Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000146
Mapping
St000146: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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
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}.$$