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Your data matches 89 different statistics following compositions of up to 3 maps.
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Matching statistic: St001020
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 2
[2,1] => [1,1,0,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> 3
[2,1,3] => [1,1,0,0,1,0]
=> 3
[2,3,1] => [1,1,0,1,0,0]
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 4
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 4
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 4
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 4
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 5
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 5
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 5
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 5
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 5
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 5
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 5
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 5
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000528
(load all 200 compositions to match this statistic)
(load all 200 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 78%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 78%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 2
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 5
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 6
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 6
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 6
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 6
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 6
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 6
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 6
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 4
[1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 6
[1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 6
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 6
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 6
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 6
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 6
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 6
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 5
[1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 7
[1,3,4,6,2,5] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 5
[1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 5
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6
Description
The height of a poset.
This equals the rank of the poset [[St000080]] plus one.
Matching statistic: St001880
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 56%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 56%
Values
[1] => [.,.]
=> [.,.]
=> ([],1)
=> ? = 1
[1,2] => [.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2
[2,1] => [[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 2
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3
[3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 4
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 4
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 5
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 5
[4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 56%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 56%
Values
[1] => [.,.]
=> [.,.]
=> ([],1)
=> ? = 1 - 1
[1,2] => [.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2 - 1
[2,1] => [[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 2 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3 - 1
[3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 4 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 4 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5 - 1
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5 - 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5 - 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5 - 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5 - 1
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5 - 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5 - 1
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5 - 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5 - 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 5 - 1
[4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 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: St001232
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 78%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 78%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5 - 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [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: St001636
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001636: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 2
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 5
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 6
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 6
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 6
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 4
[1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6
[1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 6
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 6
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 6
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5
[1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 7
Description
The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset.
Matching statistic: St000906
(load all 250 compositions to match this statistic)
(load all 250 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 56%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 56%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> ? = 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 2
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 5
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 6
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 6
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 6
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 4
[1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6
[1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 6
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 6
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 6
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St000643
(load all 250 compositions to match this statistic)
(load all 250 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000643: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 56%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000643: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 56%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> ? = 1 + 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 3 = 2 + 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 3 = 2 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 4 = 3 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 4 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 4 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5 = 4 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5 = 4 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 4 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 5 = 4 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 6 = 5 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 4 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6 = 5 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 5 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6 = 5 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 5 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6 = 5 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6 = 5 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 + 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6 = 5 + 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6 = 5 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 6 = 5 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6 = 5 + 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 + 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 + 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 + 1
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6 + 1
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6 + 1
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 6 + 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 6 + 1
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 + 1
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 + 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 6 + 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6 + 1
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6 + 1
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 4 + 1
[1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6 + 1
[1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6 + 1
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6 + 1
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6 + 1
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6 + 1
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6 + 1
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6 + 1
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6 + 1
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 + 1
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 + 1
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 + 1
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 + 1
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 + 1
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 + 1
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 6 + 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6 + 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 6 + 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 6 + 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6 + 1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6 + 1
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6 + 1
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6 + 1
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5 + 1
Description
The size of the largest orbit of antichains under Panyushev complementation.
Matching statistic: St000080
(load all 247 compositions to match this statistic)
(load all 247 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 4 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 5 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 5 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 5 - 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 - 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 5 - 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 5 - 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 - 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 - 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 5 - 1
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6 - 1
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 6 - 1
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 6 - 1
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 6 - 1
[1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
[1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 6 - 1
[1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6 - 1
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6 - 1
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 4 - 1
[1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6 - 1
[1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6 - 1
[1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6 - 1
[1,2,5,3,6,4] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6 - 1
[1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 6 - 1
[1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 6 - 1
[1,2,5,6,3,4] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6 - 1
[1,2,5,6,4,3] => [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 6 - 1
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6 - 1
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 6 - 1
[1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6 - 1
[1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 6 - 1
[1,3,2,5,6,4] => [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 6 - 1
[1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6 - 1
[1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 6 - 1
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6 - 1
[1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 6 - 1
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5 - 1
Description
The rank of the poset.
Matching statistic: St001583
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,2] => [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[2,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,2,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,2,3,4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ? = 4
[1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[3,4,1,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[3,4,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[4,1,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,2,1,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,3,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[4,3,2,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ? = 4
[1,2,3,4,5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ? = 5
[1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[1,2,5,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[1,3,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 4
[1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,4,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[1,4,3,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,4,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[1,4,5,3,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,5,3,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,5,4,2,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
[2,3,1,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[2,3,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 6
[2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[3,5,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[4,3,2,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[4,3,2,5,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[4,3,5,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[4,5,3,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,1,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? = 5
[5,1,4,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,2,4,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,3,2,1,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,3,2,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,3,4,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,4,1,3,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,4,2,1,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,4,2,3,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,4,3,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? = 5
[5,4,3,2,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? = 5
[1,2,3,4,5,6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 6
[1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? = 6
[1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? = 6
[1,2,3,5,6,4] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? = 6
[1,2,3,6,4,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? = 6
[1,2,3,6,5,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? = 6
[1,2,4,3,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? = 6
[1,2,4,3,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ? = 6
[1,2,4,5,3,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? = 6
[1,2,4,5,6,3] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? = 4
[1,2,4,6,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ? = 6
[1,2,4,6,5,3] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? = 6
[1,2,5,3,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? = 6
[1,2,5,3,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ? = 6
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
The following 79 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001875The number of simple modules with projective dimension at most 1. St001621The number of atoms of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St000097The order of the largest clique of the graph. St000288The number of ones in a binary word. St000273The domination number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000778The metric dimension of a graph. St001316The domatic number of a graph. St001429The number of negative entries in a signed permutation. St000135The number of lucky cars of the parking function. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000744The length of the path to the largest entry in a standard Young tableau. St000044The number of vertices of the unicellular map given by a perfect matching. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000327The number of cover relations in a poset. St000691The number of changes of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001581The achromatic number of a graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St001978The codimension of the alternating sign matrix variety. St000017The number of inversions of a standard tableau. St000035The number of left outer peaks of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000820The number of compositions obtained by rotating the composition. St000834The number of right outer peaks of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000884The number of isolated descents of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000924The number of topologically connected components of a perfect matching. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001480The number of simple summands of the module J^2/J^3. St001958The degree of the polynomial interpolating the values of a permutation. St001974The rank of the alternating sign matrix. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001468The smallest fixpoint of a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001409The maximal entry of a semistandard tableau. St000168The number of internal nodes of an ordered tree. St001926Sparre Andersen's position of the maximum of a signed permutation. St001555The order of a signed permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000782The indicator function of whether a given perfect matching is an L & P matching. St000741The Colin de Verdière graph invariant.
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