Processing math: 100%

Your data matches 72 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000071
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
Description
The number of maximal chains in a poset.
Matching statistic: St000909
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000910
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000910: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,2] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[3,2,1] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 5
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 3
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
Description
The number of maximal chains of minimal length in a poset.
Matching statistic: St000849
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,3,2] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[3,2,1] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 4 = 5 - 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
Description
The number of 1/3-balanced pairs in a poset. A pair of elements x,y of a poset is α-balanced if the proportion of linear extensions where x comes before y is between α and 1−α. Kislitsyn [1] conjectured that every poset which is not a chain has a 1/3-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a (1−√5)/10-balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a 1/3-balanced pair.
Matching statistic: St001633
(load all 2 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 0 = 1 - 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,3,2] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,3] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[3,2,1] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 5 - 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001330
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,3,2] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,1,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,3,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[5,1,2,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000422
(load all 3 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[1,3,2] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[2,1,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[3,1,2] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 5 + 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 5 + 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 + 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 5 + 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 5 + 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 5 + 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 5 + 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 + 1
[4,1,2,3] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 4 = 3 + 1
[4,2,1,3] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[4,2,3,1] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 4 = 3 + 1
[4,3,1,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 4 = 3 + 1
[4,3,2,1] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 4 = 3 + 1
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [2,1,3,4,5,6] => ([(4,5)],6)
 => 2 = 1 + 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => [2,1,3,4,5,6] => ([(4,5)],6)
 => 2 = 1 + 1
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph Kn equals 2n−2. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001232
(load all 8 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1] => [1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,2] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,3] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,3,1] => [1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,2] => [1,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1 + 1
[3,2,1] => [1,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,4,3] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,2,4] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,3,4,2] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,2,3] => [1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[1,4,3,2] => [1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[2,1,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,1,4,3] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,3,1,4] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,4,1,3] => [1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[2,4,3,1] => [1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[3,1,2,4] => [1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[3,1,4,2] => [1,3,4,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[3,2,1,4] => [1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[3,2,4,1] => [1,3,4,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[3,4,1,2] => [1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[3,4,2,1] => [1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 5 + 1
[4,1,2,3] => [1,4,3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[4,1,3,2] => [1,4,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 3 + 1
[4,2,1,3] => [1,4,3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => ? = 1 + 1
[4,2,3,1] => [1,4,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 3 + 1
[4,3,1,2] => [1,4,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 3 + 1
[4,3,2,1] => [1,4,2,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ? = 3 + 1
[5,1,2,3,4] => [1,5,4,3,2] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 1 + 1
[5,2,1,3,4] => [1,5,4,3,2] => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => ? = 1 + 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 1 + 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ? = 1 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001879
(load all 10 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 1
[2,1] => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 1
[1,2,3] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2
[1,3,2] => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2
[2,1,3] => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 2
[2,3,1] => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2
[3,1,2] => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1
[3,2,1] => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 5
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 5
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 5
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 5
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 5
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1
[6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[.,.],[.,.]],[[.,.],.]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001630
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1 - 1
[2,1] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 1 - 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1 = 2 - 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? = 1 - 1
[3,2,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? = 1 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 3 - 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 3 - 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 3 - 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 3 - 1
[1,4,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? = 5 - 1
[1,4,3,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? = 5 - 1
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 3 - 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 3 - 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 3 - 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2 = 3 - 1
[2,4,1,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? = 5 - 1
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? = 5 - 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 5 - 1
[3,1,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 5 - 1
[3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 5 - 1
[3,2,4,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 5 - 1
[3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 5 - 1
[3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? = 5 - 1
[4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? = 1 - 1
[4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 3 - 1
[4,2,1,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? = 1 - 1
[4,2,3,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 3 - 1
[4,3,1,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 3 - 1
[4,3,2,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? = 3 - 1
[5,1,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ? = 1 - 1
[5,2,1,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ? = 1 - 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ? = 1 - 1
[6,2,1,3,4,5] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ([],1)
 => ? = 1 - 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001623The number of doubly irreducible elements of a lattice. St000850The number of 1/2-balanced pairs in a poset. St001469The holeyness of a permutation. St001665The number of pure excedances of a permutation. St000166The depth minus 1 of an ordered tree. St000454The largest eigenvalue of a graph if it is integral. St001095The number of non-isomorphic posets with precisely one further covering relation. St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001667The maximal size of a pair of weak twins for a permutation. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000906The length of the shortest maximal chain in a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000550The number of modular elements of a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000259The diameter of a connected graph. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001060The distinguishing index of a graph. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001877Number of indecomposable injective modules with projective dimension 2. St000551The number of left modular elements of a lattice. St000219The number of occurrences of the pattern 231 in a permutation. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset.