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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
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Mp00223: Permutations —runsort⟶ 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] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,3,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,2,1,3] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,4,3,5,2] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[2,1,4,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,5,1,4] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[2,4,5,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[2,5,1,3,4] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,1,2,4] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[3,5,1,4,2] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[3,5,2,1,4] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,1,3,2] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[4,5,2,1,3] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,2,4,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[5,1,3,2,4] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[5,1,3,4,2] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,1,4,2,3] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,1,4,3,2] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,2,1,3,4] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,2,1,4,3] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,2,3,1,4] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000454
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 17%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 2
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 2
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 2
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3 - 2
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[1,3,2,4] => [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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[2,4,1,3] => [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
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[4,1,3,2] => [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
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 2
[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
[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
[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
[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
[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
[1,4,3,5,2] => [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
[2,1,4,3,5] => [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
[2,3,4,5,1] => [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
[2,3,5,1,4] => [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
[2,4,3,5,1] => [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
[2,4,5,1,3] => [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
[2,5,1,3,4] => [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
[3,4,5,1,2] => [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
[3,4,5,2,1] => [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
[3,5,1,2,4] => [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
[3,5,1,4,2] => [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
[3,5,2,1,4] => [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
[3,5,2,4,1] => [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
[4,5,1,2,3] => [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
[4,5,1,3,2] => [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
[4,5,2,1,3] => [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
[4,5,2,3,1] => [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
[4,5,3,1,2] => [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
[4,5,3,2,1] => [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
[5,1,2,3,4] => [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
[5,1,2,4,3] => [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
[5,1,3,2,4] => [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
[5,1,3,4,2] => [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
[5,1,4,2,3] => [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
[5,1,4,3,2] => [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
[5,2,1,3,4] => [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
[5,2,1,4,3] => [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
[5,2,3,1,4] => [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
[5,2,3,4,1] => [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
[5,2,4,1,3] => [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
[5,2,4,3,1] => [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
[5,3,1,2,4] => [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
[5,3,1,4,2] => [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
[5,3,2,1,4] => [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
[5,3,2,4,1] => [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
[5,3,4,1,2] => [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
[5,3,4,2,1] => [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
[5,4,1,2,3] => [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
[5,4,1,3,2] => [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
[5,4,2,1,3] => [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
[5,4,2,3,1] => [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
[5,4,3,1,2] => [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
[5,4,3,2,1] => [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
[1,2,3,4,5,6] => [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] => [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] => [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] => [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] => [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
[1,2,5,4,6,3] => [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] => [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] => [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
[1,4,5,3,6,2] => [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] => [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
[1,5,3,4,6,2] => [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
[1,5,4,6,2,3] => [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
[1,5,4,6,3,2] => [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
[2,1,4,5,3,6] => [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
[2,1,5,3,4,6] => [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
[2,1,5,4,6,3] => [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
[2,3,1,5,4,6] => [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
[2,3,4,6,1,5] => [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
[2,3,6,1,4,5] => [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
[2,6,1,3,4,5] => [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
[3,1,5,4,6,2] => [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
[3,2,1,5,4,6] => [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
[3,4,6,1,5,2] => [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
[3,4,6,2,1,5] => [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
[3,6,1,4,5,2] => [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
[3,6,2,1,4,5] => [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
[4,6,1,5,2,3] => [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
[4,6,1,5,3,2] => [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
[4,6,2,1,5,3] => [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
[4,6,2,3,1,5] => [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
[4,6,3,1,5,2] => [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
[4,6,3,2,1,5] => [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
[6,1,3,4,5,2] => [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
[6,1,4,5,2,3] => [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
[6,1,4,5,3,2] => [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
[6,1,5,2,3,4] => [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
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: St000528
(load all 108 compositions to match this statistic)
(load all 108 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%
Mp00209: Permutations —pattern poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,1,3] => [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
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,3,2] => [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
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[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
[1,3,2,4,5] => [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
[1,3,4,2,5] => [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
[1,4,2,3,5] => [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
[1,4,3,5,2] => [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
[2,1,4,3,5] => [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
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,5,1,4] => [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
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[2,4,5,1,3] => [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
[2,5,1,3,4] => [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
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,1,2,4] => [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
[3,5,1,4,2] => [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
[3,5,2,1,4] => [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
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,1,3,2] => [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
[4,5,2,1,3] => [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
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,2,4,3] => [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
[5,1,3,2,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[5,1,3,4,2] => [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
[5,1,4,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4
[5,1,4,3,2] => [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
[5,2,1,3,4] => [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
[5,2,1,4,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4
[5,2,3,1,4] => [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
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,4,1,3] => [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
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[5,3,1,2,4] => [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
[5,3,1,4,2] => [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
[5,3,2,1,4] => [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
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,1,3,2] => [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
[5,4,2,1,3] => [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
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,5,4,6] => [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] => [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] => [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] => [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
[1,2,5,4,6,3] => [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
[1,3,2,4,5,6] => [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)
=> ? = 6
[1,3,4,2,5,6] => [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)
=> ? = 5
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(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
[1,3,5,2,4,6] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6
[1,3,5,4,6,2] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6
[1,4,2,3,5,6] => [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)
=> ? = 5
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6
[1,4,3,5,6,2] => [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)
=> ? = 5
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6
[1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ? = 4
[1,4,5,3,6,2] => [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ? = 4
[1,5,2,3,4,6] => [1,5,2,3,4,6] => ([(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)
=> ? = 4
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 2
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,4,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[6,1,2,3,4,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[6,2,3,4,5,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St001631
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001631: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%
Mp00209: Permutations —pattern poset⟶ Posets
St001631: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,4,1,3] => [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
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,3,2] => [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
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[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
[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
[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
[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
[1,4,3,5,2] => [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
[2,1,4,3,5] => [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
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3,5,1,4] => [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
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[2,4,5,1,3] => [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
[2,5,1,3,4] => [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
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,1,2,4] => [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
[3,5,1,4,2] => [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
[3,5,2,1,4] => [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
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,3,2] => [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
[4,5,2,1,3] => [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
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,2,4,3] => [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
[5,1,3,2,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[5,1,3,4,2] => [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
[5,1,4,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4 - 1
[5,1,4,3,2] => [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
[5,2,1,3,4] => [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
[5,2,1,4,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4 - 1
[5,2,3,1,4] => [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
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,1,3] => [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
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[5,3,1,2,4] => [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
[5,3,1,4,2] => [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
[5,3,2,1,4] => [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
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,3,2] => [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
[5,4,2,1,3] => [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
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[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
[1,2,3,5,4,6] => [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] => [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] => [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] => [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
[1,2,5,4,6,3] => [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
[1,3,2,4,5,6] => [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)
=> ? = 6 - 1
[1,3,4,2,5,6] => [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)
=> ? = 5 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(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
[1,3,5,2,4,6] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 - 1
[1,4,2,3,5,6] => [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)
=> ? = 5 - 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 - 1
[1,4,3,5,6,2] => [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)
=> ? = 5 - 1
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 - 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,4,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,2,3,4,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,2,3,4,5,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,3,4,5,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,3,4,5,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,4,5,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,4,5,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 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 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%
Mp00209: Permutations —pattern poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,4,1,3] => [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
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,3,2] => [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
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[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
[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
[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
[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
[1,4,3,5,2] => [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
[2,1,4,3,5] => [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
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3,5,1,4] => [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
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[2,4,5,1,3] => [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
[2,5,1,3,4] => [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
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,5,1,2,4] => [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
[3,5,1,4,2] => [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
[3,5,2,1,4] => [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
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,1,3,2] => [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
[4,5,2,1,3] => [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
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,2,4,3] => [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
[5,1,3,2,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[5,1,3,4,2] => [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
[5,1,4,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4 - 1
[5,1,4,3,2] => [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
[5,2,1,3,4] => [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
[5,2,1,4,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4 - 1
[5,2,3,1,4] => [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
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,1,3] => [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
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[5,3,1,2,4] => [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
[5,3,1,4,2] => [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
[5,3,2,1,4] => [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
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,3,2] => [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
[5,4,2,1,3] => [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
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[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
[1,2,3,5,4,6] => [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] => [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] => [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] => [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
[1,2,5,4,6,3] => [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
[1,3,2,4,5,6] => [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)
=> ? = 6 - 1
[1,3,4,2,5,6] => [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)
=> ? = 5 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(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
[1,3,5,2,4,6] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 - 1
[1,4,2,3,5,6] => [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)
=> ? = 5 - 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 - 1
[1,4,3,5,6,2] => [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)
=> ? = 5 - 1
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 - 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,4,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,6,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,2,3,4,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,2,3,4,5,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,3,4,5,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,3,4,5,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,4,5,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,4,5,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001330
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 83%
Values
[1,2,3] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[2,3,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,2] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,2,1] => [1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 4
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [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] => [1,2,4,3,5] => [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] => [1,3,2,4,5] => [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] => [1,3,4,2,5] => [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] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[1,4,3,5,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[2,3,4,5,1] => [1,2,3,4,5] => [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
[2,3,5,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[2,4,3,5,1] => [1,2,4,3,5] => [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
[2,4,5,1,3] => [1,3,2,4,5] => [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
[2,5,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[3,4,5,1,2] => [1,2,3,4,5] => [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
[3,4,5,2,1] => [1,2,3,4,5] => [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
[3,5,1,2,4] => [1,2,4,3,5] => [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
[3,5,1,4,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[3,5,2,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[3,5,2,4,1] => [1,2,4,3,5] => [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
[4,5,1,2,3] => [1,2,3,4,5] => [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
[4,5,1,3,2] => [1,3,2,4,5] => [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
[4,5,2,1,3] => [1,3,2,4,5] => [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
[4,5,2,3,1] => [1,2,3,4,5] => [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
[4,5,3,1,2] => [1,2,3,4,5] => [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
[4,5,3,2,1] => [1,2,3,4,5] => [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
[5,1,2,3,4] => [1,2,3,4,5] => [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
[5,1,2,4,3] => [1,2,4,3,5] => [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
[5,1,3,2,4] => [1,3,2,4,5] => [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
[5,1,3,4,2] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,1,4,2,3] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,1,4,3,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,2,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,2,1,4,3] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,2,3,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,2,3,4,1] => [1,2,3,4,5] => [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
[5,2,4,1,3] => [1,3,2,4,5] => [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
[5,2,4,3,1] => [1,2,4,3,5] => [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
[5,3,1,2,4] => [1,2,4,3,5] => [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
[5,3,1,4,2] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,3,2,1,4] => [1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[5,3,2,4,1] => [1,2,4,3,5] => [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
[5,3,4,1,2] => [1,2,3,4,5] => [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
[5,3,4,2,1] => [1,2,3,4,5] => [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
[5,4,1,2,3] => [1,2,3,4,5] => [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
[5,4,1,3,2] => [1,3,2,4,5] => [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
[5,4,2,1,3] => [1,3,2,4,5] => [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
[5,4,2,3,1] => [1,2,3,4,5] => [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
[5,4,3,1,2] => [1,2,3,4,5] => [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
[5,4,3,2,1] => [1,2,3,4,5] => [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
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [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] => [1,2,3,5,4,6] => [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] => [1,2,4,3,5,6] => [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] => [1,2,4,5,3,6] => [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] => [1,2,5,3,4,6] => [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
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [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
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => ([(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
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [6,5,2,4,3,1] => ([(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
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [6,2,5,4,3,1] => ([(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)
=> ? = 4
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => ([(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
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[1,4,3,5,6,2] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => ([(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
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [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
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [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
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [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
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [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
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [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
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [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
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [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
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [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
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [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
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [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
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [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
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [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
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [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
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [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
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [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
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [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
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [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
[6,3,4,5,1,2] => [1,2,3,4,5,6] => [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
[6,3,4,5,2,1] => [1,2,3,4,5,6] => [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
[6,4,5,1,2,3] => [1,2,3,4,5,6] => [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
[6,4,5,2,3,1] => [1,2,3,4,5,6] => [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
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: St001875
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 67%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 67%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[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
[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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,4,1,3] => [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
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,1,3,2] => [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
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,2,3,4,5] => [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] => [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] => [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] => [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] => [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
[1,4,3,5,2] => [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
[2,1,4,3,5] => [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
[2,3,4,5,1] => [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
[2,3,5,1,4] => [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
[2,4,3,5,1] => [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
[2,4,5,1,3] => [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
[2,5,1,3,4] => [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
[3,4,5,1,2] => [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
[3,4,5,2,1] => [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
[3,5,1,2,4] => [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
[3,5,1,4,2] => [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
[3,5,2,1,4] => [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
[3,5,2,4,1] => [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
[4,5,1,2,3] => [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
[4,5,1,3,2] => [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
[4,5,2,1,3] => [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
[4,5,2,3,1] => [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
[4,5,3,1,2] => [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
[4,5,3,2,1] => [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
[5,1,2,3,4] => [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
[5,1,2,4,3] => [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
[5,1,3,2,4] => [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
[5,1,3,4,2] => [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
[5,1,4,2,3] => [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
[5,1,4,3,2] => [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
[5,2,1,3,4] => [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
[5,2,1,4,3] => [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
[5,2,3,1,4] => [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
[5,2,3,4,1] => [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
[5,2,4,1,3] => [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
[5,2,4,3,1] => [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
[5,3,1,2,4] => [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
[5,3,1,4,2] => [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
[5,3,2,1,4] => [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
[5,3,2,4,1] => [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
[5,3,4,1,2] => [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
[5,3,4,2,1] => [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
[5,4,1,2,3] => [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
[5,4,1,3,2] => [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
[5,4,2,1,3] => [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
[5,4,2,3,1] => [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
[5,4,3,1,2] => [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
[5,4,3,2,1] => [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
[1,2,3,4,5,6] => [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] => [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] => [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] => [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] => [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
[1,2,5,4,6,3] => [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
[1,3,2,4,5,6] => [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] => [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
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 6 + 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 6 + 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => ([(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
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 6 + 1
[1,4,3,5,6,2] => [1,4,2,3,5,6] => ([(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
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 6 + 1
[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)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 4 + 1
[1,4,5,3,6,2] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 4 + 1
[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)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 2 + 1
[1,5,3,4,6,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 2 + 1
[1,5,4,6,2,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[1,5,4,6,3,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[2,1,3,5,4,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 6 + 1
[2,1,4,3,5,6] => [1,4,2,3,5,6] => ([(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
[2,1,4,5,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 4 + 1
[2,1,5,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[2,1,5,4,6,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[2,3,1,5,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[2,3,4,6,1,5] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 4 + 1
[2,3,5,4,6,1] => [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
[2,3,5,6,1,4] => [1,4,2,3,5,6] => ([(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
[2,3,6,1,4,5] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 4 + 1
[2,4,1,5,3,6] => [1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,6),(2,7),(3,8),(3,10),(4,8),(4,9),(5,3),(5,4),(5,7),(6,2),(6,5),(7,9),(7,10),(8,11),(9,11),(10,11),(11,1)],12)
=> ? = 2 + 1
[2,4,3,5,6,1] => [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
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001200
Mp00223: Permutations —runsort⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1,2] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[2,4,1,3] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[3,4,1,2] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[3,4,2,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[4,1,2,3] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[4,1,3,2] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,2,1,3] => [1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,2,3,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[4,3,1,2] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,4,3,5,2] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[2,1,4,3,5] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[2,3,5,1,4] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[2,4,3,5,1] => [1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[2,4,5,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[2,5,1,3,4] => [1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[3,4,5,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[3,4,5,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[3,5,1,2,4] => [1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[3,5,1,4,2] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[3,5,2,1,4] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[3,5,2,4,1] => [1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,5,1,2,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,5,1,3,2] => [1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,5,2,1,3] => [1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,5,2,3,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,5,3,1,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[4,5,3,2,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[5,1,2,3,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[5,1,2,4,3] => [1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[5,1,3,2,4] => [1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[5,1,3,4,2] => [1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[5,1,4,2,3] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[5,1,4,3,2] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[5,2,1,3,4] => [1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[5,2,1,4,3] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[5,2,3,1,4] => [1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[5,2,3,4,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,2,5,4,6,3] => [1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,4,3,5,6,2] => [1,4,2,3,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[1,5,2,3,4,6] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,5,3,4,6,2] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,5,4,6,2,3] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[1,5,4,6,3,2] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[2,1,3,5,4,6] => [1,3,5,2,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[2,1,4,3,5,6] => [1,4,2,3,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,5,3,4,6] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[2,1,5,4,6,3] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[2,3,1,5,4,6] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[2,3,4,6,1,5] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[2,3,5,6,1,4] => [1,4,2,3,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[2,4,3,5,6,1] => [1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[2,4,5,3,6,1] => [1,2,4,5,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,4,5,6,1,3] => [1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[2,4,6,1,3,5] => [1,3,5,2,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[2,5,3,4,6,1] => [1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[2,5,4,6,3,1] => [1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,5,6,1,3,4] => [1,3,4,2,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,6,1,3,4,5] => [1,3,4,5,2,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,1,2,5,4,6] => [1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[3,1,5,4,6,2] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,2,1,5,4,6] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,2,5,4,6,1] => [1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
[3,4,6,1,2,5] => [1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 5 - 1
[3,4,6,1,5,2] => [1,5,2,3,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 1
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000906
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,1,3] => [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
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,3,2] => [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
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[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
[1,3,2,4,5] => [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
[1,3,4,2,5] => [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
[1,4,2,3,5] => [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
[1,4,3,5,2] => [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
[2,1,4,3,5] => [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
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,5,1,4] => [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
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[2,4,5,1,3] => [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
[2,5,1,3,4] => [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
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,5,1,2,4] => [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
[3,5,1,4,2] => [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
[3,5,2,1,4] => [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
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,1,3,2] => [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
[4,5,2,1,3] => [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
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,2,4,3] => [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
[5,1,3,2,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[5,1,3,4,2] => [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
[5,1,4,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4
[5,1,4,3,2] => [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
[5,2,1,3,4] => [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
[5,2,1,4,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4
[5,2,3,1,4] => [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
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,4,1,3] => [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
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[5,3,1,2,4] => [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
[5,3,1,4,2] => [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
[5,3,2,1,4] => [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
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,1,3,2] => [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
[5,4,2,1,3] => [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
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3,5,4,6] => [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] => [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] => [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] => [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
[1,2,5,4,6,3] => [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
[1,3,2,4,5,6] => [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)
=> ? = 6
[1,3,4,2,5,6] => [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)
=> ? = 5
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(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
[1,3,5,2,4,6] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6
[1,3,5,4,6,2] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6
[1,4,2,3,5,6] => [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)
=> ? = 5
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6
[1,4,3,5,6,2] => [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)
=> ? = 5
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6
[1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ? = 4
[1,4,5,3,6,2] => [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ? = 4
[1,5,2,3,4,6] => [1,5,2,3,4,6] => ([(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)
=> ? = 4
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 2
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,4,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,6,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[6,1,2,3,4,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[6,2,3,4,5,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St000643
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000643: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Mp00209: Permutations —pattern poset⟶ Posets
St000643: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[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)
=> 5 = 4 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[2,4,1,3] => [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)
=> 5 = 4 + 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,1,3,2] => [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)
=> 5 = 4 + 1
[4,2,1,3] => [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)
=> 5 = 4 + 1
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[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
[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
[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
[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
[1,4,3,5,2] => [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
[2,1,4,3,5] => [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
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[2,3,5,1,4] => [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
[2,4,3,5,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 + 1
[2,4,5,1,3] => [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
[2,5,1,3,4] => [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
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[3,5,1,2,4] => [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
[3,5,1,4,2] => [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
[3,5,2,1,4] => [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
[3,5,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 + 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[4,5,1,3,2] => [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
[4,5,2,1,3] => [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
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[5,1,2,4,3] => [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
[5,1,3,2,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 + 1
[5,1,3,4,2] => [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
[5,1,4,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4 + 1
[5,1,4,3,2] => [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
[5,2,1,3,4] => [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
[5,2,1,4,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 4 + 1
[5,2,3,1,4] => [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
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[5,2,4,1,3] => [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
[5,2,4,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 + 1
[5,3,1,2,4] => [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
[5,3,1,4,2] => [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
[5,3,2,1,4] => [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
[5,3,2,4,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 + 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[5,4,1,3,2] => [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
[5,4,2,1,3] => [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
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[1,2,3,5,4,6] => [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] => [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] => [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] => [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
[1,2,5,4,6,3] => [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
[1,3,2,4,5,6] => [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)
=> ? = 6 + 1
[1,3,4,2,5,6] => [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)
=> ? = 5 + 1
[1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(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
[1,3,5,2,4,6] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 + 1
[1,3,5,4,6,2] => [1,3,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 + 1
[1,4,2,3,5,6] => [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)
=> ? = 5 + 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 + 1
[1,4,3,5,6,2] => [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)
=> ? = 5 + 1
[1,4,3,6,2,5] => [1,4,2,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(1,17),(1,22),(2,13),(2,17),(2,22),(3,9),(3,11),(3,17),(3,22),(4,8),(4,10),(4,17),(4,22),(5,8),(5,9),(5,12),(5,13),(5,22),(6,10),(6,11),(6,12),(6,14),(6,22),(8,15),(8,19),(8,23),(9,16),(9,19),(9,23),(10,15),(10,20),(10,23),(11,16),(11,20),(11,23),(12,15),(12,16),(12,19),(12,20),(13,19),(13,23),(14,20),(14,23),(15,18),(15,21),(16,18),(16,21),(17,23),(18,7),(19,18),(19,21),(20,18),(20,21),(21,7),(22,19),(22,20),(22,23),(23,21)],24)
=> ? = 6 + 1
[1,4,5,2,3,6] => [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ? = 4 + 1
[1,4,5,3,6,2] => [1,4,5,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(1,11),(1,12),(2,8),(2,9),(2,12),(3,5),(3,7),(3,9),(3,11),(4,5),(4,7),(4,8),(4,10),(5,19),(7,13),(7,14),(7,19),(8,13),(8,15),(8,19),(9,13),(9,16),(9,19),(10,14),(10,15),(10,19),(11,14),(11,16),(11,19),(12,15),(12,16),(13,17),(13,18),(14,17),(14,18),(15,17),(15,18),(16,17),(16,18),(17,6),(18,6),(19,18)],20)
=> ? = 4 + 1
[1,5,2,3,4,6] => [1,5,2,3,4,6] => ([(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)
=> ? = 4 + 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,17),(2,10),(2,11),(2,17),(3,8),(3,9),(3,13),(3,17),(4,7),(4,9),(4,10),(4,17),(5,7),(5,8),(5,11),(5,12),(7,15),(7,20),(7,21),(8,14),(8,15),(8,20),(9,15),(9,16),(9,21),(10,21),(11,20),(11,21),(12,14),(12,20),(13,14),(13,16),(14,19),(15,18),(15,19),(16,18),(16,19),(17,16),(17,20),(17,21),(18,6),(19,6),(20,18),(20,19),(21,18)],22)
=> ? = 2 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[5,6,4,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[5,6,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[6,1,2,3,4,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
[6,2,3,4,5,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 7 = 6 + 1
Description
The size of the largest orbit of antichains under Panyushev complementation.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001557The number of inversions of the second entry of a permutation. 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. St000327The number of cover relations in a poset. St001434The number of negative sum pairs of a signed permutation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001892The flag excedance statistic of a signed permutation. St001817The number of flag weak exceedances of a signed permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000778The metric dimension of a graph.
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