Processing math: 100%

Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000225
Mapping
Mp00260: Signed permutations —Demazure product with inverse⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1] => [-1] => [-1] => [1]
 => 0
[1,-2] => [1,-2] => [1,-2] => [1]
 => 0
[-1,2] => [-1,-2] => [-1,-2] => [1,1]
 => 0
[-1,-2] => [-1,-2] => [-1,-2] => [1,1]
 => 0
[2,1] => [2,1] => [-2,1] => [2]
 => 0
[2,-1] => [-1,2] => [-1,2] => [1]
 => 0
[-2,1] => [-2,-1] => [2,-1] => [2]
 => 0
[-2,-1] => [-1,-2] => [-1,-2] => [1,1]
 => 0
[1,2,-3] => [1,2,-3] => [1,2,-3] => [1]
 => 0
[1,-2,3] => [1,-2,-3] => [1,-2,-3] => [1,1]
 => 0
[1,-2,-3] => [1,-2,-3] => [1,-2,-3] => [1,1]
 => 0
[-1,2,3] => [-1,-2,3] => [-1,-2,3] => [1,1]
 => 0
[-1,2,-3] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[-1,-2,3] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[-1,-2,-3] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[1,3,2] => [1,3,2] => [1,-3,2] => [2]
 => 0
[1,3,-2] => [1,-2,3] => [1,-2,3] => [1]
 => 0
[1,-3,2] => [1,-3,-2] => [1,3,-2] => [2]
 => 0
[1,-3,-2] => [1,-2,-3] => [1,-2,-3] => [1,1]
 => 0
[-1,3,2] => [-1,-2,3] => [-1,-2,3] => [1,1]
 => 0
[-1,3,-2] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[-1,-3,2] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[-1,-3,-2] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[2,1,3] => [2,1,3] => [-2,1,3] => [2]
 => 0
[2,1,-3] => [2,1,-3] => [-2,1,-3] => [2,1]
 => 1
[2,-1,3] => [-1,2,-3] => [-1,2,-3] => [1,1]
 => 0
[2,-1,-3] => [-1,2,-3] => [-1,2,-3] => [1,1]
 => 0
[-2,1,3] => [-2,-1,3] => [2,-1,3] => [2]
 => 0
[-2,1,-3] => [-2,-1,-3] => [2,-1,-3] => [2,1]
 => 1
[-2,-1,3] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[-2,-1,-3] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[2,3,-1] => [-1,2,3] => [-1,2,3] => [1]
 => 0
[2,-3,-1] => [-1,2,-3] => [-1,2,-3] => [1,1]
 => 0
[-2,3,1] => [-2,-1,3] => [2,-1,3] => [2]
 => 0
[-2,3,-1] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[-2,-3,1] => [-2,-1,-3] => [2,-1,-3] => [2,1]
 => 1
[-2,-3,-1] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[3,1,-2] => [3,-2,1] => [-2,3,1] => [3]
 => 0
[3,-1,2] => [-1,-2,3] => [-1,-2,3] => [1,1]
 => 0
[3,-1,-2] => [-1,-2,3] => [-1,-2,3] => [1,1]
 => 0
[-3,1,-2] => [-3,-2,-1] => [2,3,-1] => [3]
 => 0
[-3,-1,2] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[-3,-1,-2] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[3,2,-1] => [-1,3,2] => [-1,-3,2] => [2,1]
 => 1
[3,-2,1] => [-2,-1,3] => [2,-1,3] => [2]
 => 0
[3,-2,-1] => [-1,-2,3] => [-1,-2,3] => [1,1]
 => 0
[-3,2,-1] => [-1,-3,-2] => [-1,3,-2] => [2,1]
 => 1
[-3,-2,1] => [-2,-1,-3] => [2,-1,-3] => [2,1]
 => 1
[-3,-2,-1] => [-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => 0
[1,2,3,-4] => [1,2,3,-4] => [1,2,3,-4] => [1]
 => 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St001604
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
St001604: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Values
[-1] => [-1] => [1]
 => []
 => ? = 0
[1,-2] => [1,-2] => [1]
 => []
 => ? = 0
[-1,2] => [-1,-2] => [1,1]
 => [1]
 => ? = 0
[-1,-2] => [-1,-2] => [1,1]
 => [1]
 => ? = 0
[2,1] => [2,1] => []
 => ?
 => ? = 0
[2,-1] => [-1,2] => [1]
 => []
 => ? = 0
[-2,1] => [-2,-1] => []
 => ?
 => ? = 0
[-2,-1] => [-1,-2] => [1,1]
 => [1]
 => ? = 0
[1,2,-3] => [1,2,-3] => [1]
 => []
 => ? = 0
[1,-2,3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[1,-2,-3] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,2,3] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-1,2,-3] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[-1,-2,3] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[-1,-2,-3] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[1,3,2] => [1,3,2] => []
 => ?
 => ? = 0
[1,3,-2] => [1,-2,3] => [1]
 => []
 => ? = 0
[1,-3,2] => [1,-3,-2] => []
 => ?
 => ? = 0
[1,-3,-2] => [1,-2,-3] => [1,1]
 => [1]
 => ? = 0
[-1,3,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-1,3,-2] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[-1,-3,2] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[-1,-3,-2] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[2,1,3] => [2,1,3] => []
 => ?
 => ? = 0
[2,1,-3] => [2,1,-3] => [1]
 => []
 => ? = 1
[2,-1,3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[2,-1,-3] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,1,3] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,1,-3] => [-2,-1,-3] => [1]
 => []
 => ? = 1
[-2,-1,3] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[-2,-1,-3] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[2,3,-1] => [-1,2,3] => [1]
 => []
 => ? = 0
[2,-3,-1] => [-1,2,-3] => [1,1]
 => [1]
 => ? = 0
[-2,3,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[-2,3,-1] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[-2,-3,1] => [-2,-1,-3] => [1]
 => []
 => ? = 1
[-2,-3,-1] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[3,1,-2] => [3,-2,1] => [1]
 => []
 => ? = 0
[3,-1,2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[3,-1,-2] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,1,-2] => [-3,-2,-1] => [1]
 => []
 => ? = 0
[-3,-1,2] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[-3,-1,-2] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[3,2,-1] => [-1,3,2] => [1]
 => []
 => ? = 1
[3,-2,1] => [-2,-1,3] => []
 => ?
 => ? = 0
[3,-2,-1] => [-1,-2,3] => [1,1]
 => [1]
 => ? = 0
[-3,2,-1] => [-1,-3,-2] => [1]
 => []
 => ? = 1
[-3,-2,1] => [-2,-1,-3] => [1]
 => []
 => ? = 1
[-3,-2,-1] => [-1,-2,-3] => [1,1,1]
 => [1,1]
 => ? = 0
[1,2,3,-4] => [1,2,3,-4] => [1]
 => []
 => ? = 0
[-1,2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-2,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-3,2,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-3,2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-3,-2,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-3,-2,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-3,4,2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-3,4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-3,-4,2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-3,-4,-2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,4,-2,3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,4,-2,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-4,2,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-4,-2,3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-4,-2,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,4,-3,2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,4,-3,-2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-4,3,-2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-4,-3,2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-1,-4,-3,-2] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-1,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-1,3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-1,-3,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-1,-3,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-1,4,3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-1,4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-1,-4,3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-1,-4,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,3,-1,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,3,-1,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-3,-1,4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-3,-1,-4] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,3,-4,-1] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-3,4,-1] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-3,-4,-1] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,4,-1,3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,4,-1,-3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
[-2,-4,-1,3] => [-1,-2,-3,-4] => [1,1,1,1]
 => [1,1,1]
 => 0
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: St000181
(load all 4 compositions to match this statistic)
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Values
[-1] => 1 => ([(0,1)],2)
 => 1 = 0 + 1
[1,-2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-1,-2] => 11 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1] => 00 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,-1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-2,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-2,-1] => 11 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,2,-3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[1,-2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[1,-2,-3] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,2,-3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[-1,-2,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,-2,-3] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,-2] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[1,-3,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[1,-3,-2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,3,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,3,-2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[-1,-3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,-3,-2] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,3] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,-3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[2,-1,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[2,-1,-3] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,1,-3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1 + 1
[-2,-1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,-1,-3] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,3,-1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[2,-3,-1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,3,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,3,-1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[-2,-3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[-2,-3,-1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,-2] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[3,-1,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,-1,-2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-3,1,-2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[-3,-1,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-3,-1,-2] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,2,-1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[3,-2,1] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,-2,-1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-3,2,-1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1 + 1
[-3,-2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[-3,-2,-1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,3,-4] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,2,-3,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[1,2,-3,-4] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[1,-2,3,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[1,-2,3,-4] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 0 + 1
[1,-2,-3,4] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 + 1
[1,-2,-3,-4] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[-1,2,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[-1,2,3,-4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 + 1
[-1,2,-3,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 0 + 1
[-1,2,-3,-4] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[-1,-2,3,4] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[-1,-2,3,-4] => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[-1,-2,-3,4] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[-1,-2,-3,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,-3] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,2,-4,3] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[1,2,-4,-3] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[-1,-2,-4,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-1,-3,-2,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-1,-3,-4,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-1,-4,-2,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-1,-4,-3,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,3,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-1,-3,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-1,-4,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-3,-1,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,3,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-3,-4,-1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,4,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-4,-1,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,4,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-4,-3,-1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-1,-2,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-1,-4,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-2,-1,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,2,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-2,-4,-1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-4,-1,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-4,-2,-1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,1,2,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-4,-1,-2,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,1,3,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-4,-1,-3,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,2,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-4,-2,-1,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001890
(load all 4 compositions to match this statistic)
Mapping
Mp00267: Signed permutations —signs⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001890: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 25%
Values
[-1] => 1 => ([(0,1)],2)
 => 1 = 0 + 1
[1,-2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-1,-2] => 11 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1] => 00 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,-1] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-2,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[-2,-1] => 11 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,2,-3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[1,-2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[1,-2,-3] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,2,-3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[-1,-2,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,-2,-3] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,-2] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[1,-3,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[1,-3,-2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,3,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,3,-2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[-1,-3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-1,-3,-2] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,3] => 000 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,-3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[2,-1,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[2,-1,-3] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,1,-3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1 + 1
[-2,-1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,-1,-3] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,3,-1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[2,-3,-1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,3,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-2,3,-1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[-2,-3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[-2,-3,-1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,1,-2] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[3,-1,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,-1,-2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-3,1,-2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[-3,-1,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-3,-1,-2] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[3,2,-1] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[3,-2,1] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 0 + 1
[3,-2,-1] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[-3,2,-1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 1 + 1
[-3,-2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[-3,-2,-1] => 111 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,3,-4] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,2,-3,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[1,2,-3,-4] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[1,-2,3,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[1,-2,3,-4] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 0 + 1
[1,-2,-3,4] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 + 1
[1,-2,-3,-4] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[-1,2,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[-1,2,3,-4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 + 1
[-1,2,-3,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 0 + 1
[-1,2,-3,-4] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[-1,-2,3,4] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[-1,-2,3,-4] => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[-1,-2,-3,4] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[-1,-2,-3,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,-3] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[1,2,-4,3] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 + 1
[1,2,-4,-3] => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 + 1
[-1,-2,-4,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,2,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-1,-3,-2,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-1,-3,-4,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-1,-4,-2,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-1,-4,-3,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,3,4] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-1,-3,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,4,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-1,-4,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-3,-1,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,3,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-3,-4,-1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,4,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-4,-1,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,4,3,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-2,-4,-3,-1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-1,-2,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-1,-4,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-2,-1,-4] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,2,4,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-2,-4,-1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,1,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-4,-1,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[3,4,2,1] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-3,-4,-2,-1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,1,2,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-4,-1,-2,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,1,3,2] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-4,-1,-3,-2] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[4,2,1,3] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[-4,-2,-1,-3] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
Description
The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value μ(x,y) is equal to the signed sum of chains from x to y, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.