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Your data matches 67 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[2,3,1] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[3,1,2] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[3,2,1] => [3,2,1] => [-2,-3,1] => [3]
=> 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => [3]
=> 1
[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: St000284
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [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)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000681
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [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
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [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: St000704
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000901
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000901: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000901: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
Description
The cube of the number of standard Young tableaux with shape given by the partition.
Matching statistic: St001123
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
Description
The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.
Matching statistic: St001128
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001128: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 1
Description
The exponens consonantiae of a partition.
This is the quotient of the least common multiple and the greatest common divior of the parts of the partiton. See [1, Caput sextum, §19-§22].
Matching statistic: St000512
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 45%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? = 1 - 1
[2,3,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1 - 1
[2,-3,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1 - 1
[3,1,2] => [3,2,1] => [-2,-3,1] => []
=> ? = 1 - 1
[-3,1,2] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1 - 1
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? = 1 - 1
[-3,2,1] => [-3,2,-1] => [2,-3,-1] => []
=> ? = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? = 3 - 1
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
=> ? = 1 - 1
[1,2,4,-3] => [1,2,-3,4] => [1,2,-3,4] => [1]
=> ? = 1 - 1
[1,3,4,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1 - 1
[1,3,4,-2] => [1,-2,3,4] => [1,-2,3,4] => [1]
=> ? = 1 - 1
[1,3,-4,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1 - 1
[1,4,2,3] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1 - 1
[1,-4,2,3] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1 - 1
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? = 1 - 1
[1,-4,3,2] => [1,-4,3,-2] => [1,3,-4,-2] => []
=> ? = 1 - 1
[2,3,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1 - 1
[2,3,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1 - 1
[2,-3,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1 - 1
[2,-3,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1 - 1
[2,3,4,-1] => [-1,2,3,4] => [-1,2,3,4] => [1]
=> ? = 1 - 1
[2,-3,4,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1 - 1
[2,-3,-4,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1 - 1
[2,-4,1,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1 - 1
[2,4,-3,1] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1 - 1
[2,-4,-3,1] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1 - 1
[3,1,2,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1 - 1
[3,1,2,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1 - 1
[-3,1,2,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1 - 1
[-3,1,2,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1 - 1
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? = 2 - 1
[3,1,-4,2] => [3,-4,1,-2] => [3,-4,1,-2] => []
=> ? = 2 - 1
[-3,1,4,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2 - 1
[-3,1,-4,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2 - 1
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? = 1 - 1
[3,2,1,-4] => [3,2,1,-4] => [-2,-3,1,-4] => [1]
=> ? = 1 - 1
[-3,2,1,4] => [-3,2,-1,4] => [2,-3,-1,4] => []
=> ? = 1 - 1
[-3,2,1,-4] => [-3,2,-1,-4] => [2,-3,-1,-4] => [1]
=> ? = 1 - 1
[-3,2,4,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2 - 1
[-3,2,-4,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2 - 1
[3,4,1,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1 - 1
[-3,4,1,2] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2 - 1
[-3,-4,1,2] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2 - 1
[3,4,2,-1] => [-1,4,3,2] => [-1,-3,-4,2] => [1]
=> ? = 1 - 1
[3,-4,2,-1] => [-1,-4,3,-2] => [-1,3,-4,-2] => [1]
=> ? = 1 - 1
[-3,4,2,1] => [-3,4,-1,2] => [-3,4,-1,2] => []
=> ? = 2 - 1
[-3,-4,2,1] => [-3,-4,-1,-2] => [-3,-4,-1,-2] => []
=> ? = 2 - 1
[-4,1,2,-3] => [-4,2,-3,-1] => [-3,4,2,-1] => []
=> ? = 1 - 1
[4,1,3,-2] => [4,-2,3,1] => [3,-4,-2,1] => []
=> ? = 1 - 1
[1,2,3,-4,5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 0 = 1 - 1
[1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 0 = 1 - 1
[1,2,-3,4,5] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 0 = 1 - 1
[1,-2,3,4,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 0 = 1 - 1
[-1,2,3,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 0 = 1 - 1
[1,2,3,-5,4] => [1,2,3,-5,-4] => [1,2,3,5,-4] => [2]
=> 0 = 1 - 1
[1,2,3,-5,-4] => [1,2,3,-4,-5] => [1,2,3,-4,-5] => [1,1]
=> 0 = 1 - 1
[1,2,-3,5,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 0 = 1 - 1
[1,2,4,-3,5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 0 = 1 - 1
[1,2,4,-3,-5] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 0 = 1 - 1
[1,2,-4,3,5] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 0 = 1 - 1
[1,-2,4,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 0 = 1 - 1
[1,2,4,-5,-3] => [1,2,-3,4,-5] => [1,2,-3,4,-5] => [1,1]
=> 0 = 1 - 1
[1,2,-4,5,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 0 = 1 - 1
[-1,2,4,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 0 = 1 - 1
[-1,2,4,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 0 = 1 - 1
[1,2,5,-3,4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 0 = 1 - 1
[1,2,5,-3,-4] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 0 = 1 - 1
[-1,2,5,3,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-5,3,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 0 = 1 - 1
[1,2,5,-4,3] => [1,2,-4,-3,5] => [1,2,4,-3,5] => [2]
=> 0 = 1 - 1
[1,2,5,-4,-3] => [1,2,-3,-4,5] => [1,2,-3,-4,5] => [1,1]
=> 0 = 1 - 1
[-1,2,5,4,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 0 = 1 - 1
[-1,2,-5,4,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 0 = 1 - 1
[1,3,-2,4,5] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 0 = 1 - 1
[1,-3,2,4,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 0 = 1 - 1
[-1,3,2,4,5] => [-1,-2,3,4,5] => [-1,-2,3,4,5] => [1,1]
=> 0 = 1 - 1
[1,3,-2,5,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 0 = 1 - 1
[1,3,4,-2,5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 0 = 1 - 1
[1,3,4,-2,-5] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 0 = 1 - 1
[1,-3,4,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 0 = 1 - 1
[1,3,4,-5,-2] => [1,-2,3,4,-5] => [1,-2,3,4,-5] => [1,1]
=> 0 = 1 - 1
[-1,3,4,5,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 0 = 1 - 1
[-1,3,4,-5,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 0 = 1 - 1
[1,3,5,-2,4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 0 = 1 - 1
[1,3,5,-2,-4] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 0 = 1 - 1
[-1,3,5,2,4] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-5,2,4] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 0 = 1 - 1
[1,3,5,-4,-2] => [1,-2,3,-4,5] => [1,-2,3,-4,5] => [1,1]
=> 0 = 1 - 1
[-1,3,5,4,2] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 0 = 1 - 1
[-1,3,-5,4,2] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 0 = 1 - 1
[1,4,-2,3,5] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 0 = 1 - 1
[1,4,-2,5,3] => [1,-2,-3,4,5] => [1,-2,-3,4,5] => [1,1]
=> 0 = 1 - 1
[-1,4,2,5,3] => [-1,-2,5,4,3] => [-1,-2,-4,-5,3] => [1,1]
=> 0 = 1 - 1
[-1,4,2,-5,3] => [-1,-2,-5,4,-3] => [-1,-2,4,-5,-3] => [1,1]
=> 0 = 1 - 1
[1,4,-3,2,5] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 0 = 1 - 1
[1,4,-3,5,2] => [1,-3,-2,4,5] => [1,3,-2,4,5] => [2]
=> 0 = 1 - 1
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
The following 57 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000937The number of positive values of the symmetric group character corresponding to the partition. 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. 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. St000420The number of Dyck paths that are weakly above a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000478Another weight of a partition according to Alladi. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000984The number of boxes below precisely one peak. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001808The box weight or horizontal decoration of a Dyck path. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000477The weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000933The number of multipartitions of sizes given by an integer partition. St000947The major index east count of a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2.
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