Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001630
(load all 2 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St000755
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3,2,1] => ([],3)
 => [3,3]
 => 1
[1,2,3,4] => [4,3,2,1] => ([],4)
 => [4,4,4,4,4,4]
 => ? = 2
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 1
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 2
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 1
[1,2,3,4,5] => [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]
 => ? = 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 1
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 2
[1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 1
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 1
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 1
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 1
[2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 1
[2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => ? = 2
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => ? = 2
[2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => ? = 2
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => ? = 2
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => ? = 2
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 1
[3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 1
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => ? = 2
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => ? = 2
[3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 2
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => ? = 2
[3,4,5,1,2] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => 2
[3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => 1
[3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 2
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 1
[4,1,2,3,5] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 1
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => ? = 2
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 2
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 1
[4,5,1,2,3] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => 2
[4,5,2,3,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => 2
[4,5,3,1,2] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => 2
[5,1,2,3,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => ? = 2
[5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => 1
[5,3,1,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => 1
[5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => 2
[5,4,1,2,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(2,3),(2,4),(2,5)],6)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 1
[1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
 => [18,18,18,18,18,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[1,2,5,3,6,4] => [4,6,3,5,2,1] => ([(2,5),(3,4),(3,5)],6)
 => [18,18,18,18,18,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 2
[1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 1
[1,3,2,5,6,4] => [4,6,5,2,3,1] => ([(1,5),(2,3),(2,4)],6)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 1
[1,3,2,6,4,5] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 1
[1,3,4,2,6,5] => [5,6,2,4,3,1] => ([(1,5),(2,3),(2,4)],6)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 1
[1,3,4,5,2,6] => [6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 1
[1,3,4,5,6,2] => [2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
 => [24,24,24,24,24,24]
 => ? = 2
[1,3,4,6,2,5] => [5,2,6,4,3,1] => ([(1,5),(2,3),(2,4),(2,5)],6)
 => [24,24,12,12,12,12,12]
 => ? = 2
[1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(2,5),(3,4),(3,5)],6)
 => [18,18,18,18,18,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[1,3,5,2,6,4] => [4,6,2,5,3,1] => ([(1,4),(1,5),(2,3),(2,5)],6)
 => [72,24]
 => ? = 2
[1,3,5,6,2,4] => [4,2,6,5,3,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [84]
 => ? = 2
[1,3,6,2,4,5] => [5,4,2,6,3,1] => ([(1,5),(2,5),(3,4),(3,5)],6)
 => [24,24,12,12,12,12,12]
 => ? = 2
[1,4,2,3,6,5] => [5,6,3,2,4,1] => ([(1,5),(2,5),(3,4)],6)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 1
[1,4,2,5,3,6] => [6,3,5,2,4,1] => ([(2,5),(3,4),(3,5)],6)
 => [18,18,18,18,18,6,6,6,6,6,6,6,6,6,6]
 => ? = 1
[1,4,2,5,6,3] => [3,6,5,2,4,1] => ([(1,5),(2,3),(2,4),(2,5)],6)
 => [24,24,12,12,12,12,12]
 => ? = 2
[1,4,2,6,3,5] => [5,3,6,2,4,1] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [72,24]
 => ? = 2
[1,4,5,2,3,6] => [6,3,2,5,4,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 2
[1,4,5,2,6,3] => [3,6,2,5,4,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [84]
 => ? = 2
[4,3,6,5,1,2] => [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
 => [4,4,2,2]
 => 2
[4,5,6,2,3,1] => [1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [6,6]
 => 2
[4,5,6,3,1,2] => [2,1,3,6,5,4] => ([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
 => [6,6]
 => 2
[4,5,6,3,2,1] => [1,2,3,6,5,4] => ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => 1
[4,6,3,1,5,2] => [2,5,1,3,6,4] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => [10,2]
 => 2
[4,6,3,5,1,2] => [2,1,5,3,6,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => 2
[4,6,3,5,2,1] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => 1
[5,2,6,4,1,3] => [3,1,4,6,2,5] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => [10,2]
 => 2
[5,3,6,4,1,2] => [2,1,4,6,3,5] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => [6,2,2]
 => 2
[5,3,6,4,2,1] => [1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => 1
[5,6,2,1,4,3] => [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,4,2,2]
 => 2
[5,6,2,3,4,1] => [1,4,3,2,6,5] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6]
 => 2
[5,6,2,4,1,3] => [3,1,4,2,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => 2
[5,6,3,1,4,2] => [2,4,1,3,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => [6,2,2]
 => 2
[5,6,3,4,2,1] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => 2
[5,6,4,1,2,3] => [3,2,1,4,6,5] => ([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
 => [6,6]
 => 2
[5,6,4,2,3,1] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => 2
[6,3,4,5,1,2] => [2,1,5,4,3,6] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => [6,6]
 => 2
[6,3,4,5,2,1] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => [3,3]
 => 1
[6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => 1
[6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => 1
[6,4,5,1,2,3] => [3,2,1,5,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [6,6]
 => 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => 2
[6,4,5,3,1,2] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => 2
[6,5,2,3,4,1] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => 1
[6,5,2,4,1,3] => [3,1,4,2,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => 1
[6,5,3,1,4,2] => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => 1
[6,5,3,4,1,2] => [2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => 2
[6,5,4,1,2,3] => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [3,3]
 => 1
[5,4,7,6,2,3,1] => [1,3,2,6,7,4,5] => ([(0,3),(0,4),(3,5),(3,6),(4,5),(4,6),(5,2),(6,1)],7)
 => [4,4,2,2]
 => 2
[5,4,7,6,3,1,2] => [2,1,3,6,7,4,5] => ([(0,6),(1,6),(4,3),(5,2),(6,4),(6,5)],7)
 => [4,4,2,2]
 => 2
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St001876
(load all 6 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1,2,3] => ([],3)
 => ([],1)
 => ? = 1 - 1
[1,2,3,4] => ([],4)
 => ([],1)
 => ? = 2 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 1 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2 - 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2 - 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 - 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 2 - 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 1 - 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 - 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 1 - 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 - 1
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 1 - 1
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 2 - 1
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
 => ? = 2 - 1
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
 => ? = 2 - 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 1 - 1
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 1 - 1
[5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 1 - 1
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
 => ? = 2 - 1
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 1 - 1
[1,2,3,4,5,6] => ([],6)
 => ([],1)
 => ? = 1 - 1
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2 - 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 - 1
[1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2 - 1
[1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 - 1
[1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 2 - 1
[1,4,5,6,3,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 1 - 1
[1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 - 1
[1,4,6,3,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 1 - 1
[1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 2 - 1
[1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
 => ? = 2 - 1
[1,5,3,6,4,2] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 1 - 1
[1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
 => ? = 2 - 1
[1,5,6,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
 => ? = 2 - 1
[2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[3,1,2,5,4,6] => ([(1,2),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[4,1,2,3,5,6] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,3,5,7,4,6] => ([(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,3,6,4,7,5] => ([(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,3,7,4,5,6] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,4,3,6,7,5] => ([(2,3),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,4,3,7,5,6] => ([(2,3),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,4,5,3,7,6] => ([(2,3),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,4,5,6,3,7] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
[1,2,4,6,3,5,7] => ([(3,6),(4,5),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0 = 1 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001364
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001364: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [3,2,1] => ([],3)
 => [3,3]
 => 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => ([],4)
 => [4,4,4,4,4,4]
 => ? = 2 - 1
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 0 = 1 - 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 0 = 1 - 1
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 0 = 1 - 1
[1,2,3,4,5] => [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]
 => ? = 1 - 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 1 - 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 1 - 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 1 - 1
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 2 - 1
[1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 1 - 1
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 1 - 1
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 1 - 1
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 1 - 1
[2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 1 - 1
[2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => ? = 2 - 1
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => ? = 2 - 1
[2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 1 - 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => ? = 2 - 1
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => ? = 2 - 1
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => ? = 2 - 1
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 1 - 1
[3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 1 - 1
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => ? = 2 - 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => ? = 2 - 1
[3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 2 - 1
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => ? = 2 - 1
[3,4,5,1,2] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => ? = 2 - 1
[3,4,5,2,1] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => 0 = 1 - 1
[3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 2 - 1
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 0 = 1 - 1
[4,1,2,3,5] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 1 - 1
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => ? = 2 - 1
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 2 - 1
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 0 = 1 - 1
[4,5,1,2,3] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => ? = 2 - 1
[4,5,2,3,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => 1 = 2 - 1
[4,5,3,1,2] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => 1 = 2 - 1
[5,1,2,3,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => ? = 2 - 1
[5,2,3,4,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 0 = 1 - 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => 0 = 1 - 1
[5,3,1,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2]
 => 0 = 1 - 1
[5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => 1 = 2 - 1
[5,4,1,2,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => 0 = 1 - 1
[1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 1 - 1
[1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(2,3),(2,4),(2,5)],6)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 1 - 1
[1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
 => [18,18,18,18,18,6,6,6,6,6,6,6,6,6,6]
 => ? = 1 - 1
[1,2,5,3,6,4] => [4,6,3,5,2,1] => ([(2,5),(3,4),(3,5)],6)
 => [18,18,18,18,18,6,6,6,6,6,6,6,6,6,6]
 => ? = 1 - 1
[1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
 => ? = 2 - 1
[1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 1 - 1
[1,3,2,5,6,4] => [4,6,5,2,3,1] => ([(1,5),(2,3),(2,4)],6)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 1 - 1
[1,3,2,6,4,5] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 1 - 1
[1,3,4,2,6,5] => [5,6,2,4,3,1] => ([(1,5),(2,3),(2,4)],6)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 1 - 1
[1,3,4,5,2,6] => [6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6)
 => [18,18,18,18,18,18,18,18,18,18]
 => ? = 1 - 1
[1,3,4,5,6,2] => [2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
 => [24,24,24,24,24,24]
 => ? = 2 - 1
[1,3,4,6,2,5] => [5,2,6,4,3,1] => ([(1,5),(2,3),(2,4),(2,5)],6)
 => [24,24,12,12,12,12,12]
 => ? = 2 - 1
[1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(2,5),(3,4),(3,5)],6)
 => [18,18,18,18,18,6,6,6,6,6,6,6,6,6,6]
 => ? = 1 - 1
[1,3,5,2,6,4] => [4,6,2,5,3,1] => ([(1,4),(1,5),(2,3),(2,5)],6)
 => [72,24]
 => ? = 2 - 1
[1,3,5,6,2,4] => [4,2,6,5,3,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [84]
 => ? = 2 - 1
[1,3,6,2,4,5] => [5,4,2,6,3,1] => ([(1,5),(2,5),(3,4),(3,5)],6)
 => [24,24,12,12,12,12,12]
 => ? = 2 - 1
[1,4,2,3,6,5] => [5,6,3,2,4,1] => ([(1,5),(2,5),(3,4)],6)
 => [12,12,12,12,12,12,12,12,12,12]
 => ? = 1 - 1
[1,4,2,5,3,6] => [6,3,5,2,4,1] => ([(2,5),(3,4),(3,5)],6)
 => [18,18,18,18,18,6,6,6,6,6,6,6,6,6,6]
 => ? = 1 - 1
[1,4,2,5,6,3] => [3,6,5,2,4,1] => ([(1,5),(2,3),(2,4),(2,5)],6)
 => [24,24,12,12,12,12,12]
 => ? = 2 - 1
[1,4,2,6,3,5] => [5,3,6,2,4,1] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => [72,24]
 => ? = 2 - 1
[4,5,6,3,2,1] => [1,2,3,6,5,4] => ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => [3,3]
 => 0 = 1 - 1
[4,6,3,5,2,1] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => 0 = 1 - 1
[5,3,6,4,2,1] => [1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => [3,2]
 => 0 = 1 - 1
[5,6,3,4,2,1] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
 => [2,2]
 => 1 = 2 - 1
[5,6,4,2,3,1] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
 => [2,2]
 => 1 = 2 - 1
[5,6,4,3,1,2] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
 => [2,2]
 => 1 = 2 - 1
[6,3,4,5,2,1] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => [3,3]
 => 0 = 1 - 1
[6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => 0 = 1 - 1
[6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => [3,2]
 => 0 = 1 - 1
[6,4,5,2,3,1] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => [2,2]
 => 1 = 2 - 1
[6,4,5,3,1,2] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
 => [2,2]
 => 1 = 2 - 1
[6,5,2,3,4,1] => [1,4,3,2,5,6] => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => [3,3]
 => 0 = 1 - 1
[6,5,2,4,1,3] => [3,1,4,2,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => 0 = 1 - 1
[6,5,3,1,4,2] => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => [3,2]
 => 0 = 1 - 1
[6,5,3,4,1,2] => [2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
 => [2,2]
 => 1 = 2 - 1
[6,5,4,1,2,3] => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => [3,3]
 => 0 = 1 - 1
[5,6,7,4,3,2,1] => [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
 => [3,3]
 => 0 = 1 - 1
[5,7,4,6,3,2,1] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => [3,2]
 => 0 = 1 - 1
[6,4,7,5,3,2,1] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => [3,2]
 => 0 = 1 - 1
[6,7,4,5,3,2,1] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => [2,2]
 => 1 = 2 - 1
[6,7,5,3,4,2,1] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => [2,2]
 => 1 = 2 - 1
[6,7,5,4,2,3,1] => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => [2,2]
 => 1 = 2 - 1
[6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => ([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
 => [2,2]
 => 1 = 2 - 1
[7,4,5,6,3,2,1] => [1,2,3,6,5,4,7] => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => [3,3]
 => 0 = 1 - 1
[7,4,6,3,5,2,1] => [1,2,5,3,6,4,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [3,2]
 => 0 = 1 - 1
[7,5,3,6,4,2,1] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [3,2]
 => 0 = 1 - 1
[7,5,6,3,4,2,1] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => [2,2]
 => 1 = 2 - 1
[7,5,6,4,2,3,1] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => [2,2]
 => 1 = 2 - 1
[7,5,6,4,3,1,2] => [2,1,3,4,6,5,7] => ([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
 => [2,2]
 => 1 = 2 - 1
[7,6,3,4,5,2,1] => [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => [3,3]
 => 0 = 1 - 1
[7,6,3,5,2,4,1] => [1,4,2,5,3,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => [3,2]
 => 0 = 1 - 1
[7,6,4,2,5,3,1] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => [3,2]
 => 0 = 1 - 1
[7,6,4,5,2,3,1] => [1,3,2,5,4,6,7] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
 => [2,2]
 => 1 = 2 - 1
[7,6,4,5,3,1,2] => [2,1,3,5,4,6,7] => ([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
 => [2,2]
 => 1 = 2 - 1
Description
The number of permutations whose cube equals a fixed permutation of given cycle type. For example, the permutation $\pi=412365$ has cycle type $(4,2)$ and $234165$ is the unique permutation whose cube is $\pi$.
Matching statistic: St001603
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [-1,-2,-3] => []
 => ? = 1
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => []
 => ? = 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
 => 1
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
 => 1
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
 => 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
 => 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => []
 => ? = 1
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
 => 1
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
 => 1
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
 => 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
 => 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
 => 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
 => ? = 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
 => ? = 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
 => ? = 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
 => 1
[2,3,4,5,1] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => []
 => ? = 2
[2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => []
 => ? = 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
 => 1
[2,4,1,5,3] => [2,4,1,5,3] => [-2,-4,-1,-5,-3] => []
 => ? = 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
 => ? = 2
[2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => []
 => ? = 2
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
 => ? = 1
[3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => [4]
 => 1
[3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => []
 => ? = 2
[3,1,5,2,4] => [3,1,5,2,4] => [-3,-1,-5,-2,-4] => []
 => ? = 2
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [2,2]
 => 2
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [2]
 => ? = 2
[3,4,5,1,2] => [3,4,5,1,2] => [-3,-4,-5,-1,-2] => []
 => ? = 2
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [2]
 => ? = 1
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [2]
 => ? = 2
[3,5,2,4,1] => [3,5,2,4,1] => [-3,-5,-2,-4,-1] => [4]
 => 1
[4,1,2,3,5] => [4,1,2,3,5] => [-4,-1,-2,-3,-5] => [4]
 => 1
[4,1,2,5,3] => [4,1,2,5,3] => [-4,-1,-2,-5,-3] => []
 => ? = 2
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [2]
 => ? = 2
[4,2,5,3,1] => [4,2,5,3,1] => [-4,-2,-5,-3,-1] => [4]
 => 1
[4,5,1,2,3] => [4,5,1,2,3] => [-4,-5,-1,-2,-3] => []
 => ? = 2
[4,5,2,3,1] => [4,5,2,3,1] => [-4,-5,-2,-3,-1] => []
 => ? = 2
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [2,2]
 => 2
[5,1,2,3,4] => [5,1,2,3,4] => [-5,-1,-2,-3,-4] => []
 => ? = 2
[5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => [2]
 => ? = 1
[5,2,4,1,3] => [5,2,4,1,3] => [-5,-2,-4,-1,-3] => [4]
 => 1
[5,3,1,4,2] => [5,3,1,4,2] => [-5,-3,-1,-4,-2] => [4]
 => 1
[5,3,4,1,2] => [5,3,4,1,2] => [-5,-3,-4,-1,-2] => []
 => ? = 2
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [2]
 => ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [-1,-2,-3,-4,-5,-6] => ?
 => ? = 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [-1,-2,-4,-5,-6,-3] => ?
 => ? = 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [-1,-2,-4,-6,-3,-5] => ?
 => ? = 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [-1,-2,-5,-3,-6,-4] => ?
 => ? = 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [-1,-2,-5,-6,-3,-4] => ?
 => ? = 2
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [-1,-2,-6,-3,-4,-5] => ?
 => ? = 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [-1,-3,-4,-5,-2,-6] => ?
 => ? = 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [-1,-3,-4,-5,-6,-2] => ?
 => ? = 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [-1,-3,-4,-6,-2,-5] => ?
 => ? = 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [-1,-3,-5,-2,-4,-6] => ?
 => ? = 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [-1,-3,-5,-2,-6,-4] => ?
 => ? = 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 2
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [-1,-3,-6,-2,-4,-5] => ?
 => ? = 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [-1,-4,-2,-5,-3,-6] => ?
 => ? = 1
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [-1,-4,-2,-5,-6,-3] => ?
 => ? = 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [-1,-4,-2,-6,-3,-5] => ?
 => ? = 2
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [-1,-4,-5,-2,-3,-6] => ?
 => ? = 2
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 2
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [-1,-4,-5,-6,-2,-3] => ?
 => ? = 2
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001605
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2,3] => [-1,-2,-3] => []
 => ? = 1
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => []
 => ? = 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => [4]
 => 1
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => [4]
 => 1
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => [4]
 => 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => [2,2]
 => 2
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => [4]
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => []
 => ? = 1
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => [4]
 => 1
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => [4]
 => 1
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => [4]
 => 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [2,2]
 => 2
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => [4]
 => 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [2]
 => ? = 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [2]
 => ? = 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [2]
 => ? = 1
[2,3,4,1,5] => [2,3,4,1,5] => [-2,-3,-4,-1,-5] => [4]
 => 1
[2,3,4,5,1] => [2,3,4,5,1] => [-2,-3,-4,-5,-1] => []
 => ? = 2
[2,3,5,1,4] => [2,3,5,1,4] => [-2,-3,-5,-1,-4] => []
 => ? = 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,-4,-1,-3,-5] => [4]
 => 1
[2,4,1,5,3] => [2,4,1,5,3] => [-2,-4,-1,-5,-3] => []
 => ? = 2
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [2]
 => ? = 2
[2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,-1,-3,-4] => []
 => ? = 2
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [2]
 => ? = 1
[3,1,4,2,5] => [3,1,4,2,5] => [-3,-1,-4,-2,-5] => [4]
 => 1
[3,1,4,5,2] => [3,1,4,5,2] => [-3,-1,-4,-5,-2] => []
 => ? = 2
[3,1,5,2,4] => [3,1,5,2,4] => [-3,-1,-5,-2,-4] => []
 => ? = 2
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [2,2]
 => 2
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [2]
 => ? = 2
[3,4,5,1,2] => [3,4,5,1,2] => [-3,-4,-5,-1,-2] => []
 => ? = 2
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [2]
 => ? = 1
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [2]
 => ? = 2
[3,5,2,4,1] => [3,5,2,4,1] => [-3,-5,-2,-4,-1] => [4]
 => 1
[4,1,2,3,5] => [4,1,2,3,5] => [-4,-1,-2,-3,-5] => [4]
 => 1
[4,1,2,5,3] => [4,1,2,5,3] => [-4,-1,-2,-5,-3] => []
 => ? = 2
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [2]
 => ? = 2
[4,2,5,3,1] => [4,2,5,3,1] => [-4,-2,-5,-3,-1] => [4]
 => 1
[4,5,1,2,3] => [4,5,1,2,3] => [-4,-5,-1,-2,-3] => []
 => ? = 2
[4,5,2,3,1] => [4,5,2,3,1] => [-4,-5,-2,-3,-1] => []
 => ? = 2
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [2,2]
 => 2
[5,1,2,3,4] => [5,1,2,3,4] => [-5,-1,-2,-3,-4] => []
 => ? = 2
[5,2,3,4,1] => [5,2,3,4,1] => [-5,-2,-3,-4,-1] => [2]
 => ? = 1
[5,2,4,1,3] => [5,2,4,1,3] => [-5,-2,-4,-1,-3] => [4]
 => 1
[5,3,1,4,2] => [5,3,1,4,2] => [-5,-3,-1,-4,-2] => [4]
 => 1
[5,3,4,1,2] => [5,3,4,1,2] => [-5,-3,-4,-1,-2] => []
 => ? = 2
[5,4,1,2,3] => [5,4,1,2,3] => [-5,-4,-1,-2,-3] => [2]
 => ? = 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [-1,-2,-3,-4,-5,-6] => ?
 => ? = 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [-1,-2,-4,-5,-6,-3] => ?
 => ? = 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [-1,-2,-4,-6,-3,-5] => ?
 => ? = 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [-1,-2,-5,-3,-6,-4] => ?
 => ? = 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [-1,-2,-5,-6,-3,-4] => ?
 => ? = 2
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [-1,-2,-6,-3,-4,-5] => ?
 => ? = 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
 => ? = 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
 => ? = 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
 => ? = 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [-1,-3,-4,-5,-2,-6] => ?
 => ? = 1
[1,3,4,5,6,2] => [1,3,4,5,6,2] => [-1,-3,-4,-5,-6,-2] => ?
 => ? = 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [-1,-3,-4,-6,-2,-5] => ?
 => ? = 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [-1,-3,-5,-2,-4,-6] => ?
 => ? = 1
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [-1,-3,-5,-2,-6,-4] => ?
 => ? = 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [-1,-3,-5,-6,-2,-4] => ?
 => ? = 2
[1,3,6,2,4,5] => [1,3,6,2,4,5] => [-1,-3,-6,-2,-4,-5] => ?
 => ? = 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [-1,-4,-2,-3,-6,-5] => ?
 => ? = 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [-1,-4,-2,-5,-3,-6] => ?
 => ? = 1
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [-1,-4,-2,-5,-6,-3] => ?
 => ? = 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [-1,-4,-2,-6,-3,-5] => ?
 => ? = 2
[1,4,5,2,3,6] => [1,4,5,2,3,6] => [-1,-4,-5,-2,-3,-6] => ?
 => ? = 2
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [-1,-4,-5,-2,-6,-3] => ?
 => ? = 2
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [-1,-4,-5,-6,-2,-3] => ?
 => ? = 2
[1,4,5,6,3,2] => [1,4,5,6,3,2] => [-1,-4,-5,-6,-3,-2] => ?
 => ? = 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.