3,2]=>4
[5,4,3,1,1]=>1
[5,4,2,2,1]=>3
[5,4,2,1,1,1]=>-1
[5,3,3,2,1]=>3
[5,3,2,2,2]=>5
[5,2,2,2,2,1]=>4
[4,4,4,2]=>4
[4,4,3,3]=>4
[4,4,3,2,1]=>3
[4,3,2,2,2,1]=>4
[3,3,3,3,2]=>5
[3,3,3,3,1,1]=>2
[9,5,1]=>1
[8,5,2]=>3
[7,5,3]=>3
[6,5,4]=>3
[6,5,1,1,1,1]=>-2
[6,3,3,3]=>4
[6,2,2,2,2,1]=>4
[5,5,5]=>3
[5,4,3,2,1]=>3
[5,4,3,1,1,1]=>-1
[5,3,2,2,2,1]=>4
[4,4,4,3]=>4
[4,4,4,1,1,1]=>0
[3,3,3,3,3]=>5
[3,3,3,3,2,1]=>4
[8,5,3]=>3
[7,5,3,1]=>2
[5,5,3,3]=>4
[5,5,2,2,2]=>5
[5,4,3,2,1,1]=>1
[5,4,2,2,2,1]=>4
[4,4,4,4]=>4
[4,4,4,2,2]=>5
[4,3,3,3,2,1]=>4
[8,6,3]=>3
[6,5,3,3]=>4
[6,5,2,2,2]=>5
[6,4,4,3]=>4
[6,4,4,1,1,1]=>0
[6,3,3,3,2]=>5
[6,3,3,3,1,1]=>2
[5,5,4,3]=>4
[5,5,4,1,1,1]=>0
[5,5,2,2,2,1]=>4
[5,4,3,2,2,1]=>4
[5,3,3,3,2,1]=>4
[4,4,4,3,2]=>5
[4,4,4,3,1,1]=>2
[4,4,4,2,2,1]=>4
[4,4,4,3,2,1]=>4
[5,4,3,3,2,1]=>4
[6,3,3,3,2,1]=>4
[6,5,2,2,2,1]=>4
[5,5,3,3,1,1]=>2
[6,5,4,1,1,1]=>0
[5,5,3,3,2]=>5
[5,5,4,2,2]=>5
[6,4,4,2,2]=>5
[6,5,4,3]=>4
[9,6,3]=>3
[8,6,4]=>3
[5,4,4,3,2,1]=>4
[5,5,3,3,2,1]=>4
[5,5,4,2,2,1]=>4
[6,4,4,2,2,1]=>4
[5,5,4,3,1,1]=>2
[6,4,4,3,1,1]=>2
[6,5,3,3,1,1]=>2
[5,5,4,3,2]=>5
[6,4,4,3,2]=>5
[6,5,3,3,2]=>5
[6,5,4,2,2]=>5
[6,5,4,3,1]=>3
[6,5,4,1,1,1,1]=>-2
[9,6,4]=>3
[8,5,4,2]=>4
[8,5,5,1]=>2
[5,5,4,3,2,1]=>4
[6,4,4,3,2,1]=>4
[6,5,3,3,2,1]=>4
[6,5,4,2,2,1]=>4
[6,5,4,3,1,1]=>2
[6,5,4,3,2]=>5
[6,5,2,2,2,2,1]=>5
[6,5,4,2,1,1,1]=>0
[7,5,4,3,1]=>3
[8,6,4,2]=>4
[10,6,4]=>3
[10,7,3]=>3
[9,7,4]=>3
[9,5,5,1]=>2
[6,5,4,3,2,1]=>4
[6,3,3,3,3,2,1]=>5
[6,5,3,2,2,2,1]=>5
[6,5,4,3,1,1,1]=>0
[11,7,3]=>3
[4,4,4,4,3,2,1]=>5
[6,4,3,3,3,2,1]=>5
[6,5,4,2,2,2,1]=>5
[6,5,4,3,2,1,1]=>2
[9,6,4,3]=>4
[5,4,4,4,3,2,1]=>5
[6,5,3,3,3,2,1]=>5
[6,5,4,3,2,2,1]=>5
[9,6,5,3]=>4
[8,6,5,3,1]=>3
[6,4,4,4,3,2,1]=>5
[6,5,4,3,3,2,1]=>5
[11,7,5,1]=>2
[9,7,5,3]=>4
[5,5,5,4,3,2,1]=>5
[6,5,4,4,3,2,1]=>5
[9,7,5,3,1]=>3
[10,7,5,3]=>4
[6,5,5,4,3,2,1]=>5
[9,7,5,4,1]=>3
[6,6,5,4,3,2,1]=>5
[7,6,5,4,3,2]=>6
[7,6,5,4,3,2,1]=>5
[7,6,5,4,3,1,1,1]=>1
[10,7,6,4,1]=>3
[9,7,6,4,2]=>5
[10,8,5,4,1]=>3
[7,6,5,4,3,2,1,1]=>3
[7,6,5,4,2,2,2,1]=>6
[10,8,6,4,1]=>3
[9,7,5,5,3,1]=>4
[7,6,5,4,3,2,2,1]=>6
[7,6,5,3,3,3,2,1]=>6
[11,8,6,4,1]=>3
[10,8,6,4,2]=>5
[7,6,5,4,3,3,2,1]=>6
[7,6,4,4,4,3,2,1]=>6
[11,8,6,5,1]=>3
[7,6,5,4,4,3,2,1]=>6
[7,5,5,5,4,3,2,1]=>6
[7,6,5,5,4,3,2,1]=>6
[6,6,6,5,4,3,2,1]=>6
[7,6,6,5,4,3,2,1]=>6
[12,9,7,5,1]=>3
[7,7,6,5,4,3,2,1]=>6
[13,9,7,5,1]=>3
[11,9,7,5,3,1]=>4
[11,8,7,5,4,1]=>4
[8,7,6,5,4,3,2,1]=>6
[8,7,6,5,4,3,2,1,1]=>4
[8,7,6,5,4,3,2,2,1]=>7
[8,7,6,5,4,3,3,2,1]=>7
[8,7,6,5,4,4,3,2,1]=>7
[11,9,7,5,5,3]=>6
[8,7,6,5,5,4,3,2,1]=>7
[8,7,6,6,5,4,3,2,1]=>7
[8,7,7,6,5,4,3,2,1]=>7
[8,8,7,6,5,4,3,2,1]=>7
[9,8,7,6,5,4,3,2,1]=>7
[11,9,7,7,5,3,3]=>7
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Description
The Andrews-Garvan crank of a partition.
If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by
$$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$
This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank (St000145The Dyson rank of a partition.) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$
If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by
$$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$
This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank (St000145The Dyson rank of a partition.) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$
References
[1] Andrews, G. E., Garvan, F. G. Dyson's crank of a partition MathSciNet:0929094
[2] wikipedia:Ramanujan's congruences
[2] wikipedia:Ramanujan's congruences
Code
def statistic(p):
nb_ones = p.to_list().count(1)
if nb_ones == 0:
return len(p)
else:
return len([i for i in p if i > nb_ones]) - nb_ones
Created
Jul 05, 2013 at 14:36 by Olivier Mallet
Updated
Nov 29, 2021 at 10:47 by Martin Rubey
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