searching the database
Your data matches 30 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001880
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[5,1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,4,6,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,6,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,2,6,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,3,4,6,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,3,6,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,6,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,2,3,5,6] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,2,3,6,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,2,6,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,6,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,6,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,6,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,6,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,6,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,6,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,6,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,6,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,6,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[6,1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6,1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[6,1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[6,1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[6,1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[6,1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000454
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 14%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 14%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 2
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 2
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 2
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 2
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[5,1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 2
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 2
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 2
[1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,2,4,6,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 2
[1,2,6,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,3,2,6,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,3,4,6,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,3,6,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,3,6,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,4,2,3,5,6] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,4,2,3,6,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,4,2,6,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 2
[1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 2
[1,4,3,6,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 2
[1,4,6,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,4,6,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 2
[1,6,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 2
[1,6,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,6,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[1,6,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,6,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[1,6,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 2
[6,1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 2
[6,1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[6,1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 5 - 2
[6,1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[6,1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 4 - 2
[6,1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 2
[1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 6 - 2
[1,2,3,4,5,7,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 6 - 2
[1,2,3,4,7,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 6 - 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 6 - 2
[1,2,3,5,4,7,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 6 - 2
[1,2,3,5,7,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 6 - 2
[1,2,3,7,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 6 - 2
[1,2,3,7,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 6 - 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[1,2,4,3,5,7,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[1,2,4,3,7,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 5 - 2
[1,2,4,5,3,7,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 5 - 2
[1,2,4,5,7,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 5 - 2
[1,2,4,7,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[1,2,4,7,5,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 5 - 2
[1,2,5,3,4,6,7] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 5 - 2
[1,3,4,5,2,6,7] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,3,4,5,2,7,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,3,4,5,7,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,3,4,7,5,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,3,7,4,5,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,4,5,2,3,6,7] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,4,5,2,3,7,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,4,5,2,7,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,4,5,7,2,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,4,7,5,2,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,5,2,3,4,6,7] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,5,2,3,4,7,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,5,2,3,7,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,5,2,7,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,5,7,2,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,7,3,4,5,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,7,4,5,2,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,7,5,2,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[7,1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[7,1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[7,1,5,2,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
[1,5,6,2,3,8,4,7] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)
=> 2 = 4 - 2
[1,6,2,8,3,4,5,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)
=> 2 = 4 - 2
[1,6,8,2,3,4,5,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)
=> 2 = 4 - 2
[1,6,2,3,4,8,5,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)
=> 2 = 4 - 2
[1,6,2,3,4,5,7,8] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)
=> 2 = 4 - 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001875
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 57%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 57%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[5,1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 5 + 1
[1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 5 + 1
[1,2,4,6,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 5 + 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,2,6,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 5 + 1
[1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6 = 5 + 1
[1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6 = 5 + 1
[1,3,2,6,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6 = 5 + 1
[1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,3,4,6,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,3,6,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6 = 5 + 1
[1,3,6,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,4,2,3,5,6] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,4,2,3,6,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,4,2,6,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 1
[1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 1
[1,4,3,6,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 1
[1,4,6,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,4,6,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 1
[1,6,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,6,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 5 + 1
[1,6,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6 = 5 + 1
[1,6,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,6,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[1,6,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 1
[6,1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[6,1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 5 + 1
[6,1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 6 = 5 + 1
[6,1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[6,1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 4 + 1
[6,1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 1 + 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[1,2,3,5,7,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[1,2,3,7,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,4,5,7,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,4,7,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 1
[1,2,4,7,5,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,5,3,7,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,5,4,3,6,7] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 2 + 1
[1,2,5,4,3,7,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 2 + 1
[1,2,5,4,7,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 2 + 1
[1,2,5,7,3,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,5,7,4,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 2 + 1
[1,2,7,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[1,2,7,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[1,2,7,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 6 + 1
[1,2,7,4,5,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,7,5,3,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 5 + 1
[1,2,7,5,4,3,6] => [1,2,5,4,3,6] => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,5),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,6),(6,2),(6,3),(6,4),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 2 + 1
[1,3,2,4,5,6,7] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6 + 1
[1,3,2,4,5,7,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6 + 1
[1,3,2,4,7,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6 + 1
[1,3,2,7,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6 + 1
[1,3,4,2,5,6,7] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 5 + 1
[1,3,4,2,5,7,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 5 + 1
[1,3,4,2,7,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 5 + 1
[1,7,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[7,1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000528
Mp00252: Permutations —restriction⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Mp00209: Permutations —pattern poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,4,5] => [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)
=> 4
[1,3,2,5,4] => [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)
=> 4
[1,3,5,2,4] => [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)
=> 4
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,5,3,2,4] => [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)
=> 4
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[5,1,3,2,4] => [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)
=> 4
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,4,3,5,6] => [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)
=> ? = 5
[1,2,4,3,6,5] => [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)
=> ? = 5
[1,2,4,6,3,5] => [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)
=> ? = 5
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,6,4,3,5] => [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)
=> ? = 5
[1,3,2,4,5,6] => [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)
=> ? = 5
[1,3,2,4,6,5] => [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)
=> ? = 5
[1,3,2,6,4,5] => [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)
=> ? = 5
[1,3,4,2,5,6] => [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)
=> ? = 4
[1,3,4,2,6,5] => [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)
=> ? = 4
[1,3,4,6,2,5] => [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)
=> ? = 4
[1,3,6,2,4,5] => [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)
=> ? = 5
[1,3,6,4,2,5] => [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)
=> ? = 4
[1,4,2,3,5,6] => [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)
=> ? = 4
[1,4,2,3,6,5] => [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)
=> ? = 4
[1,4,2,6,3,5] => [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)
=> ? = 4
[1,4,3,2,5,6] => [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)
=> ? = 1
[1,4,3,2,6,5] => [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)
=> ? = 1
[1,4,3,6,2,5] => [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)
=> ? = 1
[1,4,6,2,3,5] => [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)
=> ? = 4
[1,4,6,3,2,5] => [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)
=> ? = 1
[1,6,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,6,2,4,3,5] => [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)
=> ? = 5
[1,6,3,2,4,5] => [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)
=> ? = 5
[1,6,3,4,2,5] => [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)
=> ? = 4
[1,6,4,2,3,5] => [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)
=> ? = 4
[1,6,4,3,2,5] => [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)
=> ? = 1
[6,1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6,1,2,4,3,5] => [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)
=> ? = 5
[6,1,3,2,4,5] => [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)
=> ? = 5
[6,1,3,4,2,5] => [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)
=> ? = 4
[6,1,4,2,3,5] => [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)
=> ? = 4
[6,1,4,3,2,5] => [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)
=> ? = 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,4,5,7,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,4,7,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6
[1,2,3,5,4,7,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6
[1,2,3,5,7,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6
[1,2,3,7,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,7,5,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6
[1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6
[1,2,4,3,5,7,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6
[1,2,4,3,7,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6
[1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,2,4,5,3,7,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,2,4,5,7,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,2,4,7,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6
[1,2,4,7,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,2,5,3,4,6,7] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,2,5,3,4,7,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,2,5,3,7,4,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,2,5,4,3,6,7] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2
[1,2,5,4,3,7,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2
[1,2,5,4,7,3,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2
[1,2,5,7,3,4,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5
[1,2,5,7,4,3,6] => [1,2,5,4,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 2
[1,2,7,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,7,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[7,1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,3,8,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,3,4,8,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,3,4,5,8,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St001631
Mp00252: Permutations —restriction⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001631: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Mp00209: Permutations —pattern poset⟶ Posets
St001631: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,3,2,4,5] => [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)
=> ? = 4 - 1
[1,3,2,5,4] => [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)
=> ? = 4 - 1
[1,3,5,2,4] => [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)
=> ? = 4 - 1
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,5,3,2,4] => [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)
=> ? = 4 - 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[5,1,3,2,4] => [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)
=> ? = 4 - 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,4,3,5,6] => [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)
=> ? = 5 - 1
[1,2,4,3,6,5] => [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)
=> ? = 5 - 1
[1,2,4,6,3,5] => [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)
=> ? = 5 - 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,6,4,3,5] => [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)
=> ? = 5 - 1
[1,3,2,4,5,6] => [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)
=> ? = 5 - 1
[1,3,2,4,6,5] => [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)
=> ? = 5 - 1
[1,3,2,6,4,5] => [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)
=> ? = 5 - 1
[1,3,4,2,5,6] => [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)
=> ? = 4 - 1
[1,3,4,2,6,5] => [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)
=> ? = 4 - 1
[1,3,4,6,2,5] => [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)
=> ? = 4 - 1
[1,3,6,2,4,5] => [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)
=> ? = 5 - 1
[1,3,6,4,2,5] => [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)
=> ? = 4 - 1
[1,4,2,3,5,6] => [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)
=> ? = 4 - 1
[1,4,2,3,6,5] => [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)
=> ? = 4 - 1
[1,4,2,6,3,5] => [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)
=> ? = 4 - 1
[1,4,3,2,5,6] => [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)
=> ? = 1 - 1
[1,4,3,2,6,5] => [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)
=> ? = 1 - 1
[1,4,3,6,2,5] => [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)
=> ? = 1 - 1
[1,4,6,2,3,5] => [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)
=> ? = 4 - 1
[1,4,6,3,2,5] => [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)
=> ? = 1 - 1
[1,6,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,6,2,4,3,5] => [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)
=> ? = 5 - 1
[1,6,3,2,4,5] => [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)
=> ? = 5 - 1
[1,6,3,4,2,5] => [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)
=> ? = 4 - 1
[1,6,4,2,3,5] => [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)
=> ? = 4 - 1
[1,6,4,3,2,5] => [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)
=> ? = 1 - 1
[6,1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[6,1,2,4,3,5] => [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)
=> ? = 5 - 1
[6,1,3,2,4,5] => [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)
=> ? = 5 - 1
[6,1,3,4,2,5] => [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)
=> ? = 4 - 1
[6,1,4,2,3,5] => [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)
=> ? = 4 - 1
[6,1,4,3,2,5] => [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)
=> ? = 1 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[1,2,3,5,7,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[1,2,3,7,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,7,5,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[1,2,4,3,7,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,4,5,7,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,4,7,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[1,2,4,7,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,5,3,7,4,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,7,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,7,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[7,1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,8,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,4,8,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,4,5,8,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
Description
The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.
Matching statistic: St001879
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Mp00209: Permutations —pattern poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,3,2,4,5] => [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)
=> ? = 4 - 1
[1,3,2,5,4] => [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)
=> ? = 4 - 1
[1,3,5,2,4] => [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)
=> ? = 4 - 1
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,5,3,2,4] => [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)
=> ? = 4 - 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[5,1,3,2,4] => [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)
=> ? = 4 - 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,4,3,5,6] => [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)
=> ? = 5 - 1
[1,2,4,3,6,5] => [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)
=> ? = 5 - 1
[1,2,4,6,3,5] => [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)
=> ? = 5 - 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,6,4,3,5] => [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)
=> ? = 5 - 1
[1,3,2,4,5,6] => [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)
=> ? = 5 - 1
[1,3,2,4,6,5] => [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)
=> ? = 5 - 1
[1,3,2,6,4,5] => [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)
=> ? = 5 - 1
[1,3,4,2,5,6] => [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)
=> ? = 4 - 1
[1,3,4,2,6,5] => [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)
=> ? = 4 - 1
[1,3,4,6,2,5] => [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)
=> ? = 4 - 1
[1,3,6,2,4,5] => [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)
=> ? = 5 - 1
[1,3,6,4,2,5] => [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)
=> ? = 4 - 1
[1,4,2,3,5,6] => [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)
=> ? = 4 - 1
[1,4,2,3,6,5] => [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)
=> ? = 4 - 1
[1,4,2,6,3,5] => [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)
=> ? = 4 - 1
[1,4,3,2,5,6] => [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)
=> ? = 1 - 1
[1,4,3,2,6,5] => [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)
=> ? = 1 - 1
[1,4,3,6,2,5] => [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)
=> ? = 1 - 1
[1,4,6,2,3,5] => [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)
=> ? = 4 - 1
[1,4,6,3,2,5] => [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)
=> ? = 1 - 1
[1,6,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,6,2,4,3,5] => [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)
=> ? = 5 - 1
[1,6,3,2,4,5] => [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)
=> ? = 5 - 1
[1,6,3,4,2,5] => [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)
=> ? = 4 - 1
[1,6,4,2,3,5] => [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)
=> ? = 4 - 1
[1,6,4,3,2,5] => [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)
=> ? = 1 - 1
[6,1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[6,1,2,4,3,5] => [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)
=> ? = 5 - 1
[6,1,3,2,4,5] => [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)
=> ? = 5 - 1
[6,1,3,4,2,5] => [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)
=> ? = 4 - 1
[6,1,4,2,3,5] => [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)
=> ? = 4 - 1
[6,1,4,3,2,5] => [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)
=> ? = 1 - 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[1,2,3,5,7,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[1,2,3,7,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,2,3,7,5,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[1,2,4,3,7,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,4,5,3,7,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,4,5,7,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,4,7,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[1,2,4,7,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,5,3,4,7,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,5,3,7,4,6] => [1,2,5,3,4,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 5 - 1
[1,2,7,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,7,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[7,1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,8,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,4,8,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,2,3,4,5,8,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 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: St001330
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
[1,2,3,4] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,4,3] => [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,4,2,3] => [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,1,2,3] => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,5,3,2,4] => [4,2,3,5,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5,1,3,2,4] => [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,3,4,6,5] => [5,6,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,3,6,4,5] => [5,4,6,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,2,4,3,6,5] => [5,6,3,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,2,4,6,3,5] => [5,3,6,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,2,6,3,4,5] => [5,4,3,6,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,2,6,4,3,5] => [5,3,4,6,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,3,2,4,5,6] => [6,5,4,2,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,3,2,4,6,5] => [5,6,4,2,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,3,2,6,4,5] => [5,4,6,2,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,3,4,2,5,6] => [6,5,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,3,4,2,6,5] => [5,6,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,3,4,6,2,5] => [5,2,6,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,3,6,2,4,5] => [5,4,2,6,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,3,6,4,2,5] => [5,2,4,6,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,2,3,5,6] => [6,5,3,2,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,2,3,6,5] => [5,6,3,2,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,2,6,3,5] => [5,3,6,2,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,3,2,5,6] => [6,5,2,3,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,4,3,2,6,5] => [5,6,2,3,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,4,3,6,2,5] => [5,2,6,3,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,4,6,2,3,5] => [5,3,2,6,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,6,3,2,5] => [5,2,3,6,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,6,2,3,4,5] => [5,4,3,2,6,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,6,2,4,3,5] => [5,3,4,2,6,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,6,3,2,4,5] => [5,4,2,3,6,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[1,6,3,4,2,5] => [5,2,4,3,6,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,6,4,2,3,5] => [5,3,2,4,6,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,6,4,3,2,5] => [5,2,3,4,6,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[6,1,2,3,4,5] => [5,4,3,2,1,6] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[6,1,2,4,3,5] => [5,3,4,2,1,6] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[6,1,3,2,4,5] => [5,4,2,3,1,6] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 5
[6,1,3,4,2,5] => [5,2,4,3,1,6] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[6,1,4,2,3,5] => [5,3,2,4,1,6] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[6,1,4,3,2,5] => [5,2,3,4,1,6] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [6,3,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[1,2,7,3,4,5,6] => [6,5,4,3,7,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,7,2,3,4,5,6] => [6,5,4,3,2,7,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,2,3,4,5,6,8,7] => [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,8,2,3,4,5,6,7] => [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,2,3,8,4,5,6,7] => [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,2,3,4,8,5,6,7] => [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[1,2,3,4,5,8,6,7] => [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
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: St001392
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 3
[1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 3
[4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4
[1,2,3,5,4] => [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)
=> [5,3]
=> 4
[1,2,5,3,4] => [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)
=> [5,3,1]
=> 4
[1,3,2,4,5] => [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)
=> [5,3,2]
=> 4
[1,3,2,5,4] => [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)
=> [5,3,2]
=> 4
[1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> 4
[1,5,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 4
[1,5,3,2,4] => [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)
=> [5,3,2,2]
=> 4
[5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4
[5,1,3,2,4] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> [5,3,2]
=> 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 5
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> [6,4]
=> 5
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 5
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 5
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ? = 5
[1,2,4,6,3,5] => [1,2,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 5
[1,2,6,3,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 5
[1,2,6,4,3,5] => [1,2,4,6,5,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ? = 5
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ? = 5
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ? = 5
[1,3,2,6,4,5] => [1,3,2,6,5,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ? = 5
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ? = 4
[1,3,4,2,6,5] => [1,4,2,3,6,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ? = 4
[1,3,4,6,2,5] => [1,6,5,2,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4
[1,3,6,2,4,5] => [1,6,5,4,2,3] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ?
=> ? = 5
[1,3,6,4,2,5] => [1,4,6,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
=> ?
=> ? = 4
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4
[1,4,2,3,6,5] => [1,4,3,2,6,5] => ([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ? = 4
[1,4,2,6,3,5] => [1,6,5,3,2,4] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ?
=> ? = 4
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ? = 1
[1,4,3,2,6,5] => [1,3,4,2,6,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ? = 1
[1,4,3,6,2,5] => [1,3,6,5,2,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ? = 1
[1,4,6,2,3,5] => [1,4,2,6,5,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ? = 4
[1,4,6,3,2,5] => [1,6,5,2,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 1
[1,6,2,3,4,5] => [1,6,5,4,3,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> [6,4]
=> 5
[1,6,2,4,3,5] => [1,4,6,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 5
[1,6,3,2,4,5] => [1,3,6,5,4,2] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ? = 5
[1,6,3,4,2,5] => [1,3,4,6,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 4
[1,6,4,2,3,5] => [1,6,5,3,4,2] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ?
=> ? = 4
[1,6,4,3,2,5] => [1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ? = 1
[6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 5
[6,1,2,4,3,5] => [4,6,5,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ? = 5
[6,1,3,2,4,5] => [3,6,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 5
[6,1,3,4,2,5] => [3,4,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4
[6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 4
[6,1,4,3,2,5] => [4,3,6,5,2,1] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ? = 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)
=> [7]
=> 6
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> 6
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> 6
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ?
=> ? = 6
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(0,3),(0,4),(0,5),(1,2),(1,11),(1,16),(2,9),(2,17),(3,6),(3,12),(4,1),(4,10),(4,12),(5,6),(5,10),(5,12),(6,15),(8,7),(9,8),(9,13),(10,11),(10,15),(10,16),(11,9),(11,14),(11,17),(12,15),(12,16),(13,7),(14,13),(15,14),(16,14),(16,17),(17,8),(17,13)],18)
=> ?
=> ? = 6
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ?
=> ? = 6
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> 6
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => ([(0,1),(0,2),(0,5),(0,6),(1,10),(1,20),(2,11),(2,20),(3,4),(3,13),(3,14),(3,21),(4,7),(4,8),(4,19),(5,10),(5,12),(5,20),(6,3),(6,11),(6,12),(6,20),(7,17),(8,17),(8,18),(10,15),(11,14),(11,21),(12,13),(12,15),(12,21),(13,8),(13,16),(13,19),(14,7),(14,19),(15,16),(16,18),(17,9),(18,9),(19,17),(19,18),(20,15),(20,21),(21,16),(21,19)],22)
=> ?
=> ? = 6
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ?
=> ? = 6
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(0,1),(0,3),(0,4),(0,5),(1,7),(1,17),(2,8),(2,9),(2,21),(3,10),(3,13),(3,17),(4,2),(4,12),(4,13),(4,17),(5,7),(5,10),(5,12),(7,19),(8,16),(8,20),(9,15),(9,16),(10,14),(10,19),(10,21),(11,6),(12,8),(12,14),(12,19),(13,9),(13,14),(13,21),(14,15),(14,16),(14,20),(15,11),(15,18),(16,11),(16,18),(17,19),(17,21),(18,6),(19,20),(20,18),(21,15),(21,20)],22)
=> ?
=> ? = 6
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => ([(0,4),(0,5),(0,6),(1,18),(2,8),(2,11),(3,7),(3,12),(3,19),(4,13),(4,14),(5,3),(5,13),(5,15),(6,2),(6,14),(6,15),(7,16),(8,1),(8,20),(9,17),(9,18),(11,9),(11,20),(12,9),(12,16),(12,20),(13,7),(13,19),(14,8),(14,19),(15,11),(15,12),(15,19),(16,17),(17,10),(18,10),(19,16),(19,20),(20,17),(20,18)],21)
=> ?
=> ? = 6
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ?
=> ? = 5
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ([(0,2),(0,4),(0,5),(0,6),(1,13),(1,22),(1,23),(2,9),(2,18),(3,10),(3,12),(3,23),(4,11),(4,14),(4,18),(5,3),(5,14),(5,15),(5,18),(6,1),(6,9),(6,11),(6,15),(8,19),(8,21),(9,22),(10,17),(10,20),(11,16),(11,22),(11,23),(12,8),(12,17),(12,24),(13,8),(13,20),(13,24),(14,10),(14,16),(14,23),(15,12),(15,13),(15,16),(15,22),(16,17),(16,20),(16,24),(17,19),(17,21),(18,22),(18,23),(19,7),(20,19),(20,21),(21,7),(22,24),(23,20),(23,24),(24,21)],25)
=> ?
=> ? = 5
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ?
=> ? = 5
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => ([(0,4),(0,5),(0,6),(1,17),(2,7),(2,11),(3,1),(3,10),(3,12),(4,13),(4,14),(5,3),(5,14),(5,15),(6,2),(6,13),(6,15),(7,19),(9,20),(9,21),(10,17),(10,18),(11,9),(11,19),(12,9),(12,17),(12,18),(13,7),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,18),(16,19),(17,21),(18,20),(18,21),(19,20),(20,8),(21,8)],22)
=> ?
=> ? = 6
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => ([(0,3),(0,4),(0,5),(0,6),(1,20),(1,27),(2,7),(2,8),(2,9),(3,11),(3,12),(3,14),(4,12),(4,13),(4,15),(5,2),(5,14),(5,15),(5,16),(6,1),(6,11),(6,13),(6,16),(7,21),(7,23),(7,28),(8,21),(8,22),(9,22),(9,23),(11,17),(11,27),(12,19),(12,20),(12,27),(13,18),(13,20),(13,27),(14,8),(14,17),(14,19),(15,9),(15,18),(15,19),(16,7),(16,17),(16,18),(16,27),(17,21),(17,28),(18,23),(18,24),(18,28),(19,22),(19,24),(19,28),(20,24),(21,25),(22,25),(22,26),(23,25),(23,26),(24,26),(25,10),(26,10),(27,24),(27,28),(28,25),(28,26)],29)
=> ?
=> ? = 5
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ?
=> ? = 5
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => ([(0,4),(0,5),(0,6),(1,9),(1,22),(2,11),(2,22),(3,10),(3,12),(4,3),(4,13),(4,15),(5,1),(5,13),(5,14),(6,2),(6,14),(6,15),(8,19),(9,17),(10,18),(10,21),(11,8),(11,21),(12,8),(12,18),(13,10),(13,16),(13,22),(14,9),(14,16),(14,22),(15,11),(15,12),(15,16),(16,17),(16,18),(16,21),(17,20),(18,19),(18,20),(19,7),(20,7),(21,19),(21,20),(22,17),(22,21)],23)
=> ?
=> ? = 5
[1,2,5,3,7,4,6] => [1,2,7,6,4,3,5] => ([(0,1),(0,3),(0,4),(0,6),(1,14),(1,21),(2,7),(2,12),(2,18),(3,15),(3,16),(3,21),(4,2),(4,16),(4,17),(4,21),(5,10),(5,11),(5,13),(6,5),(6,14),(6,15),(6,17),(7,24),(9,26),(9,27),(10,23),(11,20),(11,23),(12,9),(12,24),(12,25),(13,9),(13,20),(13,23),(14,10),(14,22),(15,11),(15,19),(15,22),(16,7),(16,18),(16,19),(17,12),(17,13),(17,19),(17,22),(18,24),(18,25),(19,20),(19,24),(19,25),(20,26),(20,27),(21,18),(21,22),(22,23),(22,25),(23,26),(24,27),(25,26),(25,27),(26,8),(27,8)],28)
=> ?
=> ? = 5
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ?
=> ? = 2
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => ([(0,2),(0,4),(0,5),(0,6),(1,13),(1,22),(1,23),(2,9),(2,18),(3,10),(3,12),(3,23),(4,11),(4,14),(4,18),(5,3),(5,14),(5,15),(5,18),(6,1),(6,9),(6,11),(6,15),(8,19),(8,21),(9,22),(10,17),(10,20),(11,16),(11,22),(11,23),(12,8),(12,17),(12,24),(13,8),(13,20),(13,24),(14,10),(14,16),(14,23),(15,12),(15,13),(15,16),(15,22),(16,17),(16,20),(16,24),(17,19),(17,21),(18,22),(18,23),(19,7),(20,19),(20,21),(21,7),(22,24),(23,20),(23,24),(24,21)],25)
=> ?
=> ? = 2
[1,2,5,4,7,3,6] => [1,2,4,7,6,3,5] => ?
=> ?
=> ? = 2
[1,2,5,7,3,4,6] => [1,2,5,3,7,6,4] => ?
=> ?
=> ? = 5
[1,2,5,7,4,3,6] => [1,2,7,6,3,5,4] => ([(0,2),(0,3),(0,4),(0,6),(1,11),(1,25),(2,12),(2,17),(3,13),(3,15),(3,17),(4,1),(4,13),(4,14),(4,17),(5,8),(5,9),(5,10),(6,5),(6,12),(6,14),(6,15),(8,19),(8,22),(9,22),(10,19),(10,22),(10,24),(11,18),(12,9),(12,23),(13,11),(13,16),(13,25),(14,10),(14,16),(14,23),(14,25),(15,8),(15,16),(15,23),(16,18),(16,19),(16,24),(17,23),(17,25),(18,21),(19,20),(19,21),(20,7),(21,7),(22,20),(23,22),(23,24),(24,20),(24,21),(25,18),(25,24)],26)
=> ?
=> ? = 2
[1,2,7,3,4,5,6] => [1,2,7,6,5,4,3] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> 6
[1,7,2,3,4,5,6] => [1,7,6,5,4,3,2] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> 6
[7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 6
[8,1,2,3,4,5,6,7] => [8,7,6,5,4,3,2,1] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> 7
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000147
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 4 = 3 + 1
[1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 4 = 3 + 1
[4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[1,2,3,5,4] => [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)
=> [5,3]
=> 5 = 4 + 1
[1,2,5,3,4] => [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)
=> [5,3,1]
=> 5 = 4 + 1
[1,3,2,4,5] => [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)
=> [5,3,2]
=> 5 = 4 + 1
[1,3,2,5,4] => [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)
=> [5,3,2]
=> 5 = 4 + 1
[1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> 5 = 4 + 1
[1,5,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 5 = 4 + 1
[1,5,3,2,4] => [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)
=> [5,3,2,2]
=> 5 = 4 + 1
[5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[5,1,3,2,4] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> [5,3,2]
=> 5 = 4 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> [6,4]
=> 6 = 5 + 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 6 = 5 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 5 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ? = 5 + 1
[1,2,4,6,3,5] => [1,2,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 5 + 1
[1,2,6,3,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 6 = 5 + 1
[1,2,6,4,3,5] => [1,2,4,6,5,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ? = 5 + 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ? = 5 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ? = 5 + 1
[1,3,2,6,4,5] => [1,3,2,6,5,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ? = 5 + 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ? = 4 + 1
[1,3,4,2,6,5] => [1,4,2,3,6,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ? = 4 + 1
[1,3,4,6,2,5] => [1,6,5,2,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4 + 1
[1,3,6,2,4,5] => [1,6,5,4,2,3] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ?
=> ? = 5 + 1
[1,3,6,4,2,5] => [1,4,6,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
=> ?
=> ? = 4 + 1
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4 + 1
[1,4,2,3,6,5] => [1,4,3,2,6,5] => ([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ? = 4 + 1
[1,4,2,6,3,5] => [1,6,5,3,2,4] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ?
=> ? = 4 + 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ? = 1 + 1
[1,4,3,2,6,5] => [1,3,4,2,6,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ? = 1 + 1
[1,4,3,6,2,5] => [1,3,6,5,2,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ? = 1 + 1
[1,4,6,2,3,5] => [1,4,2,6,5,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ? = 4 + 1
[1,4,6,3,2,5] => [1,6,5,2,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 1 + 1
[1,6,2,3,4,5] => [1,6,5,4,3,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> [6,4]
=> 6 = 5 + 1
[1,6,2,4,3,5] => [1,4,6,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 5 + 1
[1,6,3,2,4,5] => [1,3,6,5,4,2] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ? = 5 + 1
[1,6,3,4,2,5] => [1,3,4,6,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 4 + 1
[1,6,4,2,3,5] => [1,6,5,3,4,2] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ?
=> ? = 4 + 1
[1,6,4,3,2,5] => [1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ? = 1 + 1
[6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6 = 5 + 1
[6,1,2,4,3,5] => [4,6,5,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ? = 5 + 1
[6,1,3,2,4,5] => [3,6,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 5 + 1
[6,1,3,4,2,5] => [3,4,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4 + 1
[6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 4 + 1
[6,1,4,3,2,5] => [4,3,6,5,2,1] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ? = 1 + 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)
=> [7]
=> 7 = 6 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> 7 = 6 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> 7 = 6 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ?
=> ? = 6 + 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(0,3),(0,4),(0,5),(1,2),(1,11),(1,16),(2,9),(2,17),(3,6),(3,12),(4,1),(4,10),(4,12),(5,6),(5,10),(5,12),(6,15),(8,7),(9,8),(9,13),(10,11),(10,15),(10,16),(11,9),(11,14),(11,17),(12,15),(12,16),(13,7),(14,13),(15,14),(16,14),(16,17),(17,8),(17,13)],18)
=> ?
=> ? = 6 + 1
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ?
=> ? = 6 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> 7 = 6 + 1
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => ([(0,1),(0,2),(0,5),(0,6),(1,10),(1,20),(2,11),(2,20),(3,4),(3,13),(3,14),(3,21),(4,7),(4,8),(4,19),(5,10),(5,12),(5,20),(6,3),(6,11),(6,12),(6,20),(7,17),(8,17),(8,18),(10,15),(11,14),(11,21),(12,13),(12,15),(12,21),(13,8),(13,16),(13,19),(14,7),(14,19),(15,16),(16,18),(17,9),(18,9),(19,17),(19,18),(20,15),(20,21),(21,16),(21,19)],22)
=> ?
=> ? = 6 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ?
=> ? = 6 + 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(0,1),(0,3),(0,4),(0,5),(1,7),(1,17),(2,8),(2,9),(2,21),(3,10),(3,13),(3,17),(4,2),(4,12),(4,13),(4,17),(5,7),(5,10),(5,12),(7,19),(8,16),(8,20),(9,15),(9,16),(10,14),(10,19),(10,21),(11,6),(12,8),(12,14),(12,19),(13,9),(13,14),(13,21),(14,15),(14,16),(14,20),(15,11),(15,18),(16,11),(16,18),(17,19),(17,21),(18,6),(19,20),(20,18),(21,15),(21,20)],22)
=> ?
=> ? = 6 + 1
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => ([(0,4),(0,5),(0,6),(1,18),(2,8),(2,11),(3,7),(3,12),(3,19),(4,13),(4,14),(5,3),(5,13),(5,15),(6,2),(6,14),(6,15),(7,16),(8,1),(8,20),(9,17),(9,18),(11,9),(11,20),(12,9),(12,16),(12,20),(13,7),(13,19),(14,8),(14,19),(15,11),(15,12),(15,19),(16,17),(17,10),(18,10),(19,16),(19,20),(20,17),(20,18)],21)
=> ?
=> ? = 6 + 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ?
=> ? = 5 + 1
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ([(0,2),(0,4),(0,5),(0,6),(1,13),(1,22),(1,23),(2,9),(2,18),(3,10),(3,12),(3,23),(4,11),(4,14),(4,18),(5,3),(5,14),(5,15),(5,18),(6,1),(6,9),(6,11),(6,15),(8,19),(8,21),(9,22),(10,17),(10,20),(11,16),(11,22),(11,23),(12,8),(12,17),(12,24),(13,8),(13,20),(13,24),(14,10),(14,16),(14,23),(15,12),(15,13),(15,16),(15,22),(16,17),(16,20),(16,24),(17,19),(17,21),(18,22),(18,23),(19,7),(20,19),(20,21),(21,7),(22,24),(23,20),(23,24),(24,21)],25)
=> ?
=> ? = 5 + 1
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ?
=> ? = 5 + 1
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => ([(0,4),(0,5),(0,6),(1,17),(2,7),(2,11),(3,1),(3,10),(3,12),(4,13),(4,14),(5,3),(5,14),(5,15),(6,2),(6,13),(6,15),(7,19),(9,20),(9,21),(10,17),(10,18),(11,9),(11,19),(12,9),(12,17),(12,18),(13,7),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,18),(16,19),(17,21),(18,20),(18,21),(19,20),(20,8),(21,8)],22)
=> ?
=> ? = 6 + 1
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => ([(0,3),(0,4),(0,5),(0,6),(1,20),(1,27),(2,7),(2,8),(2,9),(3,11),(3,12),(3,14),(4,12),(4,13),(4,15),(5,2),(5,14),(5,15),(5,16),(6,1),(6,11),(6,13),(6,16),(7,21),(7,23),(7,28),(8,21),(8,22),(9,22),(9,23),(11,17),(11,27),(12,19),(12,20),(12,27),(13,18),(13,20),(13,27),(14,8),(14,17),(14,19),(15,9),(15,18),(15,19),(16,7),(16,17),(16,18),(16,27),(17,21),(17,28),(18,23),(18,24),(18,28),(19,22),(19,24),(19,28),(20,24),(21,25),(22,25),(22,26),(23,25),(23,26),(24,26),(25,10),(26,10),(27,24),(27,28),(28,25),(28,26)],29)
=> ?
=> ? = 5 + 1
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ?
=> ? = 5 + 1
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => ([(0,4),(0,5),(0,6),(1,9),(1,22),(2,11),(2,22),(3,10),(3,12),(4,3),(4,13),(4,15),(5,1),(5,13),(5,14),(6,2),(6,14),(6,15),(8,19),(9,17),(10,18),(10,21),(11,8),(11,21),(12,8),(12,18),(13,10),(13,16),(13,22),(14,9),(14,16),(14,22),(15,11),(15,12),(15,16),(16,17),(16,18),(16,21),(17,20),(18,19),(18,20),(19,7),(20,7),(21,19),(21,20),(22,17),(22,21)],23)
=> ?
=> ? = 5 + 1
[1,2,5,3,7,4,6] => [1,2,7,6,4,3,5] => ([(0,1),(0,3),(0,4),(0,6),(1,14),(1,21),(2,7),(2,12),(2,18),(3,15),(3,16),(3,21),(4,2),(4,16),(4,17),(4,21),(5,10),(5,11),(5,13),(6,5),(6,14),(6,15),(6,17),(7,24),(9,26),(9,27),(10,23),(11,20),(11,23),(12,9),(12,24),(12,25),(13,9),(13,20),(13,23),(14,10),(14,22),(15,11),(15,19),(15,22),(16,7),(16,18),(16,19),(17,12),(17,13),(17,19),(17,22),(18,24),(18,25),(19,20),(19,24),(19,25),(20,26),(20,27),(21,18),(21,22),(22,23),(22,25),(23,26),(24,27),(25,26),(25,27),(26,8),(27,8)],28)
=> ?
=> ? = 5 + 1
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ?
=> ? = 2 + 1
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => ([(0,2),(0,4),(0,5),(0,6),(1,13),(1,22),(1,23),(2,9),(2,18),(3,10),(3,12),(3,23),(4,11),(4,14),(4,18),(5,3),(5,14),(5,15),(5,18),(6,1),(6,9),(6,11),(6,15),(8,19),(8,21),(9,22),(10,17),(10,20),(11,16),(11,22),(11,23),(12,8),(12,17),(12,24),(13,8),(13,20),(13,24),(14,10),(14,16),(14,23),(15,12),(15,13),(15,16),(15,22),(16,17),(16,20),(16,24),(17,19),(17,21),(18,22),(18,23),(19,7),(20,19),(20,21),(21,7),(22,24),(23,20),(23,24),(24,21)],25)
=> ?
=> ? = 2 + 1
[1,2,5,4,7,3,6] => [1,2,4,7,6,3,5] => ?
=> ?
=> ? = 2 + 1
[1,2,5,7,3,4,6] => [1,2,5,3,7,6,4] => ?
=> ?
=> ? = 5 + 1
[1,2,5,7,4,3,6] => [1,2,7,6,3,5,4] => ([(0,2),(0,3),(0,4),(0,6),(1,11),(1,25),(2,12),(2,17),(3,13),(3,15),(3,17),(4,1),(4,13),(4,14),(4,17),(5,8),(5,9),(5,10),(6,5),(6,12),(6,14),(6,15),(8,19),(8,22),(9,22),(10,19),(10,22),(10,24),(11,18),(12,9),(12,23),(13,11),(13,16),(13,25),(14,10),(14,16),(14,23),(14,25),(15,8),(15,16),(15,23),(16,18),(16,19),(16,24),(17,23),(17,25),(18,21),(19,20),(19,21),(20,7),(21,7),(22,20),(23,22),(23,24),(24,20),(24,21),(25,18),(25,24)],26)
=> ?
=> ? = 2 + 1
[1,2,7,3,4,5,6] => [1,2,7,6,5,4,3] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> 7 = 6 + 1
[1,7,2,3,4,5,6] => [1,7,6,5,4,3,2] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> 7 = 6 + 1
[7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 7 = 6 + 1
[8,1,2,3,4,5,6,7] => [8,7,6,5,4,3,2,1] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> 8 = 7 + 1
Description
The largest part of an integer partition.
Matching statistic: St000380
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 71%
Values
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 5 = 3 + 2
[1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 5 = 3 + 2
[4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 6 = 4 + 2
[1,2,3,5,4] => [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)
=> [5,3]
=> 6 = 4 + 2
[1,2,5,3,4] => [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)
=> [5,3,1]
=> 6 = 4 + 2
[1,3,2,4,5] => [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)
=> [5,3,2]
=> 6 = 4 + 2
[1,3,2,5,4] => [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)
=> [5,3,2]
=> 6 = 4 + 2
[1,3,5,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> 6 = 4 + 2
[1,5,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 6 = 4 + 2
[1,5,3,2,4] => [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)
=> [5,3,2,2]
=> 6 = 4 + 2
[5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 6 = 4 + 2
[5,1,3,2,4] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> [5,3,2]
=> 6 = 4 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 7 = 5 + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> [6,4]
=> 7 = 5 + 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 7 = 5 + 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 5 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ?
=> ? = 5 + 2
[1,2,4,6,3,5] => [1,2,6,5,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 5 + 2
[1,2,6,3,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 7 = 5 + 2
[1,2,6,4,3,5] => [1,2,4,6,5,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ? = 5 + 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ? = 5 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ?
=> ? = 5 + 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,7),(2,14),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,12),(7,12),(7,13),(9,1),(9,14),(10,6),(10,14),(11,7),(11,14),(12,8),(13,8),(14,12),(14,13)],15)
=> ?
=> ? = 5 + 2
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ? = 4 + 2
[1,3,4,2,6,5] => [1,4,2,3,6,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ? = 4 + 2
[1,3,4,6,2,5] => [1,6,5,2,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4 + 2
[1,3,6,2,4,5] => [1,6,5,4,2,3] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ?
=> ? = 5 + 2
[1,3,6,4,2,5] => [1,4,6,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,19),(2,9),(2,11),(2,12),(3,8),(3,9),(3,10),(4,7),(4,8),(4,11),(5,1),(5,7),(5,10),(5,12),(7,13),(7,19),(8,13),(8,16),(9,15),(9,16),(10,13),(10,15),(10,19),(11,14),(11,16),(11,19),(12,14),(12,15),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,17),(19,18)],20)
=> ?
=> ? = 4 + 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4 + 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => ([(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,15),(3,9),(3,10),(4,1),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,14),(9,12),(9,15),(10,7),(10,12),(11,6),(11,12),(11,15),(12,13),(12,14),(13,8),(14,8),(15,13),(15,14)],16)
=> ?
=> ? = 4 + 2
[1,4,2,6,3,5] => [1,6,5,3,2,4] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ?
=> ? = 4 + 2
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ?
=> ? = 1 + 2
[1,4,3,2,6,5] => [1,3,4,2,6,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,15),(1,17),(2,7),(2,13),(3,8),(3,9),(3,13),(4,8),(4,11),(4,13),(5,1),(5,7),(5,9),(5,11),(7,17),(8,12),(8,15),(9,12),(9,15),(9,17),(10,14),(10,16),(11,10),(11,12),(11,17),(12,14),(12,16),(13,15),(13,17),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ?
=> ? = 1 + 2
[1,4,3,6,2,5] => [1,3,6,5,2,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ?
=> ? = 1 + 2
[1,4,6,2,3,5] => [1,4,2,6,5,3] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,8),(2,18),(2,22),(3,10),(3,11),(3,19),(4,12),(4,14),(4,17),(4,19),(5,10),(5,13),(5,14),(5,17),(6,2),(6,11),(6,12),(6,13),(6,19),(7,20),(8,20),(8,21),(10,15),(10,22),(11,15),(11,18),(11,22),(12,16),(12,18),(12,22),(13,8),(13,15),(13,16),(13,22),(14,7),(14,16),(14,22),(15,20),(15,21),(16,20),(16,21),(17,7),(17,22),(18,21),(19,18),(19,22),(20,9),(21,9),(22,20),(22,21)],23)
=> ?
=> ? = 4 + 2
[1,4,6,3,2,5] => [1,6,5,2,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 1 + 2
[1,6,2,3,4,5] => [1,6,5,4,3,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> [6,4]
=> 7 = 5 + 2
[1,6,2,4,3,5] => [1,4,6,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 5 + 2
[1,6,3,2,4,5] => [1,3,6,5,4,2] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ?
=> ? = 5 + 2
[1,6,3,4,2,5] => [1,3,4,6,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ?
=> ? = 4 + 2
[1,6,4,2,3,5] => [1,6,5,3,4,2] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ?
=> ? = 4 + 2
[1,6,4,3,2,5] => [1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,6),(3,7),(3,9),(3,11),(4,6),(4,7),(4,8),(4,10),(6,14),(6,19),(7,13),(7,15),(7,19),(8,13),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,15),(11,16),(12,16),(12,19),(13,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,5),(18,5),(19,17),(19,18)],20)
=> ?
=> ? = 1 + 2
[6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 7 = 5 + 2
[6,1,2,4,3,5] => [4,6,5,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ?
=> ? = 5 + 2
[6,1,3,2,4,5] => [3,6,5,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 5 + 2
[6,1,3,4,2,5] => [3,4,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ?
=> ? = 4 + 2
[6,1,4,2,3,5] => [6,5,3,4,2,1] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ?
=> ? = 4 + 2
[6,1,4,3,2,5] => [4,3,6,5,2,1] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ?
=> ? = 1 + 2
[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)
=> [7]
=> 8 = 6 + 2
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> 8 = 6 + 2
[1,2,3,4,7,5,6] => [1,2,3,4,7,6,5] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> 8 = 6 + 2
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ?
=> ? = 6 + 2
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(0,3),(0,4),(0,5),(1,2),(1,11),(1,16),(2,9),(2,17),(3,6),(3,12),(4,1),(4,10),(4,12),(5,6),(5,10),(5,12),(6,15),(8,7),(9,8),(9,13),(10,11),(10,15),(10,16),(11,9),(11,14),(11,17),(12,15),(12,16),(13,7),(14,13),(15,14),(16,14),(16,17),(17,8),(17,13)],18)
=> ?
=> ? = 6 + 2
[1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ?
=> ? = 6 + 2
[1,2,3,7,4,5,6] => [1,2,3,7,6,5,4] => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> 8 = 6 + 2
[1,2,3,7,5,4,6] => [1,2,3,5,7,6,4] => ([(0,1),(0,2),(0,5),(0,6),(1,10),(1,20),(2,11),(2,20),(3,4),(3,13),(3,14),(3,21),(4,7),(4,8),(4,19),(5,10),(5,12),(5,20),(6,3),(6,11),(6,12),(6,20),(7,17),(8,17),(8,18),(10,15),(11,14),(11,21),(12,13),(12,15),(12,21),(13,8),(13,16),(13,19),(14,7),(14,19),(15,16),(16,18),(17,9),(18,9),(19,17),(19,18),(20,15),(20,21),(21,16),(21,19)],22)
=> ?
=> ? = 6 + 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ?
=> ? = 6 + 2
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(0,1),(0,3),(0,4),(0,5),(1,7),(1,17),(2,8),(2,9),(2,21),(3,10),(3,13),(3,17),(4,2),(4,12),(4,13),(4,17),(5,7),(5,10),(5,12),(7,19),(8,16),(8,20),(9,15),(9,16),(10,14),(10,19),(10,21),(11,6),(12,8),(12,14),(12,19),(13,9),(13,14),(13,21),(14,15),(14,16),(14,20),(15,11),(15,18),(16,11),(16,18),(17,19),(17,21),(18,6),(19,20),(20,18),(21,15),(21,20)],22)
=> ?
=> ? = 6 + 2
[1,2,4,3,7,5,6] => [1,2,4,3,7,6,5] => ([(0,4),(0,5),(0,6),(1,18),(2,8),(2,11),(3,7),(3,12),(3,19),(4,13),(4,14),(5,3),(5,13),(5,15),(6,2),(6,14),(6,15),(7,16),(8,1),(8,20),(9,17),(9,18),(11,9),(11,20),(12,9),(12,16),(12,20),(13,7),(13,19),(14,8),(14,19),(15,11),(15,12),(15,19),(16,17),(17,10),(18,10),(19,16),(19,20),(20,17),(20,18)],21)
=> ?
=> ? = 6 + 2
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ?
=> ? = 5 + 2
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ([(0,2),(0,4),(0,5),(0,6),(1,13),(1,22),(1,23),(2,9),(2,18),(3,10),(3,12),(3,23),(4,11),(4,14),(4,18),(5,3),(5,14),(5,15),(5,18),(6,1),(6,9),(6,11),(6,15),(8,19),(8,21),(9,22),(10,17),(10,20),(11,16),(11,22),(11,23),(12,8),(12,17),(12,24),(13,8),(13,20),(13,24),(14,10),(14,16),(14,23),(15,12),(15,13),(15,16),(15,22),(16,17),(16,20),(16,24),(17,19),(17,21),(18,22),(18,23),(19,7),(20,19),(20,21),(21,7),(22,24),(23,20),(23,24),(24,21)],25)
=> ?
=> ? = 5 + 2
[1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ?
=> ? = 5 + 2
[1,2,4,7,3,5,6] => [1,2,7,6,5,3,4] => ([(0,4),(0,5),(0,6),(1,17),(2,7),(2,11),(3,1),(3,10),(3,12),(4,13),(4,14),(5,3),(5,14),(5,15),(6,2),(6,13),(6,15),(7,19),(9,20),(9,21),(10,17),(10,18),(11,9),(11,19),(12,9),(12,17),(12,18),(13,7),(13,16),(14,10),(14,16),(15,11),(15,12),(15,16),(16,18),(16,19),(17,21),(18,20),(18,21),(19,20),(20,8),(21,8)],22)
=> ?
=> ? = 6 + 2
[1,2,4,7,5,3,6] => [1,2,5,7,6,3,4] => ([(0,3),(0,4),(0,5),(0,6),(1,20),(1,27),(2,7),(2,8),(2,9),(3,11),(3,12),(3,14),(4,12),(4,13),(4,15),(5,2),(5,14),(5,15),(5,16),(6,1),(6,11),(6,13),(6,16),(7,21),(7,23),(7,28),(8,21),(8,22),(9,22),(9,23),(11,17),(11,27),(12,19),(12,20),(12,27),(13,18),(13,20),(13,27),(14,8),(14,17),(14,19),(15,9),(15,18),(15,19),(16,7),(16,17),(16,18),(16,27),(17,21),(17,28),(18,23),(18,24),(18,28),(19,22),(19,24),(19,28),(20,24),(21,25),(22,25),(22,26),(23,25),(23,26),(24,26),(25,10),(26,10),(27,24),(27,28),(28,25),(28,26)],29)
=> ?
=> ? = 5 + 2
[1,2,5,3,4,6,7] => [1,2,5,4,3,6,7] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ?
=> ? = 5 + 2
[1,2,5,3,4,7,6] => [1,2,5,4,3,7,6] => ([(0,4),(0,5),(0,6),(1,9),(1,22),(2,11),(2,22),(3,10),(3,12),(4,3),(4,13),(4,15),(5,1),(5,13),(5,14),(6,2),(6,14),(6,15),(8,19),(9,17),(10,18),(10,21),(11,8),(11,21),(12,8),(12,18),(13,10),(13,16),(13,22),(14,9),(14,16),(14,22),(15,11),(15,12),(15,16),(16,17),(16,18),(16,21),(17,20),(18,19),(18,20),(19,7),(20,7),(21,19),(21,20),(22,17),(22,21)],23)
=> ?
=> ? = 5 + 2
[1,2,5,3,7,4,6] => [1,2,7,6,4,3,5] => ([(0,1),(0,3),(0,4),(0,6),(1,14),(1,21),(2,7),(2,12),(2,18),(3,15),(3,16),(3,21),(4,2),(4,16),(4,17),(4,21),(5,10),(5,11),(5,13),(6,5),(6,14),(6,15),(6,17),(7,24),(9,26),(9,27),(10,23),(11,20),(11,23),(12,9),(12,24),(12,25),(13,9),(13,20),(13,23),(14,10),(14,22),(15,11),(15,19),(15,22),(16,7),(16,18),(16,19),(17,12),(17,13),(17,19),(17,22),(18,24),(18,25),(19,20),(19,24),(19,25),(20,26),(20,27),(21,18),(21,22),(22,23),(22,25),(23,26),(24,27),(25,26),(25,27),(26,8),(27,8)],28)
=> ?
=> ? = 5 + 2
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ([(0,1),(0,4),(0,5),(0,6),(1,20),(2,10),(2,12),(2,23),(3,9),(3,11),(3,23),(4,13),(4,14),(4,20),(5,3),(5,13),(5,15),(5,20),(6,2),(6,14),(6,15),(6,20),(8,19),(8,21),(9,17),(9,22),(10,18),(10,22),(11,8),(11,17),(11,22),(12,8),(12,18),(12,22),(13,9),(13,16),(13,23),(14,10),(14,16),(14,23),(15,11),(15,12),(15,16),(15,23),(16,17),(16,18),(16,22),(17,19),(17,21),(18,19),(18,21),(19,7),(20,23),(21,7),(22,21),(23,22)],24)
=> ?
=> ? = 2 + 2
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => ([(0,2),(0,4),(0,5),(0,6),(1,13),(1,22),(1,23),(2,9),(2,18),(3,10),(3,12),(3,23),(4,11),(4,14),(4,18),(5,3),(5,14),(5,15),(5,18),(6,1),(6,9),(6,11),(6,15),(8,19),(8,21),(9,22),(10,17),(10,20),(11,16),(11,22),(11,23),(12,8),(12,17),(12,24),(13,8),(13,20),(13,24),(14,10),(14,16),(14,23),(15,12),(15,13),(15,16),(15,22),(16,17),(16,20),(16,24),(17,19),(17,21),(18,22),(18,23),(19,7),(20,19),(20,21),(21,7),(22,24),(23,20),(23,24),(24,21)],25)
=> ?
=> ? = 2 + 2
[1,2,5,4,7,3,6] => [1,2,4,7,6,3,5] => ?
=> ?
=> ? = 2 + 2
[1,2,5,7,3,4,6] => [1,2,5,3,7,6,4] => ?
=> ?
=> ? = 5 + 2
[1,2,5,7,4,3,6] => [1,2,7,6,3,5,4] => ([(0,2),(0,3),(0,4),(0,6),(1,11),(1,25),(2,12),(2,17),(3,13),(3,15),(3,17),(4,1),(4,13),(4,14),(4,17),(5,8),(5,9),(5,10),(6,5),(6,12),(6,14),(6,15),(8,19),(8,22),(9,22),(10,19),(10,22),(10,24),(11,18),(12,9),(12,23),(13,11),(13,16),(13,25),(14,10),(14,16),(14,23),(14,25),(15,8),(15,16),(15,23),(16,18),(16,19),(16,24),(17,23),(17,25),(18,21),(19,20),(19,21),(20,7),(21,7),(22,20),(23,22),(23,24),(24,20),(24,21),(25,18),(25,24)],26)
=> ?
=> ? = 2 + 2
[1,2,7,3,4,5,6] => [1,2,7,6,5,4,3] => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> 8 = 6 + 2
[1,7,2,3,4,5,6] => [1,7,6,5,4,3,2] => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> [7,5]
=> 8 = 6 + 2
[7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 8 = 6 + 2
[8,1,2,3,4,5,6,7] => [8,7,6,5,4,3,2,1] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> 9 = 7 + 2
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001925The minimal number of zeros in a row of an alternating sign matrix. St000019The cardinality of the support of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000924The number of topologically connected components of a perfect matching. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000189The number of elements in the poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St000010The length of the partition. St000178Number of free entries.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!