Your data matches 60 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001603
Mp00170: Permutations to signed permutationSigned permutations
Mp00281: Signed permutations rowmotionSigned permutations
Mp00166: Signed permutations even cycle typeInteger partitions
St001603: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,3,4,1] => [2,3,4,1] => [3,1,2,-4] => [3]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,2,1,-4] => [2,1]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,3,2,-4] => [2,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [2,1,3,-4] => [2,1]
=> 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3]
=> 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,-4] => [1,1,1]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [-5,2,3,4,1] => [1,1,1]
=> 1
[1,3,5,4,2] => [1,3,5,4,2] => [-5,2,4,3,1] => [2,1]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [-5,3,2,4,1] => [2,1]
=> 1
[1,4,5,3,2] => [1,4,5,3,2] => [-5,3,4,2,1] => [3]
=> 1
[1,5,3,4,2] => [1,5,3,4,2] => [-5,4,2,3,1] => [3]
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [-5,4,3,2,1] => [2,1]
=> 1
[2,1,4,5,3] => [2,1,4,5,3] => [1,-5,3,4,2] => [1,1,1]
=> 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,-5,4,3,2] => [2,1]
=> 1
[2,3,1,5,4] => [2,3,1,5,4] => [2,1,-5,4,3] => [2,1]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [3,1,2,-5,4] => [3]
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [4,1,2,3,-5] => [4]
=> 1
[2,3,5,4,1] => [2,3,5,4,1] => [4,1,3,2,-5] => [3,1]
=> 1
[2,4,3,1,5] => [2,4,3,1,5] => [3,2,1,-5,4] => [2,1]
=> 1
[2,4,3,5,1] => [2,4,3,5,1] => [4,2,1,3,-5] => [3,1]
=> 1
[2,4,5,3,1] => [2,4,5,3,1] => [4,2,3,1,-5] => [2,1,1]
=> 2
[2,5,1,3,4] => [2,5,1,3,4] => [4,1,-5,2,3] => [3]
=> 1
[2,5,3,4,1] => [2,5,3,4,1] => [4,3,1,2,-5] => [4]
=> 1
[2,5,4,3,1] => [2,5,4,3,1] => [4,3,2,1,-5] => [2,2]
=> 2
[3,2,1,5,4] => [3,2,1,5,4] => [1,2,-5,4,3] => [1,1,1]
=> 1
[3,2,4,1,5] => [3,2,4,1,5] => [1,3,2,-5,4] => [2,1]
=> 1
[3,2,4,5,1] => [3,2,4,5,1] => [1,4,2,3,-5] => [3,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [1,4,3,2,-5] => [2,1,1]
=> 2
[3,4,2,1,5] => [3,4,2,1,5] => [2,1,3,-5,4] => [2,1]
=> 1
[3,4,2,5,1] => [3,4,2,5,1] => [2,1,4,3,-5] => [2,2]
=> 2
[3,4,5,2,1] => [3,4,5,2,1] => [3,1,2,4,-5] => [3,1]
=> 1
[3,5,1,2,4] => [3,5,1,2,4] => [4,2,-5,1,3] => [2,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,4,2,-5] => [4]
=> 1
[3,5,4,2,1] => [3,5,4,2,1] => [3,2,1,4,-5] => [2,1,1]
=> 2
[4,1,2,5,3] => [4,1,2,5,3] => [3,-5,1,4,2] => [2,1]
=> 1
[4,1,5,2,3] => [4,1,5,2,3] => [3,-5,4,1,2] => [3]
=> 1
[4,2,3,1,5] => [4,2,3,1,5] => [2,3,1,-5,4] => [3]
=> 1
[4,2,3,5,1] => [4,2,3,5,1] => [2,4,1,3,-5] => [4]
=> 1
[4,2,5,3,1] => [4,2,5,3,1] => [2,4,3,1,-5] => [3,1]
=> 1
[4,3,2,1,5] => [4,3,2,1,5] => [1,2,3,-5,4] => [1,1,1]
=> 1
[4,3,2,5,1] => [4,3,2,5,1] => [1,2,4,3,-5] => [2,1,1]
=> 2
[4,3,5,2,1] => [4,3,5,2,1] => [1,3,2,4,-5] => [2,1,1]
=> 2
[4,5,2,3,1] => [4,5,2,3,1] => [3,2,4,1,-5] => [3,1]
=> 1
[4,5,3,2,1] => [4,5,3,2,1] => [2,1,3,4,-5] => [2,1,1]
=> 2
[5,1,2,4,3] => [5,1,2,4,3] => [4,-5,1,3,2] => [3]
=> 1
[5,1,4,2,3] => [5,1,4,2,3] => [4,-5,3,1,2] => [2,1]
=> 1
[5,2,1,3,4] => [5,2,1,3,4] => [1,4,-5,2,3] => [2,1]
=> 1
[5,2,3,4,1] => [5,2,3,4,1] => [3,4,1,2,-5] => [2,2]
=> 2
[5,2,4,3,1] => [5,2,4,3,1] => [3,4,2,1,-5] => [4]
=> 1
[5,3,1,2,4] => [5,3,1,2,4] => [2,4,-5,1,3] => [3]
=> 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: St000480
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St000480: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 1
Description
The number of lower covers of a partition in dominance order. According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is $$ \frac{1}{2}(\sqrt{1+8n}-3) $$ and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000759
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St000759: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 1
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000897
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St000897: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 1
Description
The number of different multiplicities of parts of an integer partition.
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St000475: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 0 = 1 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 0 = 1 - 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 0 = 1 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0 = 1 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 0 = 1 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 0 = 1 - 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 1 - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 0 = 1 - 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2 - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 0 = 1 - 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2 - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1 - 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 1 - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 1 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 1 - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2 - 1
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1 - 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2 - 1
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3 - 1
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1 - 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1 - 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 0 = 1 - 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1 - 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2 - 1
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 0 = 1 - 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2 - 1
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2 - 1
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6 - 1
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4 - 1
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1 - 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2 - 1
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4 - 1
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1 - 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2 - 1
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1 - 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4 - 1
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 0 = 1 - 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6 - 1
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1 - 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4 - 1
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2 - 1
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000512
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St000512: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 0 = 1 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 0 = 1 - 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 0 = 1 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0 = 1 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 0 = 1 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 0 = 1 - 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 1 - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 0 = 1 - 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2 - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 0 = 1 - 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2 - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1 - 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 1 - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 1 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 1 - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2 - 1
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1 - 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2 - 1
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3 - 1
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1 - 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1 - 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 0 = 1 - 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1 - 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2 - 1
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 0 = 1 - 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2 - 1
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2 - 1
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6 - 1
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4 - 1
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1 - 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2 - 1
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4 - 1
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1 - 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2 - 1
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1 - 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4 - 1
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 0 = 1 - 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6 - 1
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1 - 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4 - 1
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2 - 1
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St000513
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St000513: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 0 = 1 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 0 = 1 - 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 0 = 1 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0 = 1 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 0 = 1 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 0 = 1 - 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 1 - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 0 = 1 - 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2 - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 0 = 1 - 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2 - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1 - 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 1 - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 1 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 1 - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2 - 1
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1 - 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2 - 1
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3 - 1
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1 - 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1 - 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 0 = 1 - 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1 - 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2 - 1
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 0 = 1 - 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2 - 1
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2 - 1
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6 - 1
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4 - 1
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1 - 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2 - 1
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4 - 1
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1 - 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2 - 1
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1 - 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4 - 1
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 0 = 1 - 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6 - 1
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1 - 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4 - 1
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2 - 1
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
Description
The number of invariant subsets of size 2 when acting with a permutation of given cycle type.
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St000929: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 0 = 1 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 0 = 1 - 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 0 = 1 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0 = 1 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 0 = 1 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 0 = 1 - 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 1 - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 0 = 1 - 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2 - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 0 = 1 - 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2 - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1 - 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 1 - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 1 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 1 - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2 - 1
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1 - 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2 - 1
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3 - 1
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1 - 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1 - 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 0 = 1 - 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1 - 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2 - 1
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 0 = 1 - 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2 - 1
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2 - 1
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6 - 1
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4 - 1
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1 - 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2 - 1
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4 - 1
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1 - 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2 - 1
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1 - 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4 - 1
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 0 = 1 - 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6 - 1
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1 - 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4 - 1
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2 - 1
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
Description
The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St001122: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 0 = 1 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 0 = 1 - 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 0 = 1 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0 = 1 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 0 = 1 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 0 = 1 - 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 1 - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 0 = 1 - 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2 - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 0 = 1 - 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2 - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1 - 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 1 - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 1 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 1 - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2 - 1
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1 - 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2 - 1
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3 - 1
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1 - 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1 - 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 0 = 1 - 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1 - 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2 - 1
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 0 = 1 - 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2 - 1
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2 - 1
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6 - 1
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4 - 1
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1 - 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2 - 1
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4 - 1
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1 - 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2 - 1
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1 - 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4 - 1
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 0 = 1 - 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6 - 1
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1 - 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4 - 1
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2 - 1
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
Description
The multiplicity of the sign 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 $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Mp00065: Permutations permutation posetPosets
Mp00307: Posets promotion cycle typeInteger partitions
St001123: Integer partitions ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 0 = 1 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 0 = 1 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 0 = 1 - 1
[3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 0 = 1 - 1
[4,3,2,1] => ([],4)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 0 = 1 - 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0 = 1 - 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0 = 1 - 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 0 = 1 - 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 0 = 1 - 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 1 - 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 1 - 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 0 = 1 - 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0 = 1 - 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 2 - 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 0 = 1 - 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 1 - 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 2 - 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 1 - 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 1 - 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 1 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 1 - 1
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 1 - 1
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 1 - 1
[4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4,4,4,4,4,4]
=> ? = 1 - 1
[4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 2 - 1
[4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 2 - 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 1 - 1
[4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 2 - 1
[5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 - 1
[5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0 = 1 - 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 2 - 1
[5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 - 1
[5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 1 - 1
[5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 1 - 1
[5,4,3,2,1] => ([],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 3 - 1
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[1,6,2,5,3,4] => ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> [15]
=> ? = 1 - 1
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> [15]
=> ? = 1 - 1
[2,1,6,3,4,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [4,4]
=> 0 = 1 - 1
[2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 1
[2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [12]
=> 0 = 1 - 1
[2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> [18]
=> ? = 1 - 1
[2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> [18,18]
=> ? = 2 - 1
[2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [12]
=> 0 = 1 - 1
[2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [12,12]
=> ? = 2 - 1
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 2 - 1
[2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 6 - 1
[2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 4 - 1
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> [7]
=> 0 = 1 - 1
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [18]
=> ? = 1 - 1
[2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [18,18]
=> ? = 2 - 1
[2,5,4,6,3,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 4 - 1
[2,5,6,3,4,1] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> [24,12]
=> ? = 1 - 1
[2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> [24]
=> ? = 2 - 1
[2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [48]
=> ? = 1 - 1
[2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 2 - 1
[2,6,5,3,4,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> [24,24,24]
=> ? = 1 - 1
[2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> [24,24,24,24,24,24]
=> ? = 4 - 1
[3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> 0 = 1 - 1
[3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> [12]
=> 0 = 1 - 1
[3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [12,12]
=> ? = 2 - 1
[3,2,5,6,4,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 6 - 1
[3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [36]
=> ? = 1 - 1
[3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [36,36]
=> ? = 4 - 1
[3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> [18]
=> ? = 2 - 1
[3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[4,1,5,2,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [5,4]
=> 0 = 1 - 1
[6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> [6]
=> 0 = 1 - 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 $\lambda\vdash n$.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000481The number of upper covers of a partition in dominance order. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000256The number of parts from which one can substract 2 and still get an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001568The smallest positive integer that does not appear twice in the partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000914The sum of the values of the Möbius function of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001301The first Betti number of the order complex associated with the poset. St001555The order of a signed permutation. St001597The Frobenius rank of a skew partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001330The hat guessing number of a graph. St000068The number of minimal elements in a poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001895The oddness of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St001396Number of triples of incomparable elements in a finite poset. St001651The Frankl number of a lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St000069The number of maximal elements of a poset. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001625The Möbius invariant of a lattice. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000189The number of elements in the poset. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset.