- FindStat

Identifier

St000146: Integer partitions ⟶ ℤ

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

>>> Load all 380 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

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 $$p(7n+5) \equiv 0 \pmod{7}.$$

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

Created

Jul 05, 2013 at 14:36 by Olivier Mallet

Updated

Nov 29, 2021 at 10:47 by Martin Rubey