searching the database
Your data matches 119 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001252
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2]
=> 1
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[3,2,1] => [2,1]
=> 1
[1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 1
[2,1,4,3] => [2,2]
=> 2
[2,3,4,1] => [4]
=> 2
[2,4,1,3] => [4]
=> 2
[3,1,4,2] => [4]
=> 2
[3,2,1,4] => [2,1,1]
=> 1
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [4]
=> 2
[4,1,2,3] => [4]
=> 2
[4,2,3,1] => [2,1,1]
=> 1
[4,3,1,2] => [4]
=> 2
[4,3,2,1] => [2,2]
=> 2
[1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> 2
[1,3,4,5,2] => [4,1]
=> 2
[1,3,5,2,4] => [4,1]
=> 2
[1,4,2,5,3] => [4,1]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> 2
[1,4,5,3,2] => [4,1]
=> 2
[1,5,2,3,4] => [4,1]
=> 2
[1,5,3,4,2] => [2,1,1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> 2
[1,5,4,3,2] => [2,2,1]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> 2
[2,1,4,3,5] => [2,2,1]
=> 2
[2,1,4,5,3] => [3,2]
=> 1
[2,1,5,3,4] => [3,2]
=> 1
[2,1,5,4,3] => [2,2,1]
=> 2
[2,3,1,5,4] => [3,2]
=> 1
[2,3,4,1,5] => [4,1]
=> 2
[2,3,5,4,1] => [4,1]
=> 2
[2,4,1,3,5] => [4,1]
=> 2
[2,4,3,5,1] => [4,1]
=> 2
[2,4,5,1,3] => [3,2]
=> 1
[2,5,1,4,3] => [4,1]
=> 2
[2,5,3,1,4] => [4,1]
=> 2
[2,5,4,3,1] => [3,2]
=> 1
[3,1,2,5,4] => [3,2]
=> 1
Description
Half the sum of the even parts of a partition.
Matching statistic: St000510
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 2
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 2
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 2
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 2
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 2
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 2
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 2
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 2
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000698
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 2
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 2
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 2
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 2
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 2
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 2
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 2
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 2
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer k, one associates a k-core to a partition by repeatedly removing all rim hooks of size k.
This statistic counts the 2-rim hooks that are removed in this process to obtain a 2-core.
Matching statistic: St000934
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 2
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 2
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 2
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 2
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 2
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 2
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 2
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 2
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 1
Description
The 2-degree of an integer partition.
For an integer partition λ, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by λ.
Matching statistic: St001601
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001601: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001601: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 2
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 2
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 2
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 2
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 2
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 2
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 2
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 2
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees.
Matching statistic: St001899
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 2
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 2
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 2
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 2
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 2
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 2
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 2
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 2
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 1
Description
The total number of irreducible representations contained in the higher Lie character for an integer partition.
Matching statistic: St001900
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001900: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001900: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 2
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 2
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 2
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 2
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 2
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 2
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 2
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 2
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 2
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 2
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 2
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 2
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 2
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 2
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 2
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 2
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 2
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 1
Description
The number of distinct irreducible representations contained in the higher Lie character for an integer partition.
Matching statistic: St000288
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2]
=> 100 => 101 => 2 = 1 + 1
[1,3,2] => [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[2,1,3] => [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[3,2,1] => [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[1,2,4,3] => [2,1,1]
=> 10110 => 10001 => 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> 10110 => 10001 => 2 = 1 + 1
[1,4,3,2] => [2,1,1]
=> 10110 => 10001 => 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> 10110 => 10001 => 2 = 1 + 1
[2,1,4,3] => [2,2]
=> 1100 => 1011 => 3 = 2 + 1
[2,3,4,1] => [4]
=> 10000 => 10101 => 3 = 2 + 1
[2,4,1,3] => [4]
=> 10000 => 10101 => 3 = 2 + 1
[3,1,4,2] => [4]
=> 10000 => 10101 => 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> 10110 => 10001 => 2 = 1 + 1
[3,4,1,2] => [2,2]
=> 1100 => 1011 => 3 = 2 + 1
[3,4,2,1] => [4]
=> 10000 => 10101 => 3 = 2 + 1
[4,1,2,3] => [4]
=> 10000 => 10101 => 3 = 2 + 1
[4,2,3,1] => [2,1,1]
=> 10110 => 10001 => 2 = 1 + 1
[4,3,1,2] => [4]
=> 10000 => 10101 => 3 = 2 + 1
[4,3,2,1] => [2,2]
=> 1100 => 1011 => 3 = 2 + 1
[1,2,3,5,4] => [2,1,1,1]
=> 101110 => 100001 => 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> 101110 => 100001 => 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> 101110 => 100001 => 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> 101110 => 100001 => 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> 11010 => 10011 => 3 = 2 + 1
[1,3,4,5,2] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[1,3,5,2,4] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[1,4,2,5,3] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[1,4,3,2,5] => [2,1,1,1]
=> 101110 => 100001 => 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> 11010 => 10011 => 3 = 2 + 1
[1,4,5,3,2] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[1,5,2,3,4] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[1,5,3,4,2] => [2,1,1,1]
=> 101110 => 100001 => 2 = 1 + 1
[1,5,4,2,3] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[1,5,4,3,2] => [2,2,1]
=> 11010 => 10011 => 3 = 2 + 1
[2,1,3,4,5] => [2,1,1,1]
=> 101110 => 100001 => 2 = 1 + 1
[2,1,3,5,4] => [2,2,1]
=> 11010 => 10011 => 3 = 2 + 1
[2,1,4,3,5] => [2,2,1]
=> 11010 => 10011 => 3 = 2 + 1
[2,1,4,5,3] => [3,2]
=> 10100 => 01001 => 2 = 1 + 1
[2,1,5,3,4] => [3,2]
=> 10100 => 01001 => 2 = 1 + 1
[2,1,5,4,3] => [2,2,1]
=> 11010 => 10011 => 3 = 2 + 1
[2,3,1,5,4] => [3,2]
=> 10100 => 01001 => 2 = 1 + 1
[2,3,4,1,5] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[2,3,5,4,1] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[2,4,1,3,5] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[2,4,3,5,1] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[2,4,5,1,3] => [3,2]
=> 10100 => 01001 => 2 = 1 + 1
[2,5,1,4,3] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[2,5,3,1,4] => [4,1]
=> 100010 => 100101 => 3 = 2 + 1
[2,5,4,3,1] => [3,2]
=> 10100 => 01001 => 2 = 1 + 1
[3,1,2,5,4] => [3,2]
=> 10100 => 01001 => 2 = 1 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000697
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000697: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000697: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 0 = 1 - 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 0 = 1 - 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 1 = 2 - 1
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 1 = 2 - 1
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 1 = 2 - 1
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 1 = 2 - 1
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 0 = 1 - 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 1 = 2 - 1
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 0 = 1 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 1 = 2 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 1 = 2 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 1 = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 1 = 2 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 0 = 1 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 0 = 1 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 1 = 2 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 0 = 1 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 1 = 2 - 1
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 1 = 2 - 1
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 1 = 2 - 1
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 1 = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 0 = 1 - 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 1 = 2 - 1
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 1 = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 0 = 1 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 0 = 1 - 1
Description
The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core.
For any positive integer k, one associates a k-core to a partition by repeatedly removing all rim hooks of size k.
This statistic counts the 3-rim hooks that are removed in this process to obtain a 3-core.
Matching statistic: St000941
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [-2,-1] => [2]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [-1,-3,-2] => [2]
=> 0 = 1 - 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => [2]
=> 0 = 1 - 1
[3,2,1] => [3,2,1] => [-3,-2,-1] => [2]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [2]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [2]
=> 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [2]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [2]
=> 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => [2,2]
=> 1 = 2 - 1
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
=> 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
=> 1 = 2 - 1
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [2]
=> 0 = 1 - 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
=> 1 = 2 - 1
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => [4]
=> 1 = 2 - 1
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
=> 1 = 2 - 1
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [2]
=> 0 = 1 - 1
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => [4]
=> 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => [2,2]
=> 1 = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [2]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [2]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [2]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [2]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [2,2]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
=> 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [2]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
=> 1 = 2 - 1
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => [4]
=> 1 = 2 - 1
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
=> 1 = 2 - 1
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [2]
=> 0 = 1 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => [4]
=> 1 = 2 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [2,2]
=> 1 = 2 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [2]
=> 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [2,2]
=> 1 = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [2,2]
=> 1 = 2 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
=> 0 = 1 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
=> 0 = 1 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [2,2]
=> 1 = 2 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
=> 0 = 1 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
=> 1 = 2 - 1
[2,3,5,4,1] => [2,3,5,4,1] => [-2,-3,-5,-4,-1] => [4]
=> 1 = 2 - 1
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
=> 1 = 2 - 1
[2,4,3,5,1] => [2,4,3,5,1] => [-2,-4,-3,-5,-1] => [4]
=> 1 = 2 - 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
=> 0 = 1 - 1
[2,5,1,4,3] => [2,5,1,4,3] => [-2,-5,-1,-4,-3] => [4]
=> 1 = 2 - 1
[2,5,3,1,4] => [2,5,3,1,4] => [-2,-5,-3,-1,-4] => [4]
=> 1 = 2 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [2]
=> 0 = 1 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
=> 0 = 1 - 1
Description
The number of characters of the symmetric group whose value on the partition is even.
The following 109 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000944The 3-degree of an integer partition. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000937The number of positive values of the symmetric group character corresponding to the partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000928The sum of the coefficients of the character polynomial of an integer partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001118The acyclic chromatic index of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000080The rank of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001779The order of promotion on the set of linear extensions of a poset. St000528The height of a poset. St000632The jump number of the poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000906The length of the shortest maximal chain in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001597The Frobenius rank of a skew partition. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001782The order of rowmotion on the set of order ideals of a poset. St000264The girth of a graph, which is not a tree. St001877Number of indecomposable injective modules with projective dimension 2. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001568The smallest positive integer that does not appear twice in the partition. St001845The number of join irreducibles minus the rank of a lattice. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!