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-----------------------------------------------------------------------------
Statistic identifier: St000146
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
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}$$ in the same way as the Dyson rank ([[St000145]]) explains the congruences $$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]]
-----------------------------------------------------------------------------
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
-----------------------------------------------------------------------------
Statistic values:
[] => 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
-----------------------------------------------------------------------------
Created: Jul 05, 2013 at 14:36 by Olivier Mallet
-----------------------------------------------------------------------------
Last Updated: Nov 29, 2021 at 10:47 by Martin Rubey