Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001820
(load all 2 compositions to match this statistic)
Mapping
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00191: Lattices —dual⟶ Lattices
St001820: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
 => 3
[1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
[2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
[2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
[3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
[3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7)
 => 3
[1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,4),(0,5),(1,8),(2,8),(3,8),(4,3),(4,7),(5,6),(5,7),(6,1),(6,2),(7,8)],9)
 => 4
[1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,2),(5,7),(6,1),(6,7),(7,3),(7,4)],9)
 => 4
[1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
 => 3
[1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,3),(0,6),(1,8),(2,8),(3,8),(4,2),(4,7),(5,1),(5,7),(6,4),(6,5),(7,8)],9)
 => 4
[2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,4),(0,5),(1,8),(2,8),(3,8),(4,3),(4,7),(5,6),(5,7),(6,1),(6,2),(7,8)],9)
 => 4
[2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => 3
[2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
 => 3
[2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,3),(0,6),(1,8),(2,8),(3,8),(4,2),(4,7),(5,1),(5,7),(6,4),(6,5),(7,8)],9)
 => 4
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
 => 3
[3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
 => 3
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,3),(0,6),(1,8),(2,8),(3,8),(4,2),(4,7),(5,1),(5,7),(6,4),(6,5),(7,8)],9)
 => 4
[3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
 => 3
[3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2)],8)
 => 3
[3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,4),(0,5),(1,8),(2,8),(3,8),(4,3),(4,7),(5,6),(5,7),(6,1),(6,2),(7,8)],9)
 => 4
[4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,3),(0,6),(1,8),(2,8),(3,8),(4,2),(4,7),(5,1),(5,7),(6,4),(6,5),(7,8)],9)
 => 4
[4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
 => 3
[4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(5,3),(6,1),(6,5)],8)
 => 3
[4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,2),(5,7),(6,1),(6,7),(7,3),(7,4)],9)
 => 4
[4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,4),(0,5),(1,8),(2,8),(3,8),(4,3),(4,7),(5,6),(5,7),(6,1),(6,2),(7,8)],9)
 => 4
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
 => 2
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
 => 2
[2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
 => 2
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8)
 => 2
[3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
[3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8)
 => 2
Description
The size of the image of the pop stack sorting operator. The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Matching statistic: St001160
(load all 8 compositions to match this statistic)
Mapping
St001160: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 2 = 3 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 2 = 3 - 1
[1,2,4,3] => 3 = 4 - 1
[1,3,2,4] => 3 = 4 - 1
[1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 3 = 4 - 1
[2,1,3,4] => 3 = 4 - 1
[2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => 3 = 4 - 1
[2,4,1,3] => 0 = 1 - 1
[2,4,3,1] => 2 = 3 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 0 = 1 - 1
[3,2,1,4] => 3 = 4 - 1
[3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 3 = 4 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 2 = 3 - 1
[4,2,1,3] => 2 = 3 - 1
[4,2,3,1] => 3 = 4 - 1
[4,3,1,2] => 3 = 4 - 1
[1,3,5,2,4] => 1 = 2 - 1
[1,4,2,5,3] => 1 = 2 - 1
[2,3,5,1,4] => 1 = 2 - 1
[2,4,1,3,5] => 1 = 2 - 1
[2,4,1,5,3] => 0 = 1 - 1
[2,4,5,1,3] => 1 = 2 - 1
[2,5,1,3,4] => 1 = 2 - 1
[2,5,1,4,3] => 1 = 2 - 1
[2,5,3,1,4] => 0 = 1 - 1
[2,5,4,1,3] => 1 = 2 - 1
[3,1,4,2,5] => 1 = 2 - 1
[3,1,4,5,2] => 1 = 2 - 1
[3,1,5,2,4] => 0 = 1 - 1
[3,1,5,4,2] => 1 = 2 - 1
[3,2,5,1,4] => 1 = 2 - 1
[3,4,1,5,2] => 1 = 2 - 1
[3,5,1,2,4] => 1 = 2 - 1
[3,5,1,4,2] => 0 = 1 - 1
[3,5,2,1,4] => 1 = 2 - 1
[2,4,1,5,7,3,6] => ? = 1 - 1
[2,4,1,6,3,7,5] => ? = 1 - 1
[2,4,1,7,5,3,6] => ? = 1 - 1
[2,4,6,1,3,7,5] => ? = 1 - 1
[2,4,6,1,5,7,3] => ? = 1 - 1
[2,4,6,1,7,3,5] => ? = 1 - 1
[2,4,6,1,7,5,3] => ? = 1 - 1
[2,4,6,3,1,7,5] => ? = 1 - 1
[2,4,6,3,7,1,5] => ? = 1 - 1
[2,4,7,1,5,3,6] => ? = 1 - 1
[2,4,7,1,6,3,5] => ? = 1 - 1
[2,4,7,3,5,1,6] => ? = 1 - 1
[2,4,7,3,6,1,5] => ? = 1 - 1
[2,4,7,5,1,3,6] => ? = 1 - 1
[2,4,7,5,1,6,3] => ? = 1 - 1
[2,4,7,5,3,1,6] => ? = 1 - 1
[2,5,1,3,7,4,6] => ? = 1 - 1
[2,5,1,4,7,3,6] => ? = 1 - 1
[2,5,1,6,3,7,4] => ? = 1 - 1
[2,5,1,6,4,7,3] => ? = 1 - 1
[2,5,1,7,3,6,4] => ? = 1 - 1
[2,5,1,7,4,6,3] => ? = 1 - 1
[2,5,3,1,7,4,6] => ? = 1 - 1
[2,5,3,6,1,7,4] => ? = 1 - 1
[2,5,3,7,1,4,6] => ? = 1 - 1
[2,5,3,7,1,6,4] => ? = 1 - 1
[2,5,3,7,4,1,6] => ? = 1 - 1
[2,5,7,1,3,6,4] => ? = 1 - 1
[2,5,7,1,4,6,3] => ? = 1 - 1
[2,5,7,3,1,4,6] => ? = 1 - 1
[2,5,7,3,1,6,4] => ? = 1 - 1
[2,5,7,3,6,1,4] => ? = 1 - 1
[2,5,7,4,1,3,6] => ? = 1 - 1
[2,5,7,4,1,6,3] => ? = 1 - 1
[2,6,1,3,5,7,4] => ? = 1 - 1
[2,6,1,4,7,3,5] => ? = 1 - 1
[2,6,1,4,7,5,3] => ? = 1 - 1
[2,6,1,5,3,7,4] => ? = 1 - 1
[2,6,3,1,4,7,5] => ? = 1 - 1
[2,6,3,1,5,7,4] => ? = 1 - 1
[2,6,3,5,1,7,4] => ? = 1 - 1
[2,6,3,5,7,1,4] => ? = 1 - 1
[2,6,3,7,4,1,5] => ? = 1 - 1
[2,6,3,7,5,1,4] => ? = 1 - 1
[2,6,4,1,3,7,5] => ? = 1 - 1
[2,6,4,1,5,7,3] => ? = 1 - 1
[2,6,4,1,7,3,5] => ? = 1 - 1
[2,6,4,1,7,5,3] => ? = 1 - 1
[2,6,4,7,1,3,5] => ? = 1 - 1
[2,6,4,7,1,5,3] => ? = 1 - 1
Description
The number of proper blocks (or intervals) of a permutations. Let $\pi = [\pi_1,\ldots,\pi_n]$ be a permutation. A block (or interval) of $\pi$ is then a consecutive subpattern $\pi_i,\ldots,\pi_{i+k}$ whose values form a set of contiguous integers.
Matching statistic: St001603
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
[1] => [1]
 => []
 => ?
 => ? = 1
[1,2] => [1,1]
 => [1]
 => []
 => ? = 1
[2,1] => [2]
 => []
 => ?
 => ? = 1
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3
[1,3,2] => [2,1]
 => [1]
 => []
 => ? = 2
[2,1,3] => [2,1]
 => [1]
 => []
 => ? = 2
[2,3,1] => [2,1]
 => [1]
 => []
 => ? = 2
[3,1,2] => [2,1]
 => [1]
 => []
 => ? = 2
[3,2,1] => [3]
 => []
 => ?
 => ? = 3
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4
[1,3,4,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3
[1,4,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3
[1,4,3,2] => [3,1]
 => [1]
 => []
 => ? = 4
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? = 3
[2,3,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3
[2,3,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4
[2,4,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 1
[2,4,3,1] => [3,1]
 => [1]
 => []
 => ? = 3
[3,1,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? = 1
[3,2,1,4] => [3,1]
 => [1]
 => []
 => ? = 4
[3,2,4,1] => [3,1]
 => [1]
 => []
 => ? = 3
[3,4,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3
[3,4,2,1] => [3,1]
 => [1]
 => []
 => ? = 4
[4,1,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4
[4,1,3,2] => [3,1]
 => [1]
 => []
 => ? = 3
[4,2,1,3] => [3,1]
 => [1]
 => []
 => ? = 3
[4,2,3,1] => [3,1]
 => [1]
 => []
 => ? = 4
[4,3,1,2] => [3,1]
 => [1]
 => []
 => ? = 4
[1,3,5,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2
[1,4,2,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 2
[2,3,5,1,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2
[2,4,1,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2
[2,4,1,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 1
[2,4,5,1,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2
[2,5,1,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2
[2,5,1,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2
[2,5,3,1,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 1
[2,5,4,1,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2
[3,1,4,2,5] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 2
[3,1,4,5,2] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 2
[3,1,5,2,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 1
[3,1,5,4,2] => [3,2]
 => [2]
 => []
 => ? = 2
[3,2,5,1,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2
[3,4,1,5,2] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 2
[3,5,1,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2
[3,5,1,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 1
[3,5,2,1,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2
[2,4,6,1,3,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,1,5,7,3,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,4,1,6,3,7,5] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
[2,4,6,1,3,7,5] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,4,6,1,5,7,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,6,1,7,3,5] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,4,6,3,7,1,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,7,1,5,3,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,7,1,6,3,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,7,3,5,1,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,7,3,6,1,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,4,7,5,1,3,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,5,1,3,7,4,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,5,1,4,7,3,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,5,1,6,3,7,4] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
[2,5,3,7,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,5,7,1,3,6,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,5,7,1,4,6,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,5,7,3,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,5,7,3,6,1,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,5,7,4,1,3,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,6,1,3,5,7,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,6,1,4,7,3,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,6,3,5,7,1,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,6,4,7,1,3,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,7,3,5,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[2,7,4,6,1,3,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,1,4,6,2,7,5] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
[3,1,5,2,7,4,6] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
[3,1,5,7,2,4,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[3,1,6,2,4,7,5] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
[3,5,1,4,7,2,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,5,1,6,2,7,4] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 1
[3,5,1,7,2,4,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[3,5,2,7,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,5,7,1,4,2,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,5,7,1,4,6,2] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,5,7,1,6,2,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,5,7,2,4,1,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,5,7,2,6,1,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,6,1,4,7,2,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,6,1,5,7,2,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,6,2,4,7,1,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,6,2,5,7,1,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,7,1,4,6,2,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,7,1,5,2,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,7,2,4,6,1,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[3,7,2,5,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[4,1,3,5,7,2,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[4,1,5,2,7,3,6] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 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: St001604
Mapping
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 25%
Values
[1] => [1]
 => []
 => ?
 => ? = 1 - 1
[1,2] => [1,1]
 => [1]
 => []
 => ? = 1 - 1
[2,1] => [2]
 => []
 => ?
 => ? = 1 - 1
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 1
[1,3,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 1
[2,1,3] => [2,1]
 => [1]
 => []
 => ? = 2 - 1
[2,3,1] => [2,1]
 => [1]
 => []
 => ? = 2 - 1
[3,1,2] => [2,1]
 => [1]
 => []
 => ? = 2 - 1
[3,2,1] => [3]
 => []
 => ?
 => ? = 3 - 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4 - 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4 - 1
[1,3,4,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 1
[1,4,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 1
[1,4,3,2] => [3,1]
 => [1]
 => []
 => ? = 4 - 1
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4 - 1
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? = 3 - 1
[2,3,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 1
[2,3,4,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4 - 1
[2,4,1,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
[2,4,3,1] => [3,1]
 => [1]
 => []
 => ? = 3 - 1
[3,1,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 1
[3,1,4,2] => [2,2]
 => [2]
 => []
 => ? = 1 - 1
[3,2,1,4] => [3,1]
 => [1]
 => []
 => ? = 4 - 1
[3,2,4,1] => [3,1]
 => [1]
 => []
 => ? = 3 - 1
[3,4,1,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 3 - 1
[3,4,2,1] => [3,1]
 => [1]
 => []
 => ? = 4 - 1
[4,1,2,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? = 4 - 1
[4,1,3,2] => [3,1]
 => [1]
 => []
 => ? = 3 - 1
[4,2,1,3] => [3,1]
 => [1]
 => []
 => ? = 3 - 1
[4,2,3,1] => [3,1]
 => [1]
 => []
 => ? = 4 - 1
[4,3,1,2] => [3,1]
 => [1]
 => []
 => ? = 4 - 1
[1,3,5,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2 - 1
[1,4,2,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 2 - 1
[2,3,5,1,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2 - 1
[2,4,1,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2 - 1
[2,4,1,5,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
[2,4,5,1,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2 - 1
[2,5,1,3,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2 - 1
[2,5,1,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,5,3,1,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
[2,5,4,1,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
[3,1,4,2,5] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 2 - 1
[3,1,4,5,2] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 2 - 1
[3,1,5,2,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 1 - 1
[3,1,5,4,2] => [3,2]
 => [2]
 => []
 => ? = 2 - 1
[3,2,5,1,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
[3,4,1,5,2] => [2,2,1]
 => [2,1]
 => [1]
 => ? = 2 - 1
[3,5,1,2,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => ? = 2 - 1
[3,5,1,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 1 - 1
[3,5,2,1,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? = 2 - 1
[2,4,6,1,3,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,1,5,7,3,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,1,6,3,7,5] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[2,4,6,1,3,7,5] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,6,1,5,7,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,6,1,7,3,5] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,6,3,7,1,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,7,1,5,3,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,7,1,6,3,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,7,3,5,1,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,7,3,6,1,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,4,7,5,1,3,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,5,1,3,7,4,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,5,1,4,7,3,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,5,1,6,3,7,4] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[2,5,3,7,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,5,7,1,3,6,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,5,7,1,4,6,3] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,5,7,3,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,5,7,3,6,1,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,5,7,4,1,3,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,6,1,3,5,7,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,6,1,4,7,3,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,6,3,5,7,1,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,6,4,7,1,3,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,7,3,5,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[2,7,4,6,1,3,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,1,4,6,2,7,5] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[3,1,5,2,7,4,6] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[3,1,5,7,2,4,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,1,6,2,4,7,5] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[3,5,1,4,7,2,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,5,1,6,2,7,4] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[3,5,1,7,2,4,6] => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,5,2,7,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,5,7,1,4,2,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,5,7,1,4,6,2] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,5,7,1,6,2,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,5,7,2,4,1,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,5,7,2,6,1,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,6,1,4,7,2,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,6,1,5,7,2,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,6,2,4,7,1,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,6,2,5,7,1,4] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,7,1,4,6,2,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,7,1,5,2,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,7,2,4,6,1,5] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[3,7,2,5,1,4,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[4,1,3,5,7,2,6] => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[4,1,5,2,7,3,6] => [2,2,2,1]
 => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
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: St001875
Mapping
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 100%
Values
[1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1 + 2
[1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 3 + 2
[1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 3 + 2
[1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 4 + 2
[1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 4 + 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 6 = 4 + 2
[2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 4 + 2
[2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 3 + 2
[2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 6 = 4 + 2
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 6 = 4 + 2
[3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 3 + 2
[3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 4 + 2
[4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 6 = 4 + 2
[4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 4 + 2
[4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 4 + 2
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,3,5,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[1,3,6,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[1,5,2,4,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[2,3,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,3,6,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,4,1,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[2,4,1,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,4,1,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,4,1,6,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,4,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,4,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,4,6,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,5,1,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,5,1,4,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,5,1,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,5,1,6,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,5,3,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[2,5,3,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,5,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,5,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,5,6,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,6,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,6,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,6,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,6,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,6,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,6,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[2,6,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[2,6,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[3,1,4,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,1,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
 => ?
 => ? = 2 + 2
[3,1,5,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,1,5,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[3,1,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[3,1,6,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[3,1,6,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1 + 2
[3,1,6,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[3,2,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[3,2,6,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[3,4,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
[3,4,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => ?
 => ? = 2 + 2
Description
The number of simple modules with projective dimension at most 1.