Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St001539
Mapping
St001539: Decorated permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => 0
[-,+] => 1
[+,-] => 1
[-,-] => 0
[2,1] => 0
[+,+,+] => 0
[-,+,+] => 1
[+,-,+] => 2
[+,+,-] => 1
[-,-,+] => 1
[-,+,-] => 2
[+,-,-] => 1
[-,-,-] => 0
[+,3,2] => 0
[-,3,2] => 0
[2,1,+] => 0
[2,1,-] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 0
[3,-,1] => 0
[+,+,+,+] => 0
[-,+,+,+] => 1
[+,-,+,+] => 2
[+,+,-,+] => 2
[+,+,+,-] => 1
[-,-,+,+] => 1
[-,+,-,+] => 3
[-,+,+,-] => 2
[+,-,-,+] => 2
[+,-,+,-] => 3
[+,+,-,-] => 1
[-,-,-,+] => 1
[-,-,+,-] => 2
[-,+,-,-] => 2
[+,-,-,-] => 1
[-,-,-,-] => 0
[+,+,4,3] => 0
[-,+,4,3] => 1
[+,-,4,3] => 1
[-,-,4,3] => 0
[+,3,2,+] => 0
[-,3,2,+] => 1
[+,3,2,-] => 1
[-,3,2,-] => 0
[+,3,4,2] => 0
[-,3,4,2] => 0
[+,4,2,3] => 0
[-,4,2,3] => 0
[+,4,+,2] => 0
Description
Number of changes in decorated fixed points.
Matching statistic: St000478
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 62%●distinct values known / distinct values provided: 40%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 1
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 1
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 1
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 1
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [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 $4$.
Matching statistic: St000512
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 62%●distinct values known / distinct values provided: 60%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 1
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000566
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 62%●distinct values known / distinct values provided: 40%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 1
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 1
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 1
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 1
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000621
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 62%●distinct values known / distinct values provided: 40%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 1
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 1
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 1
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 1
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]].
Matching statistic: St000934
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 62%●distinct values known / distinct values provided: 40%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 1
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 1
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 1
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 1
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 1
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 1
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
Description
The 2-degree of an integer partition. For an integer partition $\lambda$, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
Matching statistic: St000938
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 62%●distinct values known / distinct values provided: 40%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
Description
The number of zeros of the symmetric group character corresponding to the partition. For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000940
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 62%●distinct values known / distinct values provided: 40%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
Description
The number of characters of the symmetric group whose value on the partition is zero. The maximal value for any given size is recorded in [2].
Matching statistic: St000941
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 62%●distinct values known / distinct values provided: 40%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 1
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St001124
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 62%●distinct values known / distinct values provided: 40%
Values
[+,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,+] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[+,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[-,-] => [1,2] => [2]
 => []
 => ? ∊ {0,0,0,1,1}
[2,1] => [2,1] => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1}
[+,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,+] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,+,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,-,-] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,+] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,-] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,1] => [2,3,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[3,+,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[3,-,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => 0
[+,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,+] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,+,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,-,-] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,+,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,-,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,+] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,2,-] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[-,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[+,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[2,1,+,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,-,+] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,+,-] => [2,1,3,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 0
[2,4,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => 0
[2,4,+,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => 0
[3,1,4,2] => [3,1,4,2] => [2,2]
 => [2]
 => 0
[3,+,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => 0
[3,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 0
[3,4,2,1] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => 0
[4,1,+,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => 0
[4,+,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,-,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => 0
[4,+,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,+,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,+,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,-,-,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 0
[4,3,1,2] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 0
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => 0
[+,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[-,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 0
[+,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[-,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 0
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => 0
[+,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[-,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 0
[+,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[-,4,+,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
[+,4,-,2,+] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => 0
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$: $$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$ This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001570The minimal number of edges to add to make a graph Hamiltonian. St000455The second largest eigenvalue of a graph if it is integral. St001498The normalised height of a Nakayama algebra with magnitude 1. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph.