Your data matches 1 statistic following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000474
Mapping
St000474: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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
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 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as $$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$