- FindStat

Your data matches 60 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000456

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2,1] => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[-2,1] => [-2,-1] => [2,1] => ([(0,1)],2)
 => 1
[2,3,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,-3,1] => [-3,2,-1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,2] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,1,-2] => [3,-2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[-3,1,2] => [-3,2,-1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[-3,1,-2] => [-3,-2,-1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[-3,2,1] => [-3,2,-1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,3,-4,1] => [-4,2,3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,1,3] => [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,1,-3] => [4,2,-3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,-4,1,3] => [-4,2,3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,-4,1,-3] => [-4,2,-3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,4,3,1] => [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,-4,3,1] => [-4,2,3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,1,-4,2] => [3,-4,1,-2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[-3,1,4,2] => [-3,4,-1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[3,2,-4,1] => [-4,3,2,-1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[-3,2,4,1] => [-3,4,-1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[3,4,1,-2] => [4,-2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[3,-4,1,2] => [-4,3,2,-1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[3,-4,1,-2] => [-4,-2,3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[-3,4,1,2] => [-3,4,-1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[3,-4,2,1] => [-4,3,2,-1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[-3,4,2,1] => [-3,4,-1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[4,1,2,3] => [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,2,-3] => [4,2,-3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,-2,3] => [4,-2,-3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,-2,-3] => [4,-2,-3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[-4,1,2,3] => [-4,2,3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[-4,1,2,-3] => [-4,2,-3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[-4,1,-2,3] => [-4,-2,-3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[-4,1,-2,-3] => [-4,-2,-3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,3,2] => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[4,1,3,-2] => [4,-2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4,1,-3,2] => [4,-3,-2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[4,1,-3,-2] => [4,-2,-3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[-4,1,3,2] => [-4,3,2,-1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
[-4,1,3,-2] => [-4,-2,3,-1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4

Description

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

Matching statistic: St001632

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 22%●distinct values known / distinct values provided: 14%

Values

[2,1] => [2,1] => [2,1] => ([],2)
 => ? = 1 - 3
[-2,1] => [-2,-1] => [2,1] => ([],2)
 => ? = 1 - 3
[2,3,1] => [3,2,1] => [3,2,1] => ([],3)
 => ? = 3 - 3
[2,-3,1] => [-3,2,-1] => [3,2,1] => ([],3)
 => ? = 3 - 3
[3,1,2] => [3,2,1] => [3,2,1] => ([],3)
 => ? = 3 - 3
[3,1,-2] => [3,-2,1] => [3,2,1] => ([],3)
 => ? = 3 - 3
[-3,1,2] => [-3,2,-1] => [3,2,1] => ([],3)
 => ? = 3 - 3
[-3,1,-2] => [-3,-2,-1] => [3,2,1] => ([],3)
 => ? = 3 - 3
[3,2,1] => [3,2,1] => [3,2,1] => ([],3)
 => ? = 3 - 3
[-3,2,1] => [-3,2,-1] => [3,2,1] => ([],3)
 => ? = 3 - 3
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[2,3,-4,1] => [-4,2,3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[2,4,1,3] => [4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[2,4,1,-3] => [4,2,-3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[2,-4,1,3] => [-4,2,3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[2,-4,1,-3] => [-4,2,-3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[2,4,3,1] => [4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[2,-4,3,1] => [-4,2,3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[3,1,-4,2] => [3,-4,1,-2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[-3,1,4,2] => [-3,4,-1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[-3,1,-4,2] => [-3,-4,-1,-2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[3,2,-4,1] => [-4,3,2,-1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[-3,2,4,1] => [-3,4,-1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[-3,2,-4,1] => [-3,-4,-1,-2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[3,4,1,-2] => [4,-2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[3,-4,1,2] => [-4,3,2,-1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[3,-4,1,-2] => [-4,-2,3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[-3,4,1,2] => [-3,4,-1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[-3,-4,1,2] => [-3,-4,-1,-2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[3,-4,2,1] => [-4,3,2,-1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[-3,4,2,1] => [-3,4,-1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[-3,-4,2,1] => [-3,-4,-1,-2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? = 2 - 3
[4,1,2,3] => [4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[4,1,2,-3] => [4,2,-3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[4,1,-2,3] => [4,-2,-3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[4,1,-2,-3] => [4,-2,-3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[-4,1,2,3] => [-4,2,3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[-4,1,2,-3] => [-4,2,-3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[-4,1,-2,3] => [-4,-2,-3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[-4,1,-2,-3] => [-4,-2,-3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[4,1,3,2] => [4,3,2,1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[4,1,3,-2] => [4,-2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[4,1,-3,2] => [4,-3,-2,1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[4,1,-3,-2] => [4,-2,-3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[-4,1,3,2] => [-4,3,2,-1] => [4,3,2,1] => ([],4)
 => ? = 6 - 3
[-4,1,3,-2] => [-4,-2,3,-1] => [4,2,3,1] => ([(2,3)],4)
 => ? = 4 - 3
[2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,4,1,-5,3] => [4,2,-5,1,-3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-4,1,5,3] => [-4,2,5,-1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-4,1,-5,3] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-4,3,5,1] => [-4,2,5,-1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-4,3,-5,1] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-4,5,1,3] => [-4,2,5,-1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-4,-5,1,3] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-4,5,3,1] => [-4,2,5,-1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-4,-5,3,1] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,5,3,-4,1] => [-4,2,5,-1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-5,3,-4,1] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,5,-4,1,3] => [-4,2,5,-1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-5,-4,1,3] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,5,-4,3,1] => [-4,2,5,-1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[2,-5,-4,3,1] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0 = 3 - 3
[3,1,4,5,2] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[3,1,4,-5,2] => [3,-5,1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,1,4,5,2] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,1,4,-5,2] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[3,1,5,2,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[3,1,5,2,-4] => [3,5,1,-4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[3,1,-5,2,4] => [3,-5,1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[3,1,-5,2,-4] => [3,-5,1,-4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,1,5,2,4] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,1,5,2,-4] => [-3,5,-1,-4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,1,-5,2,4] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,1,-5,2,-4] => [-3,-5,-1,-4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[3,1,5,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[3,1,-5,4,2] => [3,-5,1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,1,5,4,2] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,1,-5,4,2] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,2,4,5,1] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,2,4,-5,1] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,2,5,1,4] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,2,5,1,-4] => [-3,5,-1,-4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,2,-5,1,4] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,2,-5,1,-4] => [-3,-5,-1,-4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,2,5,4,1] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,2,-5,4,1] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,4,1,5,2] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,4,1,-5,2] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,4,2,5,1] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,4,2,-5,1] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,4,5,1,2] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,4,-5,1,2] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,4,5,2,1] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,4,-5,2,1] => [-3,-5,-1,4,-2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,5,1,2,4] => [-3,5,-1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3
[-3,5,1,2,-4] => [-3,5,-1,-4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 0 = 3 - 3

Description

The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

Matching statistic: St001583

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00163: Signed permutations —permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 71%

Values

[2,1] => [2,1] => [2,1] => [1,2] => 1
[-2,1] => [-2,-1] => [2,1] => [1,2] => 1
[2,3,1] => [3,2,1] => [3,2,1] => [1,2,3] => 3
[2,-3,1] => [-3,2,-1] => [3,2,1] => [1,2,3] => 3
[3,1,2] => [3,2,1] => [3,2,1] => [1,2,3] => 3
[3,1,-2] => [3,-2,1] => [3,2,1] => [1,2,3] => 3
[-3,1,2] => [-3,2,-1] => [3,2,1] => [1,2,3] => 3
[-3,1,-2] => [-3,-2,-1] => [3,2,1] => [1,2,3] => 3
[3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 3
[-3,2,1] => [-3,2,-1] => [3,2,1] => [1,2,3] => 3
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 4
[2,3,-4,1] => [-4,2,3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[2,4,1,3] => [4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 4
[2,4,1,-3] => [4,2,-3,1] => [4,2,3,1] => [1,4,2,3] => 4
[2,-4,1,3] => [-4,2,3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[2,-4,1,-3] => [-4,2,-3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[2,4,3,1] => [4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 4
[2,-4,3,1] => [-4,2,3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[3,1,4,2] => [3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 2
[3,1,-4,2] => [3,-4,1,-2] => [3,4,1,2] => [4,1,3,2] => 2
[-3,1,4,2] => [-3,4,-1,2] => [3,4,1,2] => [4,1,3,2] => 2
[-3,1,-4,2] => [-3,-4,-1,-2] => [3,4,1,2] => [4,1,3,2] => 2
[3,2,4,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 6
[3,2,-4,1] => [-4,3,2,-1] => [4,3,2,1] => [1,2,3,4] => 6
[-3,2,4,1] => [-3,4,-1,2] => [3,4,1,2] => [4,1,3,2] => 2
[-3,2,-4,1] => [-3,-4,-1,-2] => [3,4,1,2] => [4,1,3,2] => 2
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 6
[3,4,1,-2] => [4,-2,3,1] => [4,2,3,1] => [1,4,2,3] => 4
[3,-4,1,2] => [-4,3,2,-1] => [4,3,2,1] => [1,2,3,4] => 6
[3,-4,1,-2] => [-4,-2,3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[-3,4,1,2] => [-3,4,-1,2] => [3,4,1,2] => [4,1,3,2] => 2
[-3,-4,1,2] => [-3,-4,-1,-2] => [3,4,1,2] => [4,1,3,2] => 2
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 6
[3,-4,2,1] => [-4,3,2,-1] => [4,3,2,1] => [1,2,3,4] => 6
[-3,4,2,1] => [-3,4,-1,2] => [3,4,1,2] => [4,1,3,2] => 2
[-3,-4,2,1] => [-3,-4,-1,-2] => [3,4,1,2] => [4,1,3,2] => 2
[4,1,2,3] => [4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 4
[4,1,2,-3] => [4,2,-3,1] => [4,2,3,1] => [1,4,2,3] => 4
[4,1,-2,3] => [4,-2,-3,1] => [4,2,3,1] => [1,4,2,3] => 4
[4,1,-2,-3] => [4,-2,-3,1] => [4,2,3,1] => [1,4,2,3] => 4
[-4,1,2,3] => [-4,2,3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[-4,1,2,-3] => [-4,2,-3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[-4,1,-2,3] => [-4,-2,-3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[-4,1,-2,-3] => [-4,-2,-3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[4,1,3,2] => [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 6
[4,1,3,-2] => [4,-2,3,1] => [4,2,3,1] => [1,4,2,3] => 4
[4,1,-3,2] => [4,-3,-2,1] => [4,3,2,1] => [1,2,3,4] => 6
[4,1,-3,-2] => [4,-2,-3,1] => [4,2,3,1] => [1,4,2,3] => 4
[-4,1,3,2] => [-4,3,2,-1] => [4,3,2,1] => [1,2,3,4] => 6
[-4,1,3,-2] => [-4,-2,3,-1] => [4,2,3,1] => [1,4,2,3] => 4
[2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,3,4,-5,1] => [-5,2,3,4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,3,5,1,4] => [5,2,3,4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,3,5,1,-4] => [5,2,3,-4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,3,-5,1,4] => [-5,2,3,4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,3,-5,1,-4] => [-5,2,3,-4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,3,5,4,1] => [5,2,3,4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,3,-5,4,1] => [-5,2,3,4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,4,1,-5,3] => [4,2,-5,1,-3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,-4,1,5,3] => [-4,2,5,-1,3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,-4,1,-5,3] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,4,3,5,1] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,4,3,-5,1] => [-5,2,4,3,-1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,-4,3,5,1] => [-4,2,5,-1,3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,-4,3,-5,1] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,4,5,1,3] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,4,5,1,-3] => [5,2,-3,4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,4,-5,1,3] => [-5,2,4,3,-1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,4,-5,1,-3] => [-5,2,-3,4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-4,5,1,3] => [-4,2,5,-1,3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,-4,-5,1,3] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,4,5,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,4,-5,3,1] => [-5,2,4,3,-1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,-4,5,3,1] => [-4,2,5,-1,3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,-4,-5,3,1] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,5,1,3,4] => [5,2,3,4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,5,1,3,-4] => [5,2,3,-4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,5,1,-3,4] => [5,2,-3,-4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,5,1,-3,-4] => [5,2,-3,-4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-5,1,3,4] => [-5,2,3,4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-5,1,3,-4] => [-5,2,3,-4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-5,1,-3,4] => [-5,2,-3,-4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-5,1,-3,-4] => [-5,2,-3,-4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,5,1,4,3] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,5,1,4,-3] => [5,2,-3,4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,5,1,-4,3] => [5,2,-4,-3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,5,1,-4,-3] => [5,2,-3,-4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-5,1,4,3] => [-5,2,4,3,-1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,-5,1,4,-3] => [-5,2,-3,4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-5,1,-4,3] => [-5,2,-4,-3,-1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,-5,1,-4,-3] => [-5,2,-3,-4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,5,3,1,4] => [5,2,3,4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,5,3,1,-4] => [5,2,3,-4,1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-5,3,1,4] => [-5,2,3,4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,-5,3,1,-4] => [-5,2,3,-4,-1] => [5,2,3,4,1] => [1,4,5,2,3] => ? = 5
[2,5,3,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,5,3,-4,1] => [-4,2,5,-1,3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3
[2,-5,3,4,1] => [-5,2,4,3,-1] => [5,2,4,3,1] => [1,4,2,3,5] => ? = 6
[2,-5,3,-4,1] => [-4,2,-5,-1,-3] => [4,2,5,1,3] => [5,3,1,4,2] => ? = 3

Description

The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.

Matching statistic: St001605
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

[2,1] => [2,1] => []
 => ? = 1 - 3
[-2,1] => [-2,-1] => []
 => ? = 1 - 3
[2,3,1] => [3,2,1] => []
 => ? = 3 - 3
[2,-3,1] => [-3,2,-1] => []
 => ? = 3 - 3
[3,1,2] => [3,2,1] => []
 => ? = 3 - 3
[3,1,-2] => [3,-2,1] => [1]
 => ? = 3 - 3
[-3,1,2] => [-3,2,-1] => []
 => ? = 3 - 3
[-3,1,-2] => [-3,-2,-1] => [1]
 => ? = 3 - 3
[3,2,1] => [3,2,1] => []
 => ? = 3 - 3
[-3,2,1] => [-3,2,-1] => []
 => ? = 3 - 3
[2,3,4,1] => [4,2,3,1] => []
 => ? = 4 - 3
[2,3,-4,1] => [-4,2,3,-1] => []
 => ? = 4 - 3
[2,4,1,3] => [4,2,3,1] => []
 => ? = 4 - 3
[2,4,1,-3] => [4,2,-3,1] => [1]
 => ? = 4 - 3
[2,-4,1,3] => [-4,2,3,-1] => []
 => ? = 4 - 3
[2,-4,1,-3] => [-4,2,-3,-1] => [1]
 => ? = 4 - 3
[2,4,3,1] => [4,2,3,1] => []
 => ? = 4 - 3
[2,-4,3,1] => [-4,2,3,-1] => []
 => ? = 4 - 3
[3,1,4,2] => [3,4,1,2] => []
 => ? = 2 - 3
[3,1,-4,2] => [3,-4,1,-2] => []
 => ? = 2 - 3
[-3,1,4,2] => [-3,4,-1,2] => []
 => ? = 2 - 3
[-3,1,-4,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 3
[3,2,4,1] => [4,3,2,1] => []
 => ? = 6 - 3
[3,2,-4,1] => [-4,3,2,-1] => []
 => ? = 6 - 3
[-3,2,4,1] => [-3,4,-1,2] => []
 => ? = 2 - 3
[-3,2,-4,1] => [-3,-4,-1,-2] => []
 => ? = 2 - 3
[3,4,1,2] => [4,3,2,1] => []
 => ? = 6 - 3
[3,4,1,-2] => [4,-2,3,1] => [1]
 => ? = 4 - 3
[3,-4,1,2] => [-4,3,2,-1] => []
 => ? = 6 - 3
[3,-4,1,-2] => [-4,-2,3,-1] => [1]
 => ? = 4 - 3
[-3,4,1,2] => [-3,4,-1,2] => []
 => ? = 2 - 3
[-3,-4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 3
[3,4,2,1] => [4,3,2,1] => []
 => ? = 6 - 3
[3,-4,2,1] => [-4,3,2,-1] => []
 => ? = 6 - 3
[-3,4,2,1] => [-3,4,-1,2] => []
 => ? = 2 - 3
[-3,-4,2,1] => [-3,-4,-1,-2] => []
 => ? = 2 - 3
[4,1,2,3] => [4,2,3,1] => []
 => ? = 4 - 3
[4,1,2,-3] => [4,2,-3,1] => [1]
 => ? = 4 - 3
[4,1,-2,3] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 3
[4,1,-2,-3] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 3
[-4,1,2,3] => [-4,2,3,-1] => []
 => ? = 4 - 3
[-4,1,2,-3] => [-4,2,-3,-1] => [1]
 => ? = 4 - 3
[-4,1,-2,3] => [-4,-2,-3,-1] => [1,1]
 => ? = 4 - 3
[-4,1,-2,-3] => [-4,-2,-3,-1] => [1,1]
 => ? = 4 - 3
[4,1,3,2] => [4,3,2,1] => []
 => ? = 6 - 3
[4,1,3,-2] => [4,-2,3,1] => [1]
 => ? = 4 - 3
[4,1,-3,2] => [4,-3,-2,1] => []
 => ? = 6 - 3
[4,1,-3,-2] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 3
[-4,1,3,2] => [-4,3,2,-1] => []
 => ? = 6 - 3
[-4,1,3,-2] => [-4,-2,3,-1] => [1]
 => ? = 4 - 3
[5,1,-2,3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-2,-3,4] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-2,-3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-2,3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-2,-3,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-2,-3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-2,4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-2,-4,3] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-2,-4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-2,4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-2,-4,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-2,-4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-3,-2,4] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-3,-2,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-3,-2,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-3,-2,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-3,4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-3,-4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-3,4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-3,-4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-4,-2,3] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-4,-2,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-4,-2,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-4,-2,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3
[5,1,-4,-3,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 2 = 5 - 3
[-5,1,-4,-3,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 2 = 5 - 3

Description

The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St001603
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

[2,1] => [2,1] => []
 => ? = 1 - 4
[-2,1] => [-2,-1] => []
 => ? = 1 - 4
[2,3,1] => [3,2,1] => []
 => ? = 3 - 4
[2,-3,1] => [-3,2,-1] => []
 => ? = 3 - 4
[3,1,2] => [3,2,1] => []
 => ? = 3 - 4
[3,1,-2] => [3,-2,1] => [1]
 => ? = 3 - 4
[-3,1,2] => [-3,2,-1] => []
 => ? = 3 - 4
[-3,1,-2] => [-3,-2,-1] => [1]
 => ? = 3 - 4
[3,2,1] => [3,2,1] => []
 => ? = 3 - 4
[-3,2,1] => [-3,2,-1] => []
 => ? = 3 - 4
[2,3,4,1] => [4,2,3,1] => []
 => ? = 4 - 4
[2,3,-4,1] => [-4,2,3,-1] => []
 => ? = 4 - 4
[2,4,1,3] => [4,2,3,1] => []
 => ? = 4 - 4
[2,4,1,-3] => [4,2,-3,1] => [1]
 => ? = 4 - 4
[2,-4,1,3] => [-4,2,3,-1] => []
 => ? = 4 - 4
[2,-4,1,-3] => [-4,2,-3,-1] => [1]
 => ? = 4 - 4
[2,4,3,1] => [4,2,3,1] => []
 => ? = 4 - 4
[2,-4,3,1] => [-4,2,3,-1] => []
 => ? = 4 - 4
[3,1,4,2] => [3,4,1,2] => []
 => ? = 2 - 4
[3,1,-4,2] => [3,-4,1,-2] => []
 => ? = 2 - 4
[-3,1,4,2] => [-3,4,-1,2] => []
 => ? = 2 - 4
[-3,1,-4,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 4
[3,2,4,1] => [4,3,2,1] => []
 => ? = 6 - 4
[3,2,-4,1] => [-4,3,2,-1] => []
 => ? = 6 - 4
[-3,2,4,1] => [-3,4,-1,2] => []
 => ? = 2 - 4
[-3,2,-4,1] => [-3,-4,-1,-2] => []
 => ? = 2 - 4
[3,4,1,2] => [4,3,2,1] => []
 => ? = 6 - 4
[3,4,1,-2] => [4,-2,3,1] => [1]
 => ? = 4 - 4
[3,-4,1,2] => [-4,3,2,-1] => []
 => ? = 6 - 4
[3,-4,1,-2] => [-4,-2,3,-1] => [1]
 => ? = 4 - 4
[-3,4,1,2] => [-3,4,-1,2] => []
 => ? = 2 - 4
[-3,-4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 4
[3,4,2,1] => [4,3,2,1] => []
 => ? = 6 - 4
[3,-4,2,1] => [-4,3,2,-1] => []
 => ? = 6 - 4
[-3,4,2,1] => [-3,4,-1,2] => []
 => ? = 2 - 4
[-3,-4,2,1] => [-3,-4,-1,-2] => []
 => ? = 2 - 4
[4,1,2,3] => [4,2,3,1] => []
 => ? = 4 - 4
[4,1,2,-3] => [4,2,-3,1] => [1]
 => ? = 4 - 4
[4,1,-2,3] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 4
[4,1,-2,-3] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 4
[-4,1,2,3] => [-4,2,3,-1] => []
 => ? = 4 - 4
[-4,1,2,-3] => [-4,2,-3,-1] => [1]
 => ? = 4 - 4
[-4,1,-2,3] => [-4,-2,-3,-1] => [1,1]
 => ? = 4 - 4
[-4,1,-2,-3] => [-4,-2,-3,-1] => [1,1]
 => ? = 4 - 4
[4,1,3,2] => [4,3,2,1] => []
 => ? = 6 - 4
[4,1,3,-2] => [4,-2,3,1] => [1]
 => ? = 4 - 4
[4,1,-3,2] => [4,-3,-2,1] => []
 => ? = 6 - 4
[4,1,-3,-2] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 4
[-4,1,3,2] => [-4,3,2,-1] => []
 => ? = 6 - 4
[-4,1,3,-2] => [-4,-2,3,-1] => [1]
 => ? = 4 - 4
[5,1,-2,3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-2,-3,4] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-2,-3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-2,3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-2,-3,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-2,-3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-2,4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-2,-4,3] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-2,-4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-2,4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-2,-4,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-2,-4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-3,-2,4] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-3,-2,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-3,-2,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-3,-2,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-3,4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-3,-4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-3,4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-3,-4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-4,-2,3] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-4,-2,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-4,-2,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-4,-2,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4
[5,1,-4,-3,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 1 = 5 - 4
[-5,1,-4,-3,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 1 = 5 - 4

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: St001604
(load all 2 compositions to match this statistic)

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

[2,1] => [2,1] => []
 => ? = 1 - 5
[-2,1] => [-2,-1] => []
 => ? = 1 - 5
[2,3,1] => [3,2,1] => []
 => ? = 3 - 5
[2,-3,1] => [-3,2,-1] => []
 => ? = 3 - 5
[3,1,2] => [3,2,1] => []
 => ? = 3 - 5
[3,1,-2] => [3,-2,1] => [1]
 => ? = 3 - 5
[-3,1,2] => [-3,2,-1] => []
 => ? = 3 - 5
[-3,1,-2] => [-3,-2,-1] => [1]
 => ? = 3 - 5
[3,2,1] => [3,2,1] => []
 => ? = 3 - 5
[-3,2,1] => [-3,2,-1] => []
 => ? = 3 - 5
[2,3,4,1] => [4,2,3,1] => []
 => ? = 4 - 5
[2,3,-4,1] => [-4,2,3,-1] => []
 => ? = 4 - 5
[2,4,1,3] => [4,2,3,1] => []
 => ? = 4 - 5
[2,4,1,-3] => [4,2,-3,1] => [1]
 => ? = 4 - 5
[2,-4,1,3] => [-4,2,3,-1] => []
 => ? = 4 - 5
[2,-4,1,-3] => [-4,2,-3,-1] => [1]
 => ? = 4 - 5
[2,4,3,1] => [4,2,3,1] => []
 => ? = 4 - 5
[2,-4,3,1] => [-4,2,3,-1] => []
 => ? = 4 - 5
[3,1,4,2] => [3,4,1,2] => []
 => ? = 2 - 5
[3,1,-4,2] => [3,-4,1,-2] => []
 => ? = 2 - 5
[-3,1,4,2] => [-3,4,-1,2] => []
 => ? = 2 - 5
[-3,1,-4,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 5
[3,2,4,1] => [4,3,2,1] => []
 => ? = 6 - 5
[3,2,-4,1] => [-4,3,2,-1] => []
 => ? = 6 - 5
[-3,2,4,1] => [-3,4,-1,2] => []
 => ? = 2 - 5
[-3,2,-4,1] => [-3,-4,-1,-2] => []
 => ? = 2 - 5
[3,4,1,2] => [4,3,2,1] => []
 => ? = 6 - 5
[3,4,1,-2] => [4,-2,3,1] => [1]
 => ? = 4 - 5
[3,-4,1,2] => [-4,3,2,-1] => []
 => ? = 6 - 5
[3,-4,1,-2] => [-4,-2,3,-1] => [1]
 => ? = 4 - 5
[-3,4,1,2] => [-3,4,-1,2] => []
 => ? = 2 - 5
[-3,-4,1,2] => [-3,-4,-1,-2] => []
 => ? = 2 - 5
[3,4,2,1] => [4,3,2,1] => []
 => ? = 6 - 5
[3,-4,2,1] => [-4,3,2,-1] => []
 => ? = 6 - 5
[-3,4,2,1] => [-3,4,-1,2] => []
 => ? = 2 - 5
[-3,-4,2,1] => [-3,-4,-1,-2] => []
 => ? = 2 - 5
[4,1,2,3] => [4,2,3,1] => []
 => ? = 4 - 5
[4,1,2,-3] => [4,2,-3,1] => [1]
 => ? = 4 - 5
[4,1,-2,3] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 5
[4,1,-2,-3] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 5
[-4,1,2,3] => [-4,2,3,-1] => []
 => ? = 4 - 5
[-4,1,2,-3] => [-4,2,-3,-1] => [1]
 => ? = 4 - 5
[-4,1,-2,3] => [-4,-2,-3,-1] => [1,1]
 => ? = 4 - 5
[-4,1,-2,-3] => [-4,-2,-3,-1] => [1,1]
 => ? = 4 - 5
[4,1,3,2] => [4,3,2,1] => []
 => ? = 6 - 5
[4,1,3,-2] => [4,-2,3,1] => [1]
 => ? = 4 - 5
[4,1,-3,2] => [4,-3,-2,1] => []
 => ? = 6 - 5
[4,1,-3,-2] => [4,-2,-3,1] => [1,1]
 => ? = 4 - 5
[-4,1,3,2] => [-4,3,2,-1] => []
 => ? = 6 - 5
[-4,1,3,-2] => [-4,-2,3,-1] => [1]
 => ? = 4 - 5
[5,1,-2,3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-2,-3,4] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-2,-3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-2,3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-2,-3,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-2,-3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-2,4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-2,-4,3] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-2,-4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-2,4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-2,-4,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-2,-4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-3,-2,4] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-3,-2,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-3,-2,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-3,-2,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-3,4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-3,-4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-3,4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-3,-4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-4,-2,3] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-4,-2,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-4,-2,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-4,-2,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5
[5,1,-4,-3,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => 0 = 5 - 5
[-5,1,-4,-3,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => 0 = 5 - 5

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Matching statistic: St000713

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

[2,1] => [2,1] => []
 => ?
 => ? = 1
[-2,1] => [-2,-1] => []
 => ?
 => ? = 1
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 3
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 3
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 3
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 3
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 3
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 3
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 3
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 3
[2,3,4,1] => [4,2,3,1] => []
 => ?
 => ? = 4
[2,3,-4,1] => [-4,2,3,-1] => []
 => ?
 => ? = 4
[2,4,1,3] => [4,2,3,1] => []
 => ?
 => ? = 4
[2,4,1,-3] => [4,2,-3,1] => [1]
 => []
 => ? = 4
[2,-4,1,3] => [-4,2,3,-1] => []
 => ?
 => ? = 4
[2,-4,1,-3] => [-4,2,-3,-1] => [1]
 => []
 => ? = 4
[2,4,3,1] => [4,2,3,1] => []
 => ?
 => ? = 4
[2,-4,3,1] => [-4,2,3,-1] => []
 => ?
 => ? = 4
[3,1,4,2] => [3,4,1,2] => []
 => ?
 => ? = 2
[3,1,-4,2] => [3,-4,1,-2] => []
 => ?
 => ? = 2
[-3,1,4,2] => [-3,4,-1,2] => []
 => ?
 => ? = 2
[-3,1,-4,2] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2
[3,2,4,1] => [4,3,2,1] => []
 => ?
 => ? = 6
[3,2,-4,1] => [-4,3,2,-1] => []
 => ?
 => ? = 6
[-3,2,4,1] => [-3,4,-1,2] => []
 => ?
 => ? = 2
[-3,2,-4,1] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2
[3,4,1,2] => [4,3,2,1] => []
 => ?
 => ? = 6
[3,4,1,-2] => [4,-2,3,1] => [1]
 => []
 => ? = 4
[3,-4,1,2] => [-4,3,2,-1] => []
 => ?
 => ? = 6
[3,-4,1,-2] => [-4,-2,3,-1] => [1]
 => []
 => ? = 4
[-3,4,1,2] => [-3,4,-1,2] => []
 => ?
 => ? = 2
[-3,-4,1,2] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2
[3,4,2,1] => [4,3,2,1] => []
 => ?
 => ? = 6
[3,-4,2,1] => [-4,3,2,-1] => []
 => ?
 => ? = 6
[-3,4,2,1] => [-3,4,-1,2] => []
 => ?
 => ? = 2
[-3,-4,2,1] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2
[4,1,2,3] => [4,2,3,1] => []
 => ?
 => ? = 4
[4,1,2,-3] => [4,2,-3,1] => [1]
 => []
 => ? = 4
[4,1,-2,3] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4
[4,1,-2,-3] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4
[-4,1,2,3] => [-4,2,3,-1] => []
 => ?
 => ? = 4
[-4,1,2,-3] => [-4,2,-3,-1] => [1]
 => []
 => ? = 4
[-4,1,-2,3] => [-4,-2,-3,-1] => [1,1]
 => [1]
 => ? = 4
[-4,1,-2,-3] => [-4,-2,-3,-1] => [1,1]
 => [1]
 => ? = 4
[4,1,3,2] => [4,3,2,1] => []
 => ?
 => ? = 6
[4,1,3,-2] => [4,-2,3,1] => [1]
 => []
 => ? = 4
[4,1,-3,2] => [4,-3,-2,1] => []
 => ?
 => ? = 6
[4,1,-3,-2] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4
[-4,1,3,2] => [-4,3,2,-1] => []
 => ?
 => ? = 6
[-4,1,3,-2] => [-4,-2,3,-1] => [1]
 => []
 => ? = 4
[5,1,-2,3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-2,-3,4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-2,-3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-2,3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-2,-3,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-2,-3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-2,4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-2,-4,3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-2,-4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-2,4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-2,-4,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-2,-4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-3,-2,4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-3,-2,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-3,-2,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-3,-2,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-3,4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-3,-4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-3,4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-3,-4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-4,-2,3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-4,-2,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-4,-2,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-4,-2,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5
[5,1,-4,-3,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 5
[-5,1,-4,-3,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 5

Description

The dimension of the irreducible representation of Sp(4) labelled by an integer partition. Consider the symplectic group

$Sp(2n)$ . Then the integer partition

$(\mu_1,\dots,\mu_k)$ of length at most

$n$ corresponds to the weight vector

$(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$ . For example, the integer partition

$(2)$ labels the symmetric square of the vector representation, whereas the integer partition

$(1,1)$ labels the second fundamental representation.

Matching statistic: St000927

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000927: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

[2,1] => [2,1] => []
 => ?
 => ? = 1 - 1
[-2,1] => [-2,-1] => []
 => ?
 => ? = 1 - 1
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 3 - 1
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 3 - 1
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 3 - 1
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 3 - 1
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 3 - 1
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 3 - 1
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 3 - 1
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 3 - 1
[2,3,4,1] => [4,2,3,1] => []
 => ?
 => ? = 4 - 1
[2,3,-4,1] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 1
[2,4,1,3] => [4,2,3,1] => []
 => ?
 => ? = 4 - 1
[2,4,1,-3] => [4,2,-3,1] => [1]
 => []
 => ? = 4 - 1
[2,-4,1,3] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 1
[2,-4,1,-3] => [-4,2,-3,-1] => [1]
 => []
 => ? = 4 - 1
[2,4,3,1] => [4,2,3,1] => []
 => ?
 => ? = 4 - 1
[2,-4,3,1] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 1
[3,1,4,2] => [3,4,1,2] => []
 => ?
 => ? = 2 - 1
[3,1,-4,2] => [3,-4,1,-2] => []
 => ?
 => ? = 2 - 1
[-3,1,4,2] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 1
[-3,1,-4,2] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 1
[3,2,4,1] => [4,3,2,1] => []
 => ?
 => ? = 6 - 1
[3,2,-4,1] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 1
[-3,2,4,1] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 1
[-3,2,-4,1] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 1
[3,4,1,2] => [4,3,2,1] => []
 => ?
 => ? = 6 - 1
[3,4,1,-2] => [4,-2,3,1] => [1]
 => []
 => ? = 4 - 1
[3,-4,1,2] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 1
[3,-4,1,-2] => [-4,-2,3,-1] => [1]
 => []
 => ? = 4 - 1
[-3,4,1,2] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 1
[-3,-4,1,2] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 1
[3,4,2,1] => [4,3,2,1] => []
 => ?
 => ? = 6 - 1
[3,-4,2,1] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 1
[-3,4,2,1] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 1
[-3,-4,2,1] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 1
[4,1,2,3] => [4,2,3,1] => []
 => ?
 => ? = 4 - 1
[4,1,2,-3] => [4,2,-3,1] => [1]
 => []
 => ? = 4 - 1
[4,1,-2,3] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 1
[4,1,-2,-3] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 1
[-4,1,2,3] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 1
[-4,1,2,-3] => [-4,2,-3,-1] => [1]
 => []
 => ? = 4 - 1
[-4,1,-2,3] => [-4,-2,-3,-1] => [1,1]
 => [1]
 => ? = 4 - 1
[-4,1,-2,-3] => [-4,-2,-3,-1] => [1,1]
 => [1]
 => ? = 4 - 1
[4,1,3,2] => [4,3,2,1] => []
 => ?
 => ? = 6 - 1
[4,1,3,-2] => [4,-2,3,1] => [1]
 => []
 => ? = 4 - 1
[4,1,-3,2] => [4,-3,-2,1] => []
 => ?
 => ? = 6 - 1
[4,1,-3,-2] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 1
[-4,1,3,2] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 1
[-4,1,3,-2] => [-4,-2,3,-1] => [1]
 => []
 => ? = 4 - 1
[5,1,-2,3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-2,-3,4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-2,-3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-2,3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-2,-3,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-2,-3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-2,4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-2,-4,3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-2,-4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-2,4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-2,-4,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-2,-4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-3,-2,4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-3,-2,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-3,-2,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-3,-2,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-3,4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-3,-4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-3,4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-3,-4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-4,-2,3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-4,-2,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-4,-2,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-4,-2,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[5,1,-4,-3,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1
[-5,1,-4,-3,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 4 = 5 - 1

Description

The alternating sum of the coefficients of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1].

Matching statistic: St000813

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000813: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

[2,1] => [2,1] => []
 => ?
 => ? = 1 - 2
[-2,1] => [-2,-1] => []
 => ?
 => ? = 1 - 2
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 3 - 2
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 3 - 2
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 3 - 2
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 3 - 2
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 3 - 2
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 3 - 2
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 3 - 2
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 3 - 2
[2,3,4,1] => [4,2,3,1] => []
 => ?
 => ? = 4 - 2
[2,3,-4,1] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 2
[2,4,1,3] => [4,2,3,1] => []
 => ?
 => ? = 4 - 2
[2,4,1,-3] => [4,2,-3,1] => [1]
 => []
 => ? = 4 - 2
[2,-4,1,3] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 2
[2,-4,1,-3] => [-4,2,-3,-1] => [1]
 => []
 => ? = 4 - 2
[2,4,3,1] => [4,2,3,1] => []
 => ?
 => ? = 4 - 2
[2,-4,3,1] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 2
[3,1,4,2] => [3,4,1,2] => []
 => ?
 => ? = 2 - 2
[3,1,-4,2] => [3,-4,1,-2] => []
 => ?
 => ? = 2 - 2
[-3,1,4,2] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 2
[-3,1,-4,2] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 2
[3,2,4,1] => [4,3,2,1] => []
 => ?
 => ? = 6 - 2
[3,2,-4,1] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 2
[-3,2,4,1] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 2
[-3,2,-4,1] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 2
[3,4,1,2] => [4,3,2,1] => []
 => ?
 => ? = 6 - 2
[3,4,1,-2] => [4,-2,3,1] => [1]
 => []
 => ? = 4 - 2
[3,-4,1,2] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 2
[3,-4,1,-2] => [-4,-2,3,-1] => [1]
 => []
 => ? = 4 - 2
[-3,4,1,2] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 2
[-3,-4,1,2] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 2
[3,4,2,1] => [4,3,2,1] => []
 => ?
 => ? = 6 - 2
[3,-4,2,1] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 2
[-3,4,2,1] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 2
[-3,-4,2,1] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 2
[4,1,2,3] => [4,2,3,1] => []
 => ?
 => ? = 4 - 2
[4,1,2,-3] => [4,2,-3,1] => [1]
 => []
 => ? = 4 - 2
[4,1,-2,3] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 2
[4,1,-2,-3] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 2
[-4,1,2,3] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 2
[-4,1,2,-3] => [-4,2,-3,-1] => [1]
 => []
 => ? = 4 - 2
[-4,1,-2,3] => [-4,-2,-3,-1] => [1,1]
 => [1]
 => ? = 4 - 2
[-4,1,-2,-3] => [-4,-2,-3,-1] => [1,1]
 => [1]
 => ? = 4 - 2
[4,1,3,2] => [4,3,2,1] => []
 => ?
 => ? = 6 - 2
[4,1,3,-2] => [4,-2,3,1] => [1]
 => []
 => ? = 4 - 2
[4,1,-3,2] => [4,-3,-2,1] => []
 => ?
 => ? = 6 - 2
[4,1,-3,-2] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 2
[-4,1,3,2] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 2
[-4,1,3,-2] => [-4,-2,3,-1] => [1]
 => []
 => ? = 4 - 2
[5,1,-2,3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-2,-3,4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-2,-3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-2,3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-2,-3,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-2,-3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-2,4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-2,-4,3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-2,-4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-2,4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-2,-4,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-2,-4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-3,-2,4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-3,-2,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-3,-2,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-3,-2,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-3,4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-3,-4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-3,4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-3,-4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-4,-2,3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-4,-2,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-4,-2,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-4,-2,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[5,1,-4,-3,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2
[-5,1,-4,-3,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 3 = 5 - 2

Description

The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. This is also the sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to monomial symmetric functions.

Matching statistic: St000514

Mapping

Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000514: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%

Values

[2,1] => [2,1] => []
 => ?
 => ? = 1 - 3
[-2,1] => [-2,-1] => []
 => ?
 => ? = 1 - 3
[2,3,1] => [3,2,1] => []
 => ?
 => ? = 3 - 3
[2,-3,1] => [-3,2,-1] => []
 => ?
 => ? = 3 - 3
[3,1,2] => [3,2,1] => []
 => ?
 => ? = 3 - 3
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 3 - 3
[-3,1,2] => [-3,2,-1] => []
 => ?
 => ? = 3 - 3
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 3 - 3
[3,2,1] => [3,2,1] => []
 => ?
 => ? = 3 - 3
[-3,2,1] => [-3,2,-1] => []
 => ?
 => ? = 3 - 3
[2,3,4,1] => [4,2,3,1] => []
 => ?
 => ? = 4 - 3
[2,3,-4,1] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 3
[2,4,1,3] => [4,2,3,1] => []
 => ?
 => ? = 4 - 3
[2,4,1,-3] => [4,2,-3,1] => [1]
 => []
 => ? = 4 - 3
[2,-4,1,3] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 3
[2,-4,1,-3] => [-4,2,-3,-1] => [1]
 => []
 => ? = 4 - 3
[2,4,3,1] => [4,2,3,1] => []
 => ?
 => ? = 4 - 3
[2,-4,3,1] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 3
[3,1,4,2] => [3,4,1,2] => []
 => ?
 => ? = 2 - 3
[3,1,-4,2] => [3,-4,1,-2] => []
 => ?
 => ? = 2 - 3
[-3,1,4,2] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 3
[-3,1,-4,2] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 3
[3,2,4,1] => [4,3,2,1] => []
 => ?
 => ? = 6 - 3
[3,2,-4,1] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 3
[-3,2,4,1] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 3
[-3,2,-4,1] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 3
[3,4,1,2] => [4,3,2,1] => []
 => ?
 => ? = 6 - 3
[3,4,1,-2] => [4,-2,3,1] => [1]
 => []
 => ? = 4 - 3
[3,-4,1,2] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 3
[3,-4,1,-2] => [-4,-2,3,-1] => [1]
 => []
 => ? = 4 - 3
[-3,4,1,2] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 3
[-3,-4,1,2] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 3
[3,4,2,1] => [4,3,2,1] => []
 => ?
 => ? = 6 - 3
[3,-4,2,1] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 3
[-3,4,2,1] => [-3,4,-1,2] => []
 => ?
 => ? = 2 - 3
[-3,-4,2,1] => [-3,-4,-1,-2] => []
 => ?
 => ? = 2 - 3
[4,1,2,3] => [4,2,3,1] => []
 => ?
 => ? = 4 - 3
[4,1,2,-3] => [4,2,-3,1] => [1]
 => []
 => ? = 4 - 3
[4,1,-2,3] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 3
[4,1,-2,-3] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 3
[-4,1,2,3] => [-4,2,3,-1] => []
 => ?
 => ? = 4 - 3
[-4,1,2,-3] => [-4,2,-3,-1] => [1]
 => []
 => ? = 4 - 3
[-4,1,-2,3] => [-4,-2,-3,-1] => [1,1]
 => [1]
 => ? = 4 - 3
[-4,1,-2,-3] => [-4,-2,-3,-1] => [1,1]
 => [1]
 => ? = 4 - 3
[4,1,3,2] => [4,3,2,1] => []
 => ?
 => ? = 6 - 3
[4,1,3,-2] => [4,-2,3,1] => [1]
 => []
 => ? = 4 - 3
[4,1,-3,2] => [4,-3,-2,1] => []
 => ?
 => ? = 6 - 3
[4,1,-3,-2] => [4,-2,-3,1] => [1,1]
 => [1]
 => ? = 4 - 3
[-4,1,3,2] => [-4,3,2,-1] => []
 => ?
 => ? = 6 - 3
[-4,1,3,-2] => [-4,-2,3,-1] => [1]
 => []
 => ? = 4 - 3
[5,1,-2,3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-2,-3,4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-2,-3,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-2,3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-2,-3,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-2,-3,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-2,4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-2,-4,3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-2,-4,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-2,4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-2,-4,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-2,-4,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-3,-2,4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-3,-2,-4] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-3,-2,4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-3,-2,-4] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-3,4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-3,-4,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-3,4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-3,-4,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-4,-2,3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-4,-2,-3] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-4,-2,3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-4,-2,-3] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[5,1,-4,-3,-2] => [5,-2,-3,-4,1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3
[-5,1,-4,-3,-2] => [-5,-2,-3,-4,-1] => [1,1,1]
 => [1,1]
 => 2 = 5 - 3

Description

The number of invariant simple graphs when acting with a permutation of given cycle type.

The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. 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

$eA$ . 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

$eA$ . 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

$eA$ . St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.