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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001560
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St001560: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 2
[1,2,3] => 6
[1,3,2] => 6
[2,1,3] => 6
[2,3,1] => 6
[3,1,2] => 6
[3,2,1] => 6
[1,2,3,4] => 24
[1,2,4,3] => 24
[1,3,2,4] => 24
[1,3,4,2] => 24
[1,4,2,3] => 24
[1,4,3,2] => 24
[2,1,3,4] => 24
[2,1,4,3] => 24
[2,3,1,4] => 24
[2,3,4,1] => 24
[2,4,1,3] => 25
[2,4,3,1] => 24
[3,1,2,4] => 24
[3,1,4,2] => 25
[3,2,1,4] => 24
[3,2,4,1] => 24
[3,4,1,2] => 24
[3,4,2,1] => 24
[4,1,2,3] => 24
[4,1,3,2] => 24
[4,2,1,3] => 24
[4,2,3,1] => 24
[4,3,1,2] => 24
[4,3,2,1] => 24
[1,2,3,4,5] => 120
[1,2,3,5,4] => 120
[1,2,4,3,5] => 120
[1,2,4,5,3] => 120
[1,2,5,3,4] => 120
[1,2,5,4,3] => 120
[1,3,2,4,5] => 120
[1,3,2,5,4] => 120
[1,3,4,2,5] => 120
[1,3,4,5,2] => 120
[1,3,5,2,4] => 125
[1,3,5,4,2] => 120
[1,4,2,3,5] => 120
[1,4,2,5,3] => 125
[1,4,3,2,5] => 120
[1,4,3,5,2] => 120
[1,4,5,2,3] => 120
[1,4,5,3,2] => 120
Description
The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order.
Let $a(\pi)$ denote this statistic, then $a(\pi)$ is the product of [1] and [2]. A result of Sidorenko [3] implies that $a(\pi) \geq n!$ when $\pi$ is a permutation of $n$, with equality if and only if $\pi$ avoids the patterns $2413$ and $3142$. See [4] for a combinatorial proof and refinement. Upper bounds for $a(\pi)$ are given in [5].
Matching statistic: St001109
Values
[1,2] => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[2,1] => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 6
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 6
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 24
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 25
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 25
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 24
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 120
[1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 120
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 120
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(0,7),(0,9),(0,10),(0,11),(1,5),(1,6),(1,8),(1,9),(1,10),(1,11),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 120
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(1,6),(1,8),(1,10),(1,11),(1,12),(1,13),(2,5),(2,8),(2,10),(2,11),(2,12),(2,13),(3,4),(3,8),(3,10),(3,11),(3,12),(3,13),(4,7),(4,9),(4,10),(4,11),(4,12),(4,13),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(6,7),(6,9),(6,10),(6,11),(6,12),(6,13),(7,11),(7,12),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 125
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(0,4),(0,6),(0,9),(0,10),(0,11),(1,3),(1,5),(1,8),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(3,7),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 120
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(0,7),(0,9),(0,10),(0,11),(1,5),(1,6),(1,8),(1,9),(1,10),(1,11),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 120
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(1,6),(1,8),(1,10),(1,11),(1,12),(1,13),(2,5),(2,8),(2,10),(2,11),(2,12),(2,13),(3,4),(3,8),(3,10),(3,11),(3,12),(3,13),(4,7),(4,9),(4,10),(4,11),(4,12),(4,13),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(6,7),(6,9),(6,10),(6,11),(6,12),(6,13),(7,11),(7,12),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 125
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(0,6),(0,8),(0,9),(0,10),(1,4),(1,5),(1,7),(1,9),(1,10),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,6),(3,7),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 120
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,6),(0,7),(0,9),(0,11),(0,12),(1,4),(1,8),(1,10),(1,11),(1,12),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 120
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(0,5),(0,6),(0,8),(0,9),(0,10),(1,2),(1,3),(1,7),(1,8),(1,9),(1,10),(2,6),(2,7),(2,8),(2,9),(2,10),(3,5),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 120
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(0,6),(0,8),(0,9),(0,10),(1,4),(1,5),(1,7),(1,9),(1,10),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,6),(3,7),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 120
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(0,4),(0,6),(0,9),(0,10),(0,11),(1,3),(1,5),(1,8),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(3,7),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 120
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,6),(0,7),(0,9),(0,11),(0,12),(1,4),(1,8),(1,10),(1,11),(1,12),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 120
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,6),(0,7),(0,9),(0,11),(0,12),(1,4),(1,8),(1,10),(1,11),(1,12),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 120
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(0,6),(0,8),(0,9),(0,10),(1,4),(1,5),(1,7),(1,9),(1,10),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,6),(3,7),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 120
[1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 120
[2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 120
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ([(0,4),(0,5),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(0,5),(0,6),(0,8),(0,9),(0,10),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(3,5),(3,7),(3,8),(3,9),(3,10),(4,6),(4,7),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 120
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(0,5),(0,6),(0,8),(0,9),(0,10),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(3,5),(3,7),(3,8),(3,9),(3,10),(4,6),(4,7),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 120
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 120
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ([(0,5),(0,6),(0,8),(0,9),(0,10),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,7),(2,8),(2,9),(2,10),(3,5),(3,7),(3,8),(3,9),(3,10),(4,6),(4,7),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 120
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 120
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,9),(1,10),(1,11),(1,12),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,6),(3,10),(3,11),(3,12),(4,7),(4,9),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 126
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ([(0,6),(0,8),(0,9),(0,10),(1,4),(1,5),(1,7),(1,9),(1,10),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,6),(3,7),(3,8),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 120
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ([(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(1,6),(1,8),(1,10),(1,11),(1,12),(1,13),(2,5),(2,8),(2,10),(2,11),(2,12),(2,13),(3,4),(3,8),(3,10),(3,11),(3,12),(3,13),(4,7),(4,9),(4,10),(4,11),(4,12),(4,13),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(6,7),(6,9),(6,10),(6,11),(6,12),(6,13),(7,11),(7,12),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 125
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ([(0,3),(0,4),(0,11),(0,12),(0,13),(1,6),(1,7),(1,10),(1,11),(1,12),(1,13),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(3,5),(3,9),(3,10),(3,11),(3,12),(3,13),(4,5),(4,8),(4,10),(4,11),(4,12),(4,13),(5,6),(5,7),(5,10),(5,11),(5,12),(5,13),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 128
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,6),(0,7),(0,9),(0,11),(0,12),(1,4),(1,8),(1,10),(1,11),(1,12),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 120
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,6),(0,7),(0,9),(0,11),(0,12),(1,4),(1,8),(1,10),(1,11),(1,12),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 120
[2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,8),(0,9),(0,10),(0,11),(0,12),(1,5),(1,6),(1,10),(1,11),(1,12),(2,3),(2,8),(2,9),(2,10),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,7),(5,9),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 126
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ([(0,4),(0,6),(0,9),(0,10),(0,11),(1,3),(1,5),(1,8),(1,10),(1,11),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(3,7),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 120
[2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,9),(1,10),(1,11),(1,12),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,6),(3,10),(3,11),(3,12),(4,7),(4,9),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 126
[2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,8),(0,9),(0,10),(0,11),(0,12),(1,5),(1,6),(1,10),(1,11),(1,12),(2,3),(2,8),(2,9),(2,10),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,7),(5,9),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 126
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(2,3),(2,14),(3,13),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,12),(5,13),(5,14),(6,7),(6,8),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,9),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,14),(12,14),(13,14)],15)
=> ([(0,8),(0,9),(0,12),(0,13),(0,14),(1,6),(1,7),(1,11),(1,13),(1,14),(2,7),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(3,7),(3,8),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(6,11),(6,12),(6,13),(6,14),(7,11),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? = 121
[2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,6),(0,7),(0,9),(0,11),(0,12),(1,4),(1,8),(1,10),(1,11),(1,12),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,9),(5,10),(5,11),(5,12),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 120
[2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ([(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,9),(1,10),(1,11),(1,12),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,6),(3,10),(3,11),(3,12),(4,7),(4,9),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 126
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ([(0,5),(0,7),(0,8),(0,9),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 120
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 120
Description
The number of proper colourings of a graph with as few colours as possible.
By definition, this is the evaluation of the chromatic polynomial at the first nonnegative integer which is not a zero of the polynomial.
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