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Matching statistic: St001770
St001770: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[-1] => 1
[1,2] => 1
[1,-2] => 2
[-1,2] => 1
[-1,-2] => 1
[2,1] => 2
[2,-1] => 1
[-2,1] => 3
[-2,-1] => 1
[1,2,3] => 1
[1,2,-3] => 3
[1,-2,3] => 4
[1,-2,-3] => 6
[-1,2,3] => 1
[-1,2,-3] => 2
[-1,-2,3] => 1
[-1,-2,-3] => 1
[1,3,2] => 3
[1,3,-2] => 3
[1,-3,2] => 6
[1,-3,-2] => 4
[-1,3,2] => 2
[-1,3,-2] => 1
[-1,-3,2] => 3
[-1,-3,-2] => 1
[2,1,3] => 3
[2,1,-3] => 9
[2,-1,3] => 1
[2,-1,-3] => 2
[-2,1,3] => 5
[-2,1,-3] => 8
[-2,-1,3] => 1
[-2,-1,-3] => 1
[2,3,1] => 3
[2,3,-1] => 1
[2,-3,1] => 8
[2,-3,-1] => 2
[-2,3,1] => 8
[-2,3,-1] => 2
[-2,-3,1] => 10
[-2,-3,-1] => 2
[3,1,2] => 6
[3,1,-2] => 8
[3,-1,2] => 2
[3,-1,-2] => 1
[-3,1,2] => 10
[-3,1,-2] => 10
[-3,-1,2] => 3
[-3,-1,-2] => 1
Description
The number of facets of a certain subword complex associated with the signed permutation.
Let $Q=[1,\dots,n,1,\dots,n,\dots,1,\dots,n]$ be the word of length $n^2$, and let $\pi$ be a signed permutation. Then this statistic yields the number of facets of the subword complex $\Delta(Q, \pi)$.
Matching statistic: St001603
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 14%●distinct values known / distinct values provided: 4%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 14%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1] => [1]
=> ? ∊ {1,1}
[-1] => [-1] => [-1] => []
=> ? ∊ {1,1}
[1,2] => [1,2] => [1,2] => [1,1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1,-2] => [1,-2] => [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [-1,-2] => [-1,-2] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => [-1,-2] => [-1,-2] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2,1] => [-2,1] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => [-1,2] => [-1,2] => [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => [-2,-1] => [2,-1] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [-1,-2] => [-1,-2] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,2,-3] => [1,2,-3] => [1,2,-3] => [1,1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,-2,-3] => [1,-2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1,-2,-3] => [1,-2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [1,3,2] => [1,-3,2] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1,-2,3] => [1,-2,3] => [1,1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1,-3,-2] => [1,3,-2] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [1,-2,-3] => [1,-2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1,3] => [-2,1,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2,1,-3] => [-2,1,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [-1,2,-3] => [-1,2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => [-1,2,-3] => [-1,2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [-2,-1,3] => [2,-1,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => [-2,-1,-3] => [2,-1,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[2,3,-1] => [-1,2,3] => [-1,2,3] => [1,1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[2,-3,-1] => [-1,2,-3] => [-1,2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => [-2,-1,3] => [2,-1,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [-2,-1,-3] => [2,-1,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[3,1,-2] => [3,-2,1] => [-2,3,1] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2,3,-1] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,2,1] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[3,2,-1] => [-1,3,2] => [-1,-3,2] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-2,1] => [-2,-1,3] => [2,-1,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-2,-1] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[-3,2,-1] => [-1,-3,-2] => [-1,3,-2] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-2,1] => [-2,-1,-3] => [2,-1,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1,1,1]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => [3,1]
=> 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1,1,1]
=> 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => [3,1]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => [3,1]
=> 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => [3,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => [3,1]
=> 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => [3,1]
=> 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => [3,1]
=> 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [3]
=> 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1,1,1]
=> 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => [4]
=> 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => [3,1]
=> 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [3]
=> 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => [2,2]
=> 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => [2,2]
=> 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => [2,2]
=> 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => [3,1]
=> 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [3]
=> 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => [2,2]
=> 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => [4]
=> 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => [2,2]
=> 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [3]
=> 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [3]
=> 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => [2,2]
=> 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => [2,2]
=> 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => [4]
=> 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => [4]
=> 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001605
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 14%●distinct values known / distinct values provided: 4%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 14%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1] => [1]
=> ? ∊ {1,1}
[-1] => [-1] => [-1] => []
=> ? ∊ {1,1}
[1,2] => [1,2] => [1,2] => [1,1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1,-2] => [1,-2] => [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [-1,-2] => [-1,-2] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => [-1,-2] => [-1,-2] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2,1] => [-2,1] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => [-1,2] => [-1,2] => [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => [-2,-1] => [2,-1] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [-1,-2] => [-1,-2] => []
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 2
[1,2,-3] => [1,2,-3] => [1,2,-3] => [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,-2,-3] => [1,-2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1,-2,-3] => [1,-2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [1,3,2] => [1,-3,2] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1,-2,3] => [1,-2,3] => [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1,-3,-2] => [1,3,-2] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [1,-2,-3] => [1,-2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1,3] => [-2,1,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2,1,-3] => [-2,1,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [-1,2,-3] => [-1,2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => [-1,2,-3] => [-1,2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [-2,-1,3] => [2,-1,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => [-2,-1,-3] => [2,-1,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[2,3,-1] => [-1,2,3] => [-1,2,3] => [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[2,-3,-1] => [-1,2,-3] => [-1,2,-3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => [-2,-1,3] => [2,-1,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [-2,-1,-3] => [2,-1,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[3,1,-2] => [3,-2,1] => [-2,3,1] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[-3,1,-2] => [-3,-2,-1] => [2,3,-1] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => [-1,-2,-3] => [-1,-2,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,2,1] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[3,2,-1] => [-1,3,2] => [-1,-3,2] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-2,1] => [-2,-1,3] => [2,-1,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-2,-1] => [-1,-2,3] => [-1,-2,3] => [1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[-3,2,-1] => [-1,-3,-2] => [-1,3,-2] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-2,1] => [-2,-1,-3] => [2,-1,-3] => []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 6
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1,1,1]
=> 2
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1,1,1]
=> 2
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => [3,1]
=> 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1,1,1]
=> 2
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => [3,1]
=> 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => [3,1]
=> 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => [3,1]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => [3,1]
=> 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => [3,1]
=> 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => [3,1]
=> 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [3]
=> 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1,1,1]
=> 2
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => [4]
=> 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => [3,1]
=> 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [3]
=> 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => [2,2]
=> 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => [2,2]
=> 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => [2,2]
=> 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => [3,1]
=> 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [3]
=> 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => [3,1]
=> 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [3]
=> 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => [2,2]
=> 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => [4]
=> 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => [2,2]
=> 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => [2,2]
=> 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [3]
=> 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [3]
=> 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => [2,2]
=> 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => [2,2]
=> 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => [4]
=> 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => [4]
=> 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 7%●distinct values known / distinct values provided: 1%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 7%●distinct values known / distinct values provided: 1%
Values
[1] => [1] => [1] => ([],1)
=> ? ∊ {1,1}
[-1] => [1] => [1] => ([],1)
=> ? ∊ {1,1}
[1,2] => [1,2] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1,2] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [1,2] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => [1,2] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => [2,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => [2,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [2,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,-3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => [1,2,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [2,1,3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,-1] => [2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => [2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,-1] => [2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => [2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => [2,3,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,-2] => [3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => [3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => [3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,-2] => [3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => [3,1,2] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,4,3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,3,-2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,-3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,4,-3,-2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,-4,3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,-4,3,-2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,-4,-3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,-4,-3,-2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,4,3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,4,3,-2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,4,-3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,4,-3,-2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-4,3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-4,3,-2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-4,-3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-1,-4,-3,-2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,-1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,-2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,-2,-1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,-3,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,-3,2,-1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,-3,-2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,-3,-2,-1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-4,3,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-4,3,2,-1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-4,3,-2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-4,3,-2,-1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-4,-3,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-4,-3,2,-1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-4,-3,-2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-4,-3,-2,-1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001629
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 7%●distinct values known / distinct values provided: 1%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 7%●distinct values known / distinct values provided: 1%
Values
[1] => [1] => [1] => [1] => ? ∊ {1,1}
[-1] => [1] => [1] => [1] => ? ∊ {1,1}
[1,2] => [1,2] => [2] => [1] => ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1,2] => [2] => [1] => ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [1,2] => [2] => [1] => ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => [1,2] => [2] => [1] => ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2,1] => [2] => [1] => ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => [2,1] => [2] => [1] => ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => [2,1] => [2] => [1] => ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [2,1] => [2] => [1] => ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,-3] => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => [1,2,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [1,3,2] => [1,2] => [1,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1,3,2] => [1,2] => [1,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1,3,2] => [1,2] => [1,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [1,3,2] => [1,2] => [1,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [1,3,2] => [1,2] => [1,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => [1,3,2] => [1,2] => [1,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => [1,3,2] => [1,2] => [1,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [1,3,2] => [1,2] => [1,1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2,1,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [2,1,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => [2,1,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [2,1,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => [2,1,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [2,1,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [2,1,3] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [2,3,1] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,-1] => [2,3,1] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => [2,3,1] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,-1] => [2,3,1] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => [2,3,1] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [2,3,1] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [2,3,1] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => [2,3,1] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3,1,2] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,-2] => [3,1,2] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => [3,1,2] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [3,1,2] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => [3,1,2] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,-2] => [3,1,2] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [3,1,2] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => [3,1,2] => [3] => [1] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,4,3,2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[1,4,3,-2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[1,4,-3,2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[1,4,-3,-2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[1,-4,3,2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[1,-4,3,-2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[1,-4,-3,2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[1,-4,-3,-2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[-1,4,3,2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[-1,4,3,-2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[-1,4,-3,2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[-1,4,-3,-2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[-1,-4,3,2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[-1,-4,3,-2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[-1,-4,-3,2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[-1,-4,-3,-2] => [1,4,3,2] => [1,2,1] => [1,1,1] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[4,3,2,-1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[4,3,-2,1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[4,3,-2,-1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[4,-3,2,1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[4,-3,2,-1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[4,-3,-2,1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[4,-3,-2,-1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[-4,3,2,1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[-4,3,2,-1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[-4,3,-2,1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[-4,3,-2,-1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[-4,-3,2,1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[-4,-3,2,-1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[-4,-3,-2,1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
[-4,-3,-2,-1] => [4,3,2,1] => [1,2,1] => [1,1,1] => 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000284
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 7%●distinct values known / distinct values provided: 1%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 7%●distinct values known / distinct values provided: 1%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,-4,-3] => [2,2]
=> [2]
=> 1
[-2,-1,4,3] => [2,2]
=> [2]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,-4,1,-2] => [2,2]
=> [2]
=> 1
[-3,4,-1,2] => [2,2]
=> [2]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 1
[4,-3,-2,1] => [2,2]
=> [2]
=> 1
[-4,3,2,-1] => [2,2]
=> [2]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 1
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000510
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,-4,-3] => [2,2]
=> [2]
=> 1
[-2,-1,4,3] => [2,2]
=> [2]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,-4,1,-2] => [2,2]
=> [2]
=> 1
[-3,4,-1,2] => [2,2]
=> [2]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 1
[4,-3,-2,1] => [2,2]
=> [2]
=> 1
[-4,3,2,-1] => [2,2]
=> [2]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000668
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,1,-4,-3] => [2,2]
=> [2]
=> 2
[-2,-1,4,3] => [2,2]
=> [2]
=> 2
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 2
[3,-4,1,-2] => [2,2]
=> [2]
=> 2
[-3,4,-1,2] => [2,2]
=> [2]
=> 2
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 2
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 2
[4,-3,-2,1] => [2,2]
=> [2]
=> 2
[-4,3,2,-1] => [2,2]
=> [2]
=> 2
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000681
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,-4,-3] => [2,2]
=> [2]
=> 1
[-2,-1,4,3] => [2,2]
=> [2]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,-4,1,-2] => [2,2]
=> [2]
=> 1
[-3,4,-1,2] => [2,2]
=> [2]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 1
[4,-3,-2,1] => [2,2]
=> [2]
=> 1
[-4,3,2,-1] => [2,2]
=> [2]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [2]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000698
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 7%●distinct values known / distinct values provided: 1%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 7%●distinct values known / distinct values provided: 1%
Values
[1] => [1]
=> []
=> ? ∊ {1,1}
[-1] => []
=> ?
=> ? ∊ {1,1}
[1,2] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[2,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,3}
[-2,-1] => [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,3}
[1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,3] => [1,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,2,-3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-2,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,3,-2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,2] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,-3,-2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,3,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-1,-3,-2] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,3] => [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,1,-3] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-1,-3] => [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[2,-3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,3,-1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,1] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-2,-3,-1] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,-1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,1,-2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,2] => [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[-3,-1,-2] => []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[3,2,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,5,6,6,6,6,7,8,8,8,8,8,9,10,10,10,15}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,-4] => [1,1,1]
=> [1,1]
=> 1
[1,2,-3,4] => [1,1,1]
=> [1,1]
=> 1
[1,-2,3,4] => [1,1,1]
=> [1,1]
=> 1
[-1,2,3,4] => [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,2,-4,-3] => [2,1,1]
=> [1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,-3,-2,4] => [2,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,-4,3,-2] => [2,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[-2,-1,3,4] => [2,1,1]
=> [1,1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,-4,-3] => [2,2]
=> [2]
=> 1
[-2,-1,4,3] => [2,2]
=> [2]
=> 1
[-2,-1,-4,-3] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[-3,2,-1,4] => [2,1,1]
=> [1,1]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> 1
[3,-4,1,-2] => [2,2]
=> [2]
=> 1
[-3,4,-1,2] => [2,2]
=> [2]
=> 1
[-3,-4,-1,-2] => [2,2]
=> [2]
=> 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 1
[-4,2,3,-1] => [2,1,1]
=> [1,1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> 1
[4,-3,-2,1] => [2,2]
=> [2]
=> 1
[-4,3,2,-1] => [2,2]
=> [2]
=> 1
[-4,-3,-2,-1] => [2,2]
=> [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.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer 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. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the 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. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000981The length of the longest zigzag subpath. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001808The box weight or horizontal decoration of a Dyck path. St001959The product of the heights of the peaks of a Dyck path.
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