Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000478
Mapping
St000478: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 1
[1,1]
 => 0
[3]
 => 2
[2,1]
 => 0
[1,1,1]
 => 0
[4]
 => 2
[3,1]
 => 1
[2,2]
 => -1
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 3
[4,1]
 => 1
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
[6]
 => 3
[5,1]
 => 2
[4,2]
 => 0
[4,1,1]
 => 0
[3,3]
 => 0
[3,2,1]
 => 0
[3,1,1,1]
 => 0
[2,2,2]
 => 1
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 4
[6,1]
 => 2
[5,2]
 => 1
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
[8]
 => 4
[7,1]
 => 3
[6,2]
 => 1
[6,1,1]
 => 0
[5,3]
 => 2
[5,2,1]
 => 0
[5,1,1,1]
 => 0
Description
Another weight of a partition according to Alladi. According to Theorem 3.4 (Alladi 2012) in [1] $$ \sum_{\pi\in GG_1(r)} w_1(\pi) $$ equals the number of partitions of $r$ whose odd parts are all distinct. $GG_1(r)$ is the set of partitions of $r$ where consecutive entries differ by at least $2$, and consecutive even entries differ by at least $4$.