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Your data matches 75 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
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Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000672
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[3,1,2] => [1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 2 = 3 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 3 = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 3 = 4 - 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => 3 = 4 - 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,5,6,1,3] => 4 = 5 - 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,5,6,1,3] => 4 = 5 - 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => 4 = 5 - 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [5,4,2,6,1,3] => 3 = 4 - 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [5,4,2,6,1,3] => 3 = 4 - 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => 4 = 5 - 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [5,4,2,6,1,3] => 3 = 4 - 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [5,4,2,6,1,3] => 3 = 4 - 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => 4 = 5 - 1
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [2,1,3,6,7,4,5] => ? = 4 - 1
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,1,5,6,7,3,4] => ? = 5 - 1
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [2,1,5,6,7,3,4] => ? = 5 - 1
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [6,2,1,5,7,3,4] => ? = 6 - 1
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [6,2,1,5,7,3,4] => ? = 6 - 1
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,5,6,7,1,3] => ? = 6 - 1
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,5,6,7,1,3] => ? = 6 - 1
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,5,6,7,1,3] => ? = 6 - 1
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,5,6,7,1,3] => ? = 6 - 1
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,5,6,7,1,3] => ? = 6 - 1
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [4,2,5,6,7,1,3] => ? = 6 - 1
[3,1,2,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,6,3,7,1,4,5] => ? = 5 - 1
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,2,3,6,7,1,4] => ? = 6 - 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,2,3,6,7,1,4] => ? = 6 - 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [6,2,1,4,7,3,5] => ? = 6 - 1
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [2,1,6,5,7,3,4] => ? = 4 - 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [2,1,6,5,7,3,4] => ? = 4 - 1
[3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,1,6,2,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,1,6,4,2,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,2,1,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [2,6,3,7,1,4,5] => ? = 5 - 1
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,2,3,6,7,1,4] => ? = 6 - 1
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [5,2,3,6,7,1,4] => ? = 6 - 1
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [6,2,1,4,7,3,5] => ? = 6 - 1
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [2,1,6,5,7,3,4] => ? = 4 - 1
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [2,1,6,5,7,3,4] => ? = 4 - 1
[3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[3,2,6,5,4,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [5,4,2,6,7,1,3] => ? = 5 - 1
[4,1,2,3,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ? = 6 - 1
[4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [6,5,2,3,7,1,4] => ? = 5 - 1
[4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [6,5,2,3,7,1,4] => ? = 5 - 1
[4,1,3,2,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ? = 6 - 1
[4,1,3,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [6,5,2,3,7,1,4] => ? = 5 - 1
[4,1,3,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [6,5,2,3,7,1,4] => ? = 5 - 1
[4,1,6,2,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [4,6,5,2,7,1,3] => ? = 4 - 1
[4,1,6,2,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [4,6,5,2,7,1,3] => ? = 4 - 1
[4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [4,6,5,2,7,1,3] => ? = 4 - 1
[4,1,6,3,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [4,6,5,2,7,1,3] => ? = 4 - 1
[4,1,6,5,2,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [4,6,5,2,7,1,3] => ? = 4 - 1
[4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [4,6,5,2,7,1,3] => ? = 4 - 1
[4,2,1,3,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [6,2,3,4,7,1,5] => ? = 6 - 1
[4,2,1,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [6,5,2,3,7,1,4] => ? = 5 - 1
[4,2,1,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [6,5,2,3,7,1,4] => ? = 5 - 1
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 5 = 6 - 1
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 5 = 6 - 1
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 5 = 6 - 1
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 5 = 6 - 1
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 5 = 6 - 1
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 5 = 6 - 1
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 5 = 6 - 1
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => 5 = 6 - 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[3,1,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 3 - 1
[3,2,1] => [1,2,3] => [1,3,2] => ([(1,2)],3)
=> 2 = 3 - 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 4 - 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,1,2] => [1,2,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,1,5,3,4] => [4,5,1,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5 - 1
[2,1,5,4,3] => [4,5,1,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5 - 1
[3,1,2,5,4] => [3,5,4,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5 - 1
[3,1,5,2,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,1,5,4,2] => [3,5,1,2,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,2,1,5,4] => [3,4,5,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5 - 1
[3,2,5,1,4] => [3,4,1,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,2,5,4,1] => [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,1,2,4,3] => [1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,1,3,2,4] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,1,3,4,2] => [1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,1,4,2,3] => [1,5,2,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,2,1,3,4] => [1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,2,1,4,3] => [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,2,3,1,4] => [1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,2,4,3,1] => [1,4,2,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,3,1,2,4] => [1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,3,1,4,2] => [1,3,5,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,3,2,1,4] => [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,3,2,4,1] => [1,3,4,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,3,4,1,2] => [1,3,2,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,3,4,2,1] => [1,3,2,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,1,2,3] => [1,2,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,1,3,2] => [1,2,5,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,2,1,3] => [1,2,4,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,2,3,1] => [1,2,4,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,3,1,2] => [1,2,3,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[2,1,3,4,6,5] => [5,6,4,3,1,2] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[2,1,3,6,4,5] => [5,6,4,1,3,2] => [5,6,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 1
[2,1,3,6,5,4] => [5,6,4,1,2,3] => [5,6,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 1
[2,1,4,6,3,5] => [5,6,3,1,4,2] => [5,6,3,1,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[2,1,4,6,5,3] => [5,6,3,1,2,4] => [5,6,3,1,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[2,1,6,3,4,5] => [5,6,1,4,3,2] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[2,1,6,3,5,4] => [5,6,1,4,2,3] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[2,1,6,4,3,5] => [5,6,1,3,4,2] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[2,1,6,4,5,3] => [5,6,1,3,2,4] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[2,1,6,5,3,4] => [5,6,1,2,4,3] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[2,1,6,5,4,3] => [5,6,1,2,3,4] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[3,1,2,4,6,5] => [4,6,5,3,1,2] => [4,6,5,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,1,2,6,4,5] => [4,6,5,1,3,2] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[3,1,2,6,5,4] => [4,6,5,1,2,3] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[3,1,4,2,6,5] => [4,6,3,5,1,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[3,1,4,6,2,5] => [4,6,3,1,5,2] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,1,4,6,5,2] => [4,6,3,1,2,5] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,1,6,2,4,5] => [4,6,1,5,3,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,1,6,2,5,4] => [4,6,1,5,2,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,1,6,4,2,5] => [4,6,1,3,5,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,1,6,4,5,2] => [4,6,1,3,2,5] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,1,6,5,2,4] => [4,6,1,2,5,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,1,6,5,4,2] => [4,6,1,2,3,5] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,2,1,4,6,5] => [4,5,6,3,1,2] => [4,6,5,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,2,1,6,4,5] => [4,5,6,1,3,2] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[3,2,1,6,5,4] => [4,5,6,1,2,3] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 1
[3,2,4,1,6,5] => [4,5,3,6,1,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[3,2,4,6,1,5] => [4,5,3,1,6,2] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,2,4,6,5,1] => [4,5,3,1,2,6] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,2,6,1,4,5] => [4,5,1,6,3,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,2,6,1,5,4] => [4,5,1,6,2,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,2,6,4,1,5] => [4,5,1,3,6,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,2,6,4,5,1] => [4,5,1,3,2,6] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,2,6,5,1,4] => [4,5,1,2,6,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[3,2,6,5,4,1] => [4,5,1,2,3,6] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[4,1,2,3,6,5] => [3,6,5,4,1,2] => [3,6,5,4,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[4,1,2,6,3,5] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[4,1,2,6,5,3] => [3,6,5,1,2,4] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[4,1,3,2,6,5] => [3,6,4,5,1,2] => [3,6,5,4,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 1
[4,1,3,6,2,5] => [3,6,4,1,5,2] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[4,1,3,6,5,2] => [3,6,4,1,2,5] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,2,3,5,4] => [1,6,5,4,2,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,2,4,3,5] => [1,6,5,3,4,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,2,4,5,3] => [1,6,5,3,2,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,2,5,3,4] => [1,6,5,2,4,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,2,5,4,3] => [1,6,5,2,3,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,3,2,4,5] => [1,6,4,5,3,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,3,2,5,4] => [1,6,4,5,2,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,3,4,2,5] => [1,6,4,3,5,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,3,4,5,2] => [1,6,4,3,2,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,3,5,2,4] => [1,6,4,2,5,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,3,5,4,2] => [1,6,4,2,3,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,4,2,3,5] => [1,6,3,5,4,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,4,2,5,3] => [1,6,3,5,2,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,4,3,2,5] => [1,6,3,4,5,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,4,3,5,2] => [1,6,3,4,2,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
[6,1,4,5,2,3] => [1,6,3,2,5,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 6 - 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[3,1,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1 = 3 - 2
[3,2,1] => [1,2,3] => [1,3,2] => ([(1,2)],3)
=> 1 = 3 - 2
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[4,2,3,1] => [1,3,2,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[4,3,1,2] => [1,2,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[4,3,2,1] => [1,2,3,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 4 - 2
[2,1,3,5,4] => [4,5,3,1,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[2,1,5,3,4] => [4,5,1,3,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5 - 2
[2,1,5,4,3] => [4,5,1,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5 - 2
[3,1,2,5,4] => [3,5,4,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5 - 2
[3,1,5,2,4] => [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,1,5,4,2] => [3,5,1,2,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,2,1,5,4] => [3,4,5,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 5 - 2
[3,2,5,1,4] => [3,4,1,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,2,5,4,1] => [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 2
[5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,1,2,4,3] => [1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,1,3,2,4] => [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,1,3,4,2] => [1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,1,4,2,3] => [1,5,2,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,1,4,3,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,2,1,3,4] => [1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,2,1,4,3] => [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,2,3,1,4] => [1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,2,4,3,1] => [1,4,2,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,3,1,2,4] => [1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,3,1,4,2] => [1,3,5,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,3,2,1,4] => [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,3,2,4,1] => [1,3,4,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,3,4,1,2] => [1,3,2,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,3,4,2,1] => [1,3,2,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,4,1,2,3] => [1,2,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,4,1,3,2] => [1,2,5,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,4,2,1,3] => [1,2,4,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,4,2,3,1] => [1,2,4,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,4,3,1,2] => [1,2,3,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 5 - 2
[2,1,3,4,6,5] => [5,6,4,3,1,2] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 2
[2,1,3,6,4,5] => [5,6,4,1,3,2] => [5,6,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 2
[2,1,3,6,5,4] => [5,6,4,1,2,3] => [5,6,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 2
[2,1,4,6,3,5] => [5,6,3,1,4,2] => [5,6,3,1,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[2,1,4,6,5,3] => [5,6,3,1,2,4] => [5,6,3,1,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[2,1,6,3,4,5] => [5,6,1,4,3,2] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[2,1,6,3,5,4] => [5,6,1,4,2,3] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[2,1,6,4,3,5] => [5,6,1,3,4,2] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[2,1,6,4,5,3] => [5,6,1,3,2,4] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[2,1,6,5,3,4] => [5,6,1,2,4,3] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[2,1,6,5,4,3] => [5,6,1,2,3,4] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[3,1,2,4,6,5] => [4,6,5,3,1,2] => [4,6,5,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,1,2,6,4,5] => [4,6,5,1,3,2] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 2
[3,1,2,6,5,4] => [4,6,5,1,2,3] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 2
[3,1,4,2,6,5] => [4,6,3,5,1,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[3,1,4,6,2,5] => [4,6,3,1,5,2] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 2
[3,1,4,6,5,2] => [4,6,3,1,2,5] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 2
[3,1,6,2,4,5] => [4,6,1,5,3,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,1,6,2,5,4] => [4,6,1,5,2,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,1,6,4,2,5] => [4,6,1,3,5,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,1,6,4,5,2] => [4,6,1,3,2,5] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,1,6,5,2,4] => [4,6,1,2,5,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,1,6,5,4,2] => [4,6,1,2,3,5] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,2,1,4,6,5] => [4,5,6,3,1,2] => [4,6,5,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,2,1,6,4,5] => [4,5,6,1,3,2] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 2
[3,2,1,6,5,4] => [4,5,6,1,2,3] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 2
[3,2,4,1,6,5] => [4,5,3,6,1,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[3,2,4,6,1,5] => [4,5,3,1,6,2] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 2
[3,2,4,6,5,1] => [4,5,3,1,2,6] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 2
[3,2,6,1,4,5] => [4,5,1,6,3,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,2,6,1,5,4] => [4,5,1,6,2,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,2,6,4,1,5] => [4,5,1,3,6,2] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,2,6,4,5,1] => [4,5,1,3,2,6] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,2,6,5,1,4] => [4,5,1,2,6,3] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,2,6,5,4,1] => [4,5,1,2,3,6] => [4,6,1,5,3,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[4,1,2,3,6,5] => [3,6,5,4,1,2] => [3,6,5,4,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[4,1,2,6,3,5] => [3,6,5,1,4,2] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[4,1,2,6,5,3] => [3,6,5,1,2,4] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[4,1,3,2,6,5] => [3,6,4,5,1,2] => [3,6,5,4,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 6 - 2
[4,1,3,6,2,5] => [3,6,4,1,5,2] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[4,1,3,6,5,2] => [3,6,4,1,2,5] => [3,6,5,1,4,2] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 2
[6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,2,3,5,4] => [1,6,5,4,2,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,2,4,3,5] => [1,6,5,3,4,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,2,4,5,3] => [1,6,5,3,2,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,2,5,3,4] => [1,6,5,2,4,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,2,5,4,3] => [1,6,5,2,3,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,3,2,4,5] => [1,6,4,5,3,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,3,2,5,4] => [1,6,4,5,2,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,3,4,2,5] => [1,6,4,3,5,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,3,4,5,2] => [1,6,4,3,2,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,3,5,2,4] => [1,6,4,2,5,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,3,5,4,2] => [1,6,4,2,3,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,4,2,3,5] => [1,6,3,5,4,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,4,2,5,3] => [1,6,3,5,2,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,4,3,2,5] => [1,6,3,4,5,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,4,3,5,2] => [1,6,3,4,2,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 2
[6,1,4,5,2,3] => [1,6,3,2,5,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 6 - 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: St001232
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 5 - 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 5 - 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 5 - 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 5 - 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 4 - 1
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 6 - 1
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 6 - 1
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 6 - 1
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 6 - 1
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 6 - 1
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 6 - 1
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 6 - 1
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 6 - 1
[3,1,2,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 6 - 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 6 - 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,1,6,2,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,1,6,4,2,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,2,1,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 5 - 1
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 6 - 1
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 6 - 1
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[3,2,6,5,4,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[4,1,2,3,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 6 - 1
[4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 5 - 1
[4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 5 - 1
[4,1,3,2,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 6 - 1
[4,1,3,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 5 - 1
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[6,1,4,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001879
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ? = 4 - 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ? = 5 - 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ? = 5 - 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? = 5 - 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? = 5 - 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 - 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 - 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => ([(2,5),(3,4)],6)
=> ? = 4 - 1
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => ([(1,3),(2,4),(4,5)],6)
=> ? = 5 - 1
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => ([(1,3),(2,4),(4,5)],6)
=> ? = 5 - 1
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ? = 6 - 1
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ? = 6 - 1
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 - 1
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 - 1
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 - 1
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 - 1
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 - 1
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 - 1
[3,1,2,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,6,4,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> ? = 5 - 1
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ? = 6 - 1
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ? = 6 - 1
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,6,3,1,2,4] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ? = 6 - 1
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,1,6,2,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,1,6,4,2,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,2,1,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [5,6,4,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> ? = 5 - 1
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ? = 6 - 1
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> ? = 6 - 1
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,6,3,1,2,4] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ? = 6 - 1
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 4 - 1
[3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[3,2,6,5,4,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[4,1,2,3,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 - 1
[4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[4,1,3,2,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 6 - 1
[4,1,3,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 5 - 1
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[6,1,4,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [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: St001480
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 60%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 60%
Values
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? = 3 - 2
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? = 3 - 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4 = 6 - 2
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4 = 6 - 2
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[3,1,2,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 4 = 6 - 2
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,2,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,4,2,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,1,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 4 = 6 - 2
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,5,4,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[4,1,2,3,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,1,3,2,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[4,1,3,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[6,1,4,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St001526
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? = 3 - 2
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? = 3 - 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 2
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 2
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 2
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 2
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 2
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 2
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 2
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 2
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 2
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 2
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 2
[3,1,2,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 2
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 2
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 2
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 6 - 2
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,2,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,4,2,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,1,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 5 - 2
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 2
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6 - 2
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[3,2,6,5,4,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,1,3,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,1,3,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,1,6,2,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,1,6,2,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,1,6,3,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,1,6,5,2,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,2,1,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,2,1,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,2,3,6,1,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,2,3,6,5,1] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,2,6,1,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,2,6,3,1,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,2,6,3,5,1] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,2,6,5,1,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,2,6,5,3,1] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,3,2,6,1,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[4,3,2,6,5,1] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001596
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 40%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 40%
Values
[3,1,2] => [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 3 - 3
[3,2,1] => [1,1,1,0,0,0]
=> []
=> [[],[]]
=> ? = 3 - 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 4 - 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 4 - 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 4 - 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 4 - 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 4 - 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> [[],[]]
=> ? = 4 - 3
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 4 - 3
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> [[],[]]
=> ? = 5 - 3
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [[4,4,3,2],[]]
=> ? = 4 - 3
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[3,3,3,2],[]]
=> ? = 5 - 3
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [[3,3,3,2],[]]
=> ? = 5 - 3
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ? = 6 - 3
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ? = 6 - 3
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 6 - 3
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 6 - 3
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 6 - 3
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 6 - 3
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 6 - 3
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 6 - 3
[3,1,2,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [[4,4,3],[]]
=> ? = 5 - 3
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[3,3,3],[]]
=> ? = 6 - 3
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [[3,3,3],[]]
=> ? = 6 - 3
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [[4,4,2],[]]
=> ? = 6 - 3
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 4 - 3
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 4 - 3
[3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,1,6,2,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,1,6,4,2,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[3,2,6,5,4,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [[2,2,2],[]]
=> 2 = 5 - 3
[4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,1,3,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,1,3,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,1,6,2,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,1,6,2,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,1,6,3,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,1,6,5,2,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,2,1,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,2,1,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,2,3,6,1,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,2,3,6,5,1] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,2,6,1,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,2,6,1,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,2,6,3,1,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,2,6,3,5,1] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,2,6,5,1,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,2,6,5,3,1] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,3,1,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,3,2,6,1,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,3,2,6,5,1] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [[3,3],[]]
=> 2 = 5 - 3
[4,3,6,1,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,3,6,1,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,3,6,2,1,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,3,6,2,5,1] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
[4,3,6,5,1,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [[2,2],[]]
=> 1 = 4 - 3
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St000373
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 80%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 80%
Values
[3,1,2] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,1,3] => 1 = 3 - 2
[3,2,1] => [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,1,3] => 1 = 3 - 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,5,3,4,2] => 2 = 4 - 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,1,4] => 2 = 4 - 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,1,4] => 2 = 4 - 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,1,4] => 2 = 4 - 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,1,4] => 2 = 4 - 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,1,4] => 2 = 4 - 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,1,4] => 2 = 4 - 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,2,5,1,3] => 2 = 4 - 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,6,3,4,5,2] => 3 = 5 - 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,6,3,4,5,2] => 3 = 5 - 2
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6,2,1,4,5,3] => 3 = 5 - 2
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,6,4,3,5,2] => 2 = 4 - 2
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,6,4,3,5,2] => 2 = 4 - 2
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6,2,1,4,5,3] => 3 = 5 - 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,6,4,3,5,2] => 2 = 4 - 2
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,6,4,3,5,2] => 2 = 4 - 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,1,5] => 3 = 5 - 2
[2,1,3,4,6,5] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => [7,5,2,3,6,1,4] => ? = 4 - 2
[2,1,3,6,4,5] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [7,4,2,5,6,1,3] => ? = 5 - 2
[2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [7,4,2,5,6,1,3] => ? = 5 - 2
[2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [7,5,4,1,2,6,3] => ? = 6 - 2
[2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,6,1,5,3,7,4] => [7,5,4,1,2,6,3] => ? = 6 - 2
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 6 - 2
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 6 - 2
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 6 - 2
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 6 - 2
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 6 - 2
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 6 - 2
[3,1,2,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [7,2,5,3,6,1,4] => ? = 5 - 2
[3,1,2,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [7,2,1,4,5,6,3] => ? = 6 - 2
[3,1,2,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [7,2,1,4,5,6,3] => ? = 6 - 2
[3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [7,5,3,2,6,1,4] => ? = 6 - 2
[3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [7,5,4,2,6,1,3] => ? = 4 - 2
[3,1,4,6,5,2] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [7,5,4,2,6,1,3] => ? = 4 - 2
[3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,1,6,2,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,1,6,4,2,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,1,6,5,4,2] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,2,1,4,6,5] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => [7,2,5,3,6,1,4] => ? = 5 - 2
[3,2,1,6,4,5] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [7,2,1,4,5,6,3] => ? = 6 - 2
[3,2,1,6,5,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [7,2,1,4,5,6,3] => ? = 6 - 2
[3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,6,4,1,3,7,5] => [7,5,3,2,6,1,4] => ? = 6 - 2
[3,2,4,6,1,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [7,5,4,2,6,1,3] => ? = 4 - 2
[3,2,4,6,5,1] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,6,5,1,3,7,4] => [7,5,4,2,6,1,3] => ? = 4 - 2
[3,2,6,1,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,2,6,1,5,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,2,6,4,1,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,2,6,4,5,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,2,6,5,1,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[3,2,6,5,4,1] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,7,4,3,5,6,2] => ? = 5 - 2
[4,1,2,3,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [7,2,3,1,5,6,4] => ? = 6 - 2
[4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [7,2,1,5,4,6,3] => ? = 5 - 2
[4,1,2,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [7,2,1,5,4,6,3] => ? = 5 - 2
[4,1,3,2,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [7,2,3,1,5,6,4] => ? = 6 - 2
[4,1,3,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [7,2,1,5,4,6,3] => ? = 5 - 2
[4,1,3,6,5,2] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [7,2,1,5,4,6,3] => ? = 5 - 2
[4,1,6,2,3,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [7,5,3,1,4,6,2] => ? = 4 - 2
[4,1,6,2,5,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [7,5,3,1,4,6,2] => ? = 4 - 2
[4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [7,5,3,1,4,6,2] => ? = 4 - 2
[4,1,6,3,5,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [7,5,3,1,4,6,2] => ? = 4 - 2
[4,1,6,5,2,3] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [7,5,3,1,4,6,2] => ? = 4 - 2
[4,1,6,5,3,2] => [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => [7,5,3,1,4,6,2] => ? = 4 - 2
[4,2,1,3,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => [7,2,3,1,5,6,4] => ? = 6 - 2
[4,2,1,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [7,2,1,5,4,6,3] => ? = 5 - 2
[4,2,1,6,5,3] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => [7,2,1,5,4,6,3] => ? = 5 - 2
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => 4 = 6 - 2
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => 4 = 6 - 2
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => 4 = 6 - 2
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => 4 = 6 - 2
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => 4 = 6 - 2
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => 4 = 6 - 2
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => 4 = 6 - 2
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [7,2,3,4,5,1,6] => 4 = 6 - 2
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000718The largest Laplacian eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001488The number of corners of a skew partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000080The rank of the poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000216The absolute length of a permutation. St000443The number of long tunnels of a Dyck path. St000831The number of indices that are either descents or recoils. St000863The length of the first row of the shifted shape of a permutation. St000956The maximal displacement of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001391The disjunction number of a graph. St001649The length of a longest trail in a graph. St000062The length of the longest increasing subsequence of the permutation. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. 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. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000039The number of crossings of a permutation. St000837The number of ascents of distance 2 of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001556The number of inversions of the third entry of a permutation. St001875The number of simple modules with projective dimension at most 1. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000784The maximum of the length and the largest part of the integer partition. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000455The second largest eigenvalue of a graph if it is integral. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001434The number of negative sum pairs of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001896The number of right descents of a signed permutations. St001866The nesting alignments of a signed permutation. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St000189The number of elements in the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001645The pebbling number of a connected graph.
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