- FindStat

Your data matches 42 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000708

Mapping

Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The product of the parts of an integer partition.

Matching statistic: St001632
(load all 6 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 18%

Values

[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,1,4] => [1,3,2,4] => [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
[3,2,4,1] => [1,3,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2
[3,4,2,1] => [1,3,2,4] => [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)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,3,1] => [1,4,2,3] => [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)
 => ? = 3
[4,3,2,1] => [1,4,2,3] => [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
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(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
[1,4,3,5,2] => [1,2,4,5,3] => [4,5,1,2,3] => ([(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
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => ([(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)
 => ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(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
[1,5,3,4,2] => [1,2,5,3,4] => [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)
 => ? = 3
[1,5,4,3,2] => [1,2,5,3,4] => [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
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,4,1,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(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)
 => ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(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
[2,4,3,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => ([(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
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(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)
 => ? = 3
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(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
[2,5,3,4,1] => [1,2,5,3,4] => [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)
 => ? = 3
[2,5,4,1,3] => [1,2,5,3,4] => [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)
 => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => [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
[3,1,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => ([(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)
 => ? = 1
[3,1,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => ([(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
[3,2,1,4,5] => [1,3,2,4,5] => [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
[3,2,1,5,4] => [1,3,2,4,5] => [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
[3,2,4,1,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(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
[3,2,4,5,1] => [1,3,4,5,2] => [3,4,5,1,2] => ([(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
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,4,1,2] => ([(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
[3,2,5,4,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(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)
 => ? = 4
[3,4,2,1,5] => [1,3,2,4,5] => [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)
 => ? = 3
[3,4,2,5,1] => [1,3,2,4,5] => [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)
 => ? = 3
[3,4,5,2,1] => [1,3,5,2,4] => [5,3,1,2,4] => ([(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)
 => ? = 4
[3,5,1,2,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(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)
 => ? = 3
[3,5,2,4,1] => [1,3,2,5,4] => [3,5,1,2,4] => ([(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)
 => ? = 4
[3,5,4,2,1] => [1,3,4,2,5] => [3,4,1,2,5] => ([(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)
 => ? = 3
[4,1,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(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
[4,1,2,5,3] => [1,4,5,3,2] => [4,5,3,1,2] => ([(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
[4,1,5,2,3] => [1,4,2,3,5] => [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)
 => ? = 3
[4,2,3,1,5] => [1,4,2,3,5] => [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)
 => ? = 3
[4,2,3,5,1] => [1,4,5,2,3] => [4,1,5,2,3] => ([(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)
 => ? = 3
[4,2,5,3,1] => [1,4,3,5,2] => [4,3,5,1,2] => ([(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)
 => ? = 4
[4,3,1,5,2] => [1,4,5,2,3] => [4,1,5,2,3] => ([(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)
 => ? = 1
[4,3,2,1,5] => [1,4,2,3,5] => [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
[4,3,2,5,1] => [1,4,5,2,3] => [4,1,5,2,3] => ([(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
[4,3,5,2,1] => [1,4,2,3,5] => [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)
 => ? = 3
[4,5,1,3,2] => [1,4,3,2,5] => [4,3,1,2,5] => ([(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
[4,5,2,1,3] => [1,4,2,5,3] => [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
[4,5,2,3,1] => [1,4,3,2,5] => [4,3,1,2,5] => ([(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)
 => ? = 4
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,1,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,1,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,5,1,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1

Description

The number of indecomposable injective modules

$I$ with

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

Matching statistic: St000307
(load all 5 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%

Values

[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,1,4] => [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
[3,2,4,1] => [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
[3,4,2,1] => [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)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,3,1] => [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)
 => ? = 3
[4,3,2,1] => [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
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,4,3] => [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
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,4,2] => [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
[1,4,3,2,5] => [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
[1,4,3,5,2] => [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
[1,4,5,3,2] => [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)
 => ? = 3
[1,5,2,3,4] => [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
[1,5,3,4,2] => [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)
 => ? = 3
[1,5,4,3,2] => [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
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,4,3] => [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
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [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)
 => ? = 1
[2,3,5,4,1] => [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
[2,4,1,5,3] => [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)
 => ? = 1
[2,4,3,1,5] => [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
[2,4,3,5,1] => [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
[2,4,5,3,1] => [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)
 => ? = 3
[2,5,1,3,4] => [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
[2,5,3,4,1] => [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)
 => ? = 3
[2,5,4,1,3] => [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)
 => ? = 1
[2,5,4,3,1] => [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
[3,1,4,5,2] => [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)
 => ? = 1
[3,1,5,4,2] => [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
[3,2,1,4,5] => [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
[3,2,1,5,4] => [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
[3,2,4,1,5] => [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
[3,2,4,5,1] => [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
[3,2,5,1,4] => [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
[3,2,5,4,1] => [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)
 => ? = 4
[3,4,2,1,5] => [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)
 => ? = 3
[3,4,2,5,1] => [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)
 => ? = 3
[3,4,5,2,1] => [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)
 => ? = 4
[3,5,1,2,4] => [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)
 => ? = 3
[3,5,2,4,1] => [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)
 => ? = 4
[3,5,4,2,1] => [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)
 => ? = 3
[4,1,2,3,5] => [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
[4,1,2,5,3] => [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
[4,1,5,2,3] => [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)
 => ? = 3
[4,2,3,1,5] => [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)
 => ? = 3
[4,2,3,5,1] => [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)
 => ? = 3
[4,2,5,3,1] => [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)
 => ? = 4
[4,3,1,5,2] => [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)
 => ? = 1
[4,3,2,1,5] => [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
[4,3,2,5,1] => [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
[4,3,5,2,1] => [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)
 => ? = 3
[4,5,1,3,2] => [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
[4,5,2,1,3] => [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
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1

Description

The number of rowmotion orbits of a poset. Rowmotion is an operation on order ideals in a poset

$P$ . It sends an order ideal

$I$ to the order ideal generated by the minimal antichain of

$P \setminus I$ .

Matching statistic: St000298

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%

Values

[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,1,4] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,4,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,4,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,3,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,3,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [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] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,4,2] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,4,3,2,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,3,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,5,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,2,3,4] => [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)
 => ([(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
[1,5,3,4,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,4,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[2,3,5,4,1] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,4,1,5,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[2,4,3,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,3,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,5,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,1,3,4] => [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)
 => ([(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
[2,5,3,4,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,4,1,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[2,5,4,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,1,4,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[3,1,5,4,2] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2
[3,2,1,4,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,1,5,4] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,4,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,2,4,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,5,1,4] => [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)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,2,5,4,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4
[3,4,2,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[3,4,2,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[3,4,5,2,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4
[3,5,1,2,4] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 3
[3,5,2,4,1] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4
[3,5,4,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,1,2,3,5] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,1,2,5,3] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,1,5,2,3] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,2,3,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,2,3,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 3
[4,2,5,3,1] => [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)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4
[4,3,1,5,2] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 1
[4,3,2,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[4,3,2,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 2
[4,3,5,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,5,1,3,2] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,5,2,1,3] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2
[1,2,3,5,6,4] => [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
[1,2,4,3,6,5] => [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
[1,2,4,5,3,6] => [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] => [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
[1,3,2,4,6,5] => [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
[1,3,2,5,4,6] => [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
[1,3,2,5,6,4] => [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
[1,3,4,2,5,6] => [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
[1,3,4,2,6,5] => [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
[1,3,4,5,2,6] => [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
[1,3,4,5,6,2] => [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
[2,1,3,4,6,5] => [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
[2,1,3,5,4,6] => [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
[2,1,3,5,6,4] => [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
[2,1,4,3,5,6] => [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
[2,1,4,3,6,5] => [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
[2,1,4,5,3,6] => [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
[2,1,4,5,6,3] => [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
[2,3,1,4,5,6] => [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
[2,3,1,4,6,5] => [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
[2,3,1,5,4,6] => [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
[2,3,1,5,6,4] => [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
[2,3,4,1,5,6] => [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
[2,3,4,1,6,5] => [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
[2,3,4,5,1,6] => [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
[2,3,4,5,6,1] => [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
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1

Description

The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.

Matching statistic: St000845

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%

Values

[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,1,4] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,4,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,4,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,3,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,3,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [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] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,4,2] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,4,3,2,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,3,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,5,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,2,3,4] => [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)
 => ([(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
[1,5,3,4,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,4,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[2,3,5,4,1] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,4,1,5,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[2,4,3,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,3,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,5,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,1,3,4] => [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)
 => ([(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
[2,5,3,4,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,4,1,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[2,5,4,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,1,4,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[3,1,5,4,2] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2
[3,2,1,4,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,1,5,4] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,4,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,2,4,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,5,1,4] => [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)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,2,5,4,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4
[3,4,2,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[3,4,2,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[3,4,5,2,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4
[3,5,1,2,4] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 3
[3,5,2,4,1] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4
[3,5,4,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,1,2,3,5] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,1,2,5,3] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,1,5,2,3] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,2,3,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,2,3,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 3
[4,2,5,3,1] => [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)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4
[4,3,1,5,2] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 1
[4,3,2,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[4,3,2,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 2
[4,3,5,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,5,1,3,2] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,5,2,1,3] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2
[1,2,3,5,6,4] => [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
[1,2,4,3,6,5] => [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
[1,2,4,5,3,6] => [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] => [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
[1,3,2,4,6,5] => [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
[1,3,2,5,4,6] => [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
[1,3,2,5,6,4] => [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
[1,3,4,2,5,6] => [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
[1,3,4,2,6,5] => [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
[1,3,4,5,2,6] => [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
[1,3,4,5,6,2] => [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
[2,1,3,4,6,5] => [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
[2,1,3,5,4,6] => [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
[2,1,3,5,6,4] => [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
[2,1,4,3,5,6] => [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
[2,1,4,3,6,5] => [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
[2,1,4,5,3,6] => [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
[2,1,4,5,6,3] => [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
[2,3,1,4,5,6] => [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
[2,3,1,4,6,5] => [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
[2,3,1,5,4,6] => [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
[2,3,1,5,6,4] => [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
[2,3,4,1,5,6] => [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
[2,3,4,1,6,5] => [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
[2,3,4,5,1,6] => [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
[2,3,4,5,6,1] => [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
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1

Description

The maximal number of elements covered by an element in a poset.

Matching statistic: St000846

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%

Values

[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,1,4] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,4,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,4,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,3,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,3,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [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] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,4,2] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,4,3,2,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,3,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,5,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,2,3,4] => [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)
 => ([(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
[1,5,3,4,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,4,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[2,3,5,4,1] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,4,1,5,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[2,4,3,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,3,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,5,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,1,3,4] => [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)
 => ([(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
[2,5,3,4,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,4,1,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[2,5,4,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,1,4,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[3,1,5,4,2] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2
[3,2,1,4,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,1,5,4] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,4,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,2,4,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,5,1,4] => [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)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,2,5,4,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4
[3,4,2,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[3,4,2,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[3,4,5,2,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4
[3,5,1,2,4] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 3
[3,5,2,4,1] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4
[3,5,4,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,1,2,3,5] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,1,2,5,3] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,1,5,2,3] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,2,3,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,2,3,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 3
[4,2,5,3,1] => [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)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4
[4,3,1,5,2] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 1
[4,3,2,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[4,3,2,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 2
[4,3,5,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,5,1,3,2] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,5,2,1,3] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2
[1,2,3,5,6,4] => [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
[1,2,4,3,6,5] => [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
[1,2,4,5,3,6] => [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] => [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
[1,3,2,4,6,5] => [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
[1,3,2,5,4,6] => [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
[1,3,2,5,6,4] => [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
[1,3,4,2,5,6] => [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
[1,3,4,2,6,5] => [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
[1,3,4,5,2,6] => [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
[1,3,4,5,6,2] => [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
[2,1,3,4,6,5] => [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
[2,1,3,5,4,6] => [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
[2,1,3,5,6,4] => [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
[2,1,4,3,5,6] => [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
[2,1,4,3,6,5] => [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
[2,1,4,5,3,6] => [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
[2,1,4,5,6,3] => [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
[2,3,1,4,5,6] => [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
[2,3,1,4,6,5] => [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
[2,3,1,5,4,6] => [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
[2,3,1,5,6,4] => [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
[2,3,4,1,5,6] => [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
[2,3,4,1,6,5] => [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
[2,3,4,5,1,6] => [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
[2,3,4,5,6,1] => [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
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1

Description

The maximal number of elements covering an element of a poset.

Matching statistic: St000632

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%

Values

[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 2 - 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 2 - 1
[3,2,1,4] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[3,2,4,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[3,4,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 2 - 1
[4,2,3,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[4,3,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,2,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 - 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,3,5,4,2] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 - 1
[1,4,3,2,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[1,4,3,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[1,4,5,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[1,5,2,3,4] => [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)
 => ([(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 - 1
[1,5,3,4,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[1,5,4,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,1,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 - 1
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,3,5,1,4] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[2,3,5,4,1] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 - 1
[2,4,1,5,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 - 1
[2,4,3,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[2,4,3,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[2,4,5,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[2,5,1,3,4] => [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)
 => ([(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 - 1
[2,5,3,4,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[2,5,4,1,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 - 1
[2,5,4,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[3,1,4,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 - 1
[3,1,5,4,2] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2 - 1
[3,2,1,4,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[3,2,1,5,4] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[3,2,4,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2 - 1
[3,2,4,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2 - 1
[3,2,5,1,4] => [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)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2 - 1
[3,2,5,4,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4 - 1
[3,4,2,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[3,4,2,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[3,4,5,2,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4 - 1
[3,5,1,2,4] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 3 - 1
[3,5,2,4,1] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4 - 1
[3,5,4,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3 - 1
[4,1,2,3,5] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2 - 1
[4,1,2,5,3] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2 - 1
[4,1,5,2,3] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3 - 1
[4,2,3,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3 - 1
[4,2,3,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 3 - 1
[4,2,5,3,1] => [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)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4 - 1
[4,3,1,5,2] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 1 - 1
[4,3,2,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2 - 1
[4,3,2,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 2 - 1
[4,3,5,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3 - 1
[4,5,1,3,2] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2 - 1
[4,5,2,1,3] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2 - 1
[1,2,3,5,6,4] => [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)
 => 0 = 1 - 1
[1,2,4,3,6,5] => [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)
 => 0 = 1 - 1
[1,2,4,5,3,6] => [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)
 => 0 = 1 - 1
[1,2,4,5,6,3] => [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)
 => 0 = 1 - 1
[1,3,2,4,6,5] => [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)
 => 0 = 1 - 1
[1,3,2,5,4,6] => [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)
 => 0 = 1 - 1
[1,3,2,5,6,4] => [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)
 => 0 = 1 - 1
[1,3,4,2,5,6] => [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)
 => 0 = 1 - 1
[1,3,4,2,6,5] => [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)
 => 0 = 1 - 1
[1,3,4,5,2,6] => [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)
 => 0 = 1 - 1
[1,3,4,5,6,2] => [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)
 => 0 = 1 - 1
[2,1,3,4,6,5] => [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)
 => 0 = 1 - 1
[2,1,3,5,4,6] => [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)
 => 0 = 1 - 1
[2,1,3,5,6,4] => [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)
 => 0 = 1 - 1
[2,1,4,3,5,6] => [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)
 => 0 = 1 - 1
[2,1,4,3,6,5] => [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)
 => 0 = 1 - 1
[2,1,4,5,3,6] => [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)
 => 0 = 1 - 1
[2,1,4,5,6,3] => [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)
 => 0 = 1 - 1
[2,3,1,4,5,6] => [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)
 => 0 = 1 - 1
[2,3,1,4,6,5] => [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)
 => 0 = 1 - 1
[2,3,1,5,4,6] => [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)
 => 0 = 1 - 1
[2,3,1,5,6,4] => [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)
 => 0 = 1 - 1
[2,3,4,1,5,6] => [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)
 => 0 = 1 - 1
[2,3,4,1,6,5] => [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)
 => 0 = 1 - 1
[2,3,4,5,1,6] => [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)
 => 0 = 1 - 1
[2,3,4,5,6,1] => [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)
 => 0 = 1 - 1
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0 = 1 - 1

Description

The jump number of the poset. A jump in a linear extension

$e_1, \dots, e_n$ of a poset

$P$ is a pair

$(e_i, e_{i+1})$ so that

$e_{i+1}$ does not cover

$e_i$ in

$P$ . The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

Matching statistic: St000640

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000640: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%

Values

[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,2,1,4] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,2,4,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[3,4,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,2,3,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3
[4,3,2,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => [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] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,5,4,2] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[1,4,3,2,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,3,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[1,4,5,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,2,3,4] => [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)
 => ([(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
[1,5,3,4,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[1,5,4,3,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,1,5,4,3] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,5,1,4] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[2,3,5,4,1] => [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)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2
[2,4,1,5,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[2,4,3,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,3,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[2,4,5,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,1,3,4] => [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)
 => ([(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
[2,5,3,4,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[2,5,4,1,3] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[2,5,4,3,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,1,4,5,2] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1
[3,1,5,4,2] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2
[3,2,1,4,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,1,5,4] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,4,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,2,4,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 2
[3,2,5,1,4] => [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)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 2
[3,2,5,4,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4
[3,4,2,1,5] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[3,4,2,5,1] => [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)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3
[3,4,5,2,1] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 4
[3,5,1,2,4] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 3
[3,5,2,4,1] => [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)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4
[3,5,4,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,1,2,3,5] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,1,2,5,3] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,1,5,2,3] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,2,3,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,2,3,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 3
[4,2,5,3,1] => [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)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4
[4,3,1,5,2] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 1
[4,3,2,1,5] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 2
[4,3,2,5,1] => [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)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 2
[4,3,5,2,1] => [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)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 3
[4,5,1,3,2] => [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)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 2
[4,5,2,1,3] => [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)
 => ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
 => ? = 2
[1,2,3,5,6,4] => [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
[1,2,4,3,6,5] => [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
[1,2,4,5,3,6] => [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] => [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
[1,3,2,4,6,5] => [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
[1,3,2,5,4,6] => [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
[1,3,2,5,6,4] => [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
[1,3,4,2,5,6] => [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
[1,3,4,2,6,5] => [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
[1,3,4,5,2,6] => [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
[1,3,4,5,6,2] => [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
[2,1,3,4,6,5] => [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
[2,1,3,5,4,6] => [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
[2,1,3,5,6,4] => [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
[2,1,4,3,5,6] => [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
[2,1,4,3,6,5] => [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
[2,1,4,5,3,6] => [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
[2,1,4,5,6,3] => [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
[2,3,1,4,5,6] => [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
[2,3,1,4,6,5] => [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
[2,3,1,5,4,6] => [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
[2,3,1,5,6,4] => [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
[2,3,4,1,5,6] => [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
[2,3,4,1,6,5] => [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
[2,3,4,5,1,6] => [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
[2,3,4,5,6,1] => [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

Description

The rank of the largest boolean interval in a poset.

Matching statistic: St000771

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 9%

Values

[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 3
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[3,1,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 4
[3,4,2,1,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[3,4,2,5,1] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 4
[3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 3
[3,5,2,4,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 4
[3,5,4,2,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[4,1,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 3
[4,2,5,3,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[4,3,1,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 1
[4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,1,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,1,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,1,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,1,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,1,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,1,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,5,1,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([],7)
 => ([],1)
 => 1

Description

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$0$ , which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$2$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore statistic

$1$ .

Matching statistic: St000772

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 9%

Values

[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 3
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 3
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 1
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[3,1,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 4
[3,4,2,1,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[3,4,2,5,1] => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 4
[3,5,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 3
[3,5,2,4,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 4
[3,5,4,2,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[4,1,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 2
[4,1,5,2,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3
[4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 3
[4,2,5,3,1] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
[4,3,1,5,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 1
[4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,1,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,1,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,1,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,1,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,1,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,1,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,1,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,5,1,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 1
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([],7)
 => ([],1)
 => 1

Description

The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$1$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic

$1$ . The graphs with statistic

$n-1$ ,

$n-2$ and

$n-3$ have been characterised, see [1].

The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001890The maximum magnitude of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001964The interval resolution global dimension of a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000189The number of elements in the poset. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.