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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St001195
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Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 0
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 0
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 0
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 0
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 0
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 0
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> 1
Description
The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
Matching statistic: St000260
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? = 0
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 0
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 0
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 0
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,4),(2,6),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[(1,4),(2,5),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,5),(2,4),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,6),(2,4),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 1
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 1
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? = 0
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? = 0
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? = 0
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 0
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 0
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 0
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? = 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001199
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,5),(2,10),(3,6),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,5),(2,9),(3,6),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,6),(2,9),(3,5),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,7),(2,9),(3,5),(4,6),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,8),(2,9),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,9),(2,8),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,10),(2,8),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,10),(2,7),(3,5),(4,6),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,9),(2,7),(3,5),(4,6),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,8),(2,7),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,7),(2,8),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,6),(2,8),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,5),(2,8),(3,6),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,5),(2,7),(3,6),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,6),(2,7),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,7),(2,6),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,8),(2,6),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,9),(2,6),(3,5),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,10),(2,6),(3,5),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,10),(2,5),(3,6),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[(1,9),(2,5),(3,6),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
Description
The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
Matching statistic: St001198
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,10),(3,6),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,9),(3,6),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,9),(3,5),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,9),(3,5),(4,6),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,9),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,8),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,8),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,7),(3,5),(4,6),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,7),(3,5),(4,6),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,7),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,8),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,8),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,8),(3,6),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,6),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,6),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,6),(3,5),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,6),(3,5),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,5),(3,6),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,5),(3,6),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
Matching statistic: St001206
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,10),(3,6),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,9),(3,6),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,9),(3,5),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,9),(3,5),(4,6),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,9),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,8),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,8),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,7),(3,5),(4,6),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,7),(3,5),(4,6),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,7),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,8),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,8),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,8),(3,6),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,6),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,8),(2,6),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,6),(3,5),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,6),(3,5),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,10),(2,5),(3,6),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[(1,9),(2,5),(3,6),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA.
Matching statistic: St001498
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 - 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 - 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 0 - 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,10),(3,6),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,9),(3,6),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,9),(3,5),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,9),(3,5),(4,6),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,9),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,9),(2,8),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,10),(2,8),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,10),(2,7),(3,5),(4,6),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,9),(2,7),(3,5),(4,6),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,7),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,8),(3,5),(4,6),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,8),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,8),(3,6),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,5),(2,7),(3,6),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,6),(2,7),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,7),(2,6),(3,5),(4,8),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,8),(2,6),(3,5),(4,7),(9,10)]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,9),(2,6),(3,5),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,10),(2,6),(3,5),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,10),(2,5),(3,6),(4,7),(8,9)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
[(1,9),(2,5),(3,6),(4,7),(8,10)]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0 = 1 - 1
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 0 + 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 0 + 3
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 0 + 3
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1 + 3
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1 + 3
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1 + 3
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? = 0 + 3
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 0 + 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 0 + 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 3
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,4),(2,6),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,6),(2,4),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,7),(2,4),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,8),(2,4),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,8),(2,4),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,7),(2,4),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,6),(2,4),(3,8),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,4),(2,6),(3,8),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,4),(2,7),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,4),(2,8),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 1 + 3
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,9),(2,10),(3,5),(4,6),(7,8)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,8),(2,10),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,4),(2,10),(3,6),(5,7),(8,9)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,4),(2,9),(3,6),(5,7),(8,10)]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,8),(2,9),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,9),(2,8),(3,5),(4,6),(7,10)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,10),(2,8),(3,5),(4,6),(7,9)]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 1 + 3
[(1,4),(2,8),(3,6),(5,7),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,4),(2,7),(3,6),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,4),(2,6),(3,7),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
[(1,6),(2,4),(3,7),(5,8),(9,10)]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 1 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000374
Mp00144: Perfect matchings —rotation⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,6,2,5,4,1] => 1 = 0 + 1
[(1,3),(2,4),(5,6)]
=> [(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [4,6,5,2,3,1] => 1 = 0 + 1
[(1,4),(2,3),(5,6)]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,6,5,3,2,1] => 1 = 0 + 1
[(1,5),(2,3),(4,6)]
=> [(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [4,6,5,3,1,2] => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => 2 = 1 + 1
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,6,5,1,3,2] => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
=> [(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,6,5,1,3,2] => 2 = 1 + 1
[(1,3),(2,5),(4,6)]
=> [(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [4,6,5,2,1,3] => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,6,2,5,1,4] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,6,1,5,4,2] => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> [(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
=> [(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,6,1,5,4,2] => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [3,8,2,7,6,5,4,1] => ? = 0 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [4,8,7,2,6,5,3,1] => ? = 0 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,8,7,3,2,6,5,1] => ? = 0 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,8,7,3,6,2,5,1] => ? = 0 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => ? = 0 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [4,8,7,3,6,5,1,2] => ? = 1 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [5,8,7,6,3,4,1,2] => ? = 1 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [5,8,7,6,3,4,2,1] => ? = 1 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [5,8,7,6,3,2,4,1] => ? = 1 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [5,8,7,6,2,4,3,1] => ? = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,8),(2,4),(3,6),(5,7)]
=> [4,6,7,2,8,3,5,1] => [4,8,7,2,6,5,3,1] => ? = 0 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,8),(2,3),(4,6),(5,7)]
=> [3,6,2,7,8,4,5,1] => [3,8,2,7,6,5,4,1] => ? = 0 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [3,8,2,7,6,5,4,1] => ? = 0 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [4,6,7,2,8,5,3,1] => [4,8,7,2,6,5,3,1] => ? = 0 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,8),(2,5),(3,7),(4,6)]
=> [5,6,7,8,2,4,3,1] => [5,8,7,6,2,4,3,1] => ? = 1 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,8),(2,6),(3,7),(4,5)]
=> [5,6,7,8,4,2,3,1] => [5,8,7,6,4,2,3,1] => ? = 1 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [5,8,7,6,4,3,2,1] => ? = 1 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,7),(2,8),(3,6),(4,5)]
=> [5,6,7,8,4,3,1,2] => [5,8,7,6,4,3,1,2] => ? = 1 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,6),(2,8),(3,7),(4,5)]
=> [5,6,7,8,4,1,3,2] => [5,8,7,6,4,1,3,2] => ? = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,6),(2,7),(3,8),(4,5)]
=> [5,6,7,8,4,1,2,3] => [5,8,7,6,4,1,3,2] => ? = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,7),(2,6),(3,8),(4,5)]
=> [5,6,7,8,4,2,1,3] => [5,8,7,6,4,2,1,3] => ? = 1 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,7),(2,5),(3,8),(4,6)]
=> [5,6,7,8,2,4,1,3] => [5,8,7,6,2,4,1,3] => ? = 1 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,7),(2,4),(3,8),(5,6)]
=> [4,6,7,2,8,5,1,3] => [4,8,7,2,6,5,1,3] => ? = 1 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,7),(2,3),(4,8),(5,6)]
=> [3,6,2,7,8,5,1,4] => [3,8,2,7,6,5,1,4] => ? = 1 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => ? = 1 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,5),(2,8),(3,4),(6,7)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,6),(2,8),(3,4),(5,7)]
=> [(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,7),(2,8),(3,4),(5,6)]
=> [(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,8),(2,7),(3,4),(5,6)]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,8),(2,7),(3,5),(4,6)]
=> [(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,7),(2,8),(3,5),(4,6)]
=> [(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,6),(2,8),(3,5),(4,7)]
=> [(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,5),(2,8),(3,6),(4,7)]
=> [(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,4),(2,8),(3,6),(5,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,3),(2,8),(4,6),(5,7)]
=> [(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => [3,8,1,7,6,5,4,2] => 2 = 1 + 1
[(1,2),(3,8),(4,6),(5,7)]
=> [(1,4),(2,3),(5,7),(6,8)]
=> [3,4,2,1,7,8,5,6] => [3,8,2,1,7,6,5,4] => ? = 1 + 1
[(1,2),(3,7),(4,6),(5,8)]
=> [(1,6),(2,3),(4,8),(5,7)]
=> [3,6,2,7,8,1,5,4] => [3,8,2,7,6,1,5,4] => ? = 1 + 1
[(1,3),(2,7),(4,6),(5,8)]
=> [(1,6),(2,4),(3,8),(5,7)]
=> [4,6,7,2,8,1,5,3] => [4,8,7,2,6,1,5,3] => ? = 1 + 1
[(1,4),(2,7),(3,6),(5,8)]
=> [(1,6),(2,5),(3,8),(4,7)]
=> [5,6,7,8,2,1,4,3] => [5,8,7,6,2,1,4,3] => ? = 1 + 1
[(1,5),(2,7),(3,6),(4,8)]
=> [(1,5),(2,6),(3,8),(4,7)]
=> [5,6,7,8,1,2,4,3] => [5,8,7,6,1,4,3,2] => 2 = 1 + 1
[(1,6),(2,7),(3,5),(4,8)]
=> [(1,5),(2,7),(3,8),(4,6)]
=> [5,6,7,8,1,4,2,3] => [5,8,7,6,1,4,3,2] => 2 = 1 + 1
[(1,7),(2,6),(3,5),(4,8)]
=> [(1,5),(2,8),(3,7),(4,6)]
=> [5,6,7,8,1,4,3,2] => [5,8,7,6,1,4,3,2] => 2 = 1 + 1
[(1,8),(2,6),(3,5),(4,7)]
=> [(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,8),(2,5),(3,6),(4,7)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,7),(2,5),(3,6),(4,8)]
=> [(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [5,8,7,6,1,4,3,2] => 2 = 1 + 1
[(1,6),(2,5),(3,7),(4,8)]
=> [(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [5,8,7,6,1,4,3,2] => 2 = 1 + 1
[(1,5),(2,6),(3,7),(4,8)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => 2 = 1 + 1
[(1,4),(2,6),(3,7),(5,8)]
=> [(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [5,8,7,6,2,1,4,3] => ? = 1 + 1
[(1,3),(2,6),(4,7),(5,8)]
=> [(1,6),(2,4),(3,7),(5,8)]
=> [4,6,7,2,8,1,3,5] => [4,8,7,2,6,1,5,3] => ? = 1 + 1
[(1,2),(3,6),(4,7),(5,8)]
=> [(1,6),(2,3),(4,7),(5,8)]
=> [3,6,2,7,8,1,4,5] => [3,8,2,7,6,1,5,4] => ? = 1 + 1
[(1,2),(3,5),(4,7),(6,8)]
=> [(1,7),(2,3),(4,6),(5,8)]
=> [3,6,2,7,8,4,1,5] => [3,8,2,7,6,5,1,4] => ? = 1 + 1
[(1,3),(2,5),(4,7),(6,8)]
=> [(1,7),(2,4),(3,6),(5,8)]
=> [4,6,7,2,8,3,1,5] => [4,8,7,2,6,5,1,3] => ? = 1 + 1
[(1,4),(2,5),(3,7),(6,8)]
=> [(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [5,8,7,6,2,4,1,3] => ? = 1 + 1
[(1,5),(2,4),(3,7),(6,8)]
=> [(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [5,8,7,6,3,2,1,4] => ? = 1 + 1
[(1,6),(2,4),(3,7),(5,8)]
=> [(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [5,8,7,6,3,1,4,2] => ? = 1 + 1
[(1,7),(2,4),(3,6),(5,8)]
=> [(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [5,8,7,6,3,1,4,2] => ? = 1 + 1
[(1,8),(2,4),(3,6),(5,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,8),(2,3),(4,6),(5,7)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,7),(2,3),(4,6),(5,8)]
=> [(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [4,8,7,3,6,1,5,2] => ? = 1 + 1
[(1,6),(2,3),(4,7),(5,8)]
=> [(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [4,8,7,3,6,1,5,2] => ? = 1 + 1
[(1,5),(2,3),(4,7),(6,8)]
=> [(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,8,7,3,6,2,1,5] => ? = 1 + 1
[(1,4),(2,3),(5,7),(6,8)]
=> [(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,8,7,3,2,6,1,5] => ? = 1 + 1
[(1,3),(2,4),(5,7),(6,8)]
=> [(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [4,8,7,2,6,5,1,3] => ? = 1 + 1
[(1,2),(3,4),(5,7),(6,8)]
=> [(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [3,8,2,7,6,5,1,4] => ? = 1 + 1
[(1,2),(3,4),(5,8),(6,7)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => ? = 1 + 1
[(1,3),(2,4),(5,8),(6,7)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [4,8,7,2,6,1,5,3] => ? = 1 + 1
[(1,4),(2,3),(5,8),(6,7)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => ? = 1 + 1
[(1,5),(2,3),(4,8),(6,7)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [4,8,7,3,1,6,5,2] => ? = 1 + 1
[(1,6),(2,3),(4,8),(5,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [4,8,7,3,1,6,5,2] => ? = 1 + 1
[(1,7),(2,3),(4,8),(5,6)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [4,8,7,3,1,6,5,2] => ? = 1 + 1
[(1,8),(2,3),(4,7),(5,6)]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,8),(2,4),(3,7),(5,6)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,7,3,8,6,4] => [2,1,8,7,6,5,4,3] => 2 = 1 + 1
[(1,7),(2,4),(3,8),(5,6)]
=> [(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [4,8,7,1,6,5,3,2] => 2 = 1 + 1
[(1,6),(2,4),(3,8),(5,7)]
=> [(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [4,8,7,1,6,5,3,2] => 2 = 1 + 1
[(1,5),(2,4),(3,8),(6,7)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,8,7,1,6,5,3,2] => 2 = 1 + 1
[(1,4),(2,5),(3,8),(6,7)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,8,7,1,6,5,3,2] => 2 = 1 + 1
[(1,4),(2,6),(3,8),(5,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [4,8,7,1,6,5,3,2] => 2 = 1 + 1
[(1,5),(2,6),(3,8),(4,7)]
=> [(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [4,8,7,1,6,5,3,2] => 2 = 1 + 1
[(1,6),(2,5),(3,8),(4,7)]
=> [(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [4,8,7,1,6,5,3,2] => 2 = 1 + 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation π=[π1,…,πn], this statistic counts the number of position j such that πj<j and there do not exist indices i,k with i<j<k and πi>πj>πk.
See also [[St000213]] and [[St000119]].
Matching statistic: St001004
Mp00144: Perfect matchings —rotation⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [3,6,2,5,4,1] => 3 = 0 + 3
[(1,3),(2,4),(5,6)]
=> [(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [4,6,5,2,3,1] => 3 = 0 + 3
[(1,4),(2,3),(5,6)]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [4,6,5,3,2,1] => 3 = 0 + 3
[(1,5),(2,3),(4,6)]
=> [(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [4,6,5,3,1,2] => 4 = 1 + 3
[(1,6),(2,3),(4,5)]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 4 = 1 + 3
[(1,6),(2,4),(3,5)]
=> [(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => 4 = 1 + 3
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,6,5,1,3,2] => 4 = 1 + 3
[(1,4),(2,5),(3,6)]
=> [(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,6,5,1,3,2] => 4 = 1 + 3
[(1,3),(2,5),(4,6)]
=> [(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [4,6,5,2,1,3] => 4 = 1 + 3
[(1,2),(3,5),(4,6)]
=> [(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [3,6,2,5,1,4] => 4 = 1 + 3
[(1,2),(3,6),(4,5)]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => 4 = 1 + 3
[(1,3),(2,6),(4,5)]
=> [(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,6,1,5,4,2] => 4 = 1 + 3
[(1,4),(2,6),(3,5)]
=> [(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => 4 = 1 + 3
[(1,5),(2,6),(3,4)]
=> [(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,6,1,5,4,2] => 4 = 1 + 3
[(1,6),(2,5),(3,4)]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 4 = 1 + 3
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [3,8,2,7,6,5,4,1] => ? = 0 + 3
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [4,8,7,2,6,5,3,1] => ? = 0 + 3
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [4,8,7,3,2,6,5,1] => ? = 0 + 3
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [4,8,7,3,6,2,5,1] => ? = 0 + 3
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => ? = 0 + 3
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [4,8,7,3,6,5,1,2] => ? = 1 + 3
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [5,8,7,6,3,4,1,2] => ? = 1 + 3
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [5,8,7,6,3,4,2,1] => ? = 1 + 3
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [5,8,7,6,3,2,4,1] => ? = 1 + 3
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [5,8,7,6,2,4,3,1] => ? = 1 + 3
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,8),(2,4),(3,6),(5,7)]
=> [4,6,7,2,8,3,5,1] => [4,8,7,2,6,5,3,1] => ? = 0 + 3
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,8),(2,3),(4,6),(5,7)]
=> [3,6,2,7,8,4,5,1] => [3,8,2,7,6,5,4,1] => ? = 0 + 3
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [3,8,2,7,6,5,4,1] => ? = 0 + 3
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [4,6,7,2,8,5,3,1] => [4,8,7,2,6,5,3,1] => ? = 0 + 3
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,8),(2,5),(3,7),(4,6)]
=> [5,6,7,8,2,4,3,1] => [5,8,7,6,2,4,3,1] => ? = 1 + 3
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,8),(2,6),(3,7),(4,5)]
=> [5,6,7,8,4,2,3,1] => [5,8,7,6,4,2,3,1] => ? = 1 + 3
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [5,8,7,6,4,3,2,1] => ? = 1 + 3
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,7),(2,8),(3,6),(4,5)]
=> [5,6,7,8,4,3,1,2] => [5,8,7,6,4,3,1,2] => ? = 1 + 3
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,6),(2,8),(3,7),(4,5)]
=> [5,6,7,8,4,1,3,2] => [5,8,7,6,4,1,3,2] => ? = 1 + 3
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,6),(2,7),(3,8),(4,5)]
=> [5,6,7,8,4,1,2,3] => [5,8,7,6,4,1,3,2] => ? = 1 + 3
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,7),(2,6),(3,8),(4,5)]
=> [5,6,7,8,4,2,1,3] => [5,8,7,6,4,2,1,3] => ? = 1 + 3
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,7),(2,5),(3,8),(4,6)]
=> [5,6,7,8,2,4,1,3] => [5,8,7,6,2,4,1,3] => ? = 1 + 3
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,7),(2,4),(3,8),(5,6)]
=> [4,6,7,2,8,5,1,3] => [4,8,7,2,6,5,1,3] => ? = 1 + 3
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,7),(2,3),(4,8),(5,6)]
=> [3,6,2,7,8,5,1,4] => [3,8,2,7,6,5,1,4] => ? = 1 + 3
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => ? = 1 + 3
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,5),(2,8),(3,4),(6,7)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,6),(2,8),(3,4),(5,7)]
=> [(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,7),(2,8),(3,4),(5,6)]
=> [(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,8),(2,7),(3,4),(5,6)]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,8),(2,7),(3,5),(4,6)]
=> [(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,7),(2,8),(3,5),(4,6)]
=> [(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,6),(2,8),(3,5),(4,7)]
=> [(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,5),(2,8),(3,6),(4,7)]
=> [(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,4),(2,8),(3,6),(5,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,3),(2,8),(4,6),(5,7)]
=> [(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => [3,8,1,7,6,5,4,2] => 4 = 1 + 3
[(1,2),(3,8),(4,6),(5,7)]
=> [(1,4),(2,3),(5,7),(6,8)]
=> [3,4,2,1,7,8,5,6] => [3,8,2,1,7,6,5,4] => ? = 1 + 3
[(1,2),(3,7),(4,6),(5,8)]
=> [(1,6),(2,3),(4,8),(5,7)]
=> [3,6,2,7,8,1,5,4] => [3,8,2,7,6,1,5,4] => ? = 1 + 3
[(1,3),(2,7),(4,6),(5,8)]
=> [(1,6),(2,4),(3,8),(5,7)]
=> [4,6,7,2,8,1,5,3] => [4,8,7,2,6,1,5,3] => ? = 1 + 3
[(1,4),(2,7),(3,6),(5,8)]
=> [(1,6),(2,5),(3,8),(4,7)]
=> [5,6,7,8,2,1,4,3] => [5,8,7,6,2,1,4,3] => ? = 1 + 3
[(1,5),(2,7),(3,6),(4,8)]
=> [(1,5),(2,6),(3,8),(4,7)]
=> [5,6,7,8,1,2,4,3] => [5,8,7,6,1,4,3,2] => 4 = 1 + 3
[(1,6),(2,7),(3,5),(4,8)]
=> [(1,5),(2,7),(3,8),(4,6)]
=> [5,6,7,8,1,4,2,3] => [5,8,7,6,1,4,3,2] => 4 = 1 + 3
[(1,7),(2,6),(3,5),(4,8)]
=> [(1,5),(2,8),(3,7),(4,6)]
=> [5,6,7,8,1,4,3,2] => [5,8,7,6,1,4,3,2] => 4 = 1 + 3
[(1,8),(2,6),(3,5),(4,7)]
=> [(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,8),(2,5),(3,6),(4,7)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,7),(2,5),(3,6),(4,8)]
=> [(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [5,8,7,6,1,4,3,2] => 4 = 1 + 3
[(1,6),(2,5),(3,7),(4,8)]
=> [(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [5,8,7,6,1,4,3,2] => 4 = 1 + 3
[(1,5),(2,6),(3,7),(4,8)]
=> [(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => 4 = 1 + 3
[(1,4),(2,6),(3,7),(5,8)]
=> [(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [5,8,7,6,2,1,4,3] => ? = 1 + 3
[(1,3),(2,6),(4,7),(5,8)]
=> [(1,6),(2,4),(3,7),(5,8)]
=> [4,6,7,2,8,1,3,5] => [4,8,7,2,6,1,5,3] => ? = 1 + 3
[(1,2),(3,6),(4,7),(5,8)]
=> [(1,6),(2,3),(4,7),(5,8)]
=> [3,6,2,7,8,1,4,5] => [3,8,2,7,6,1,5,4] => ? = 1 + 3
[(1,2),(3,5),(4,7),(6,8)]
=> [(1,7),(2,3),(4,6),(5,8)]
=> [3,6,2,7,8,4,1,5] => [3,8,2,7,6,5,1,4] => ? = 1 + 3
[(1,3),(2,5),(4,7),(6,8)]
=> [(1,7),(2,4),(3,6),(5,8)]
=> [4,6,7,2,8,3,1,5] => [4,8,7,2,6,5,1,3] => ? = 1 + 3
[(1,4),(2,5),(3,7),(6,8)]
=> [(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [5,8,7,6,2,4,1,3] => ? = 1 + 3
[(1,5),(2,4),(3,7),(6,8)]
=> [(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [5,8,7,6,3,2,1,4] => ? = 1 + 3
[(1,6),(2,4),(3,7),(5,8)]
=> [(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [5,8,7,6,3,1,4,2] => ? = 1 + 3
[(1,7),(2,4),(3,6),(5,8)]
=> [(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [5,8,7,6,3,1,4,2] => ? = 1 + 3
[(1,8),(2,4),(3,6),(5,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,8),(2,3),(4,6),(5,7)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,7),(2,3),(4,6),(5,8)]
=> [(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [4,8,7,3,6,1,5,2] => ? = 1 + 3
[(1,6),(2,3),(4,7),(5,8)]
=> [(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [4,8,7,3,6,1,5,2] => ? = 1 + 3
[(1,5),(2,3),(4,7),(6,8)]
=> [(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [4,8,7,3,6,2,1,5] => ? = 1 + 3
[(1,4),(2,3),(5,7),(6,8)]
=> [(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [4,8,7,3,2,6,1,5] => ? = 1 + 3
[(1,3),(2,4),(5,7),(6,8)]
=> [(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [4,8,7,2,6,5,1,3] => ? = 1 + 3
[(1,2),(3,4),(5,7),(6,8)]
=> [(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [3,8,2,7,6,5,1,4] => ? = 1 + 3
[(1,2),(3,4),(5,8),(6,7)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => ? = 1 + 3
[(1,3),(2,4),(5,8),(6,7)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [4,8,7,2,6,1,5,3] => ? = 1 + 3
[(1,4),(2,3),(5,8),(6,7)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => ? = 1 + 3
[(1,5),(2,3),(4,8),(6,7)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [4,8,7,3,1,6,5,2] => ? = 1 + 3
[(1,6),(2,3),(4,8),(5,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [4,8,7,3,1,6,5,2] => ? = 1 + 3
[(1,7),(2,3),(4,8),(5,6)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [4,8,7,3,1,6,5,2] => ? = 1 + 3
[(1,8),(2,3),(4,7),(5,6)]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,8),(2,4),(3,7),(5,6)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [2,1,5,7,3,8,6,4] => [2,1,8,7,6,5,4,3] => 4 = 1 + 3
[(1,7),(2,4),(3,8),(5,6)]
=> [(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [4,8,7,1,6,5,3,2] => 4 = 1 + 3
[(1,6),(2,4),(3,8),(5,7)]
=> [(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [4,8,7,1,6,5,3,2] => 4 = 1 + 3
[(1,5),(2,4),(3,8),(6,7)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,8,7,1,6,5,3,2] => 4 = 1 + 3
[(1,4),(2,5),(3,8),(6,7)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,8,7,1,6,5,3,2] => 4 = 1 + 3
[(1,4),(2,6),(3,8),(5,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [4,8,7,1,6,5,3,2] => 4 = 1 + 3
[(1,5),(2,6),(3,8),(4,7)]
=> [(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [4,8,7,1,6,5,3,2] => 4 = 1 + 3
[(1,6),(2,5),(3,8),(4,7)]
=> [(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [4,8,7,1,6,5,3,2] => 4 = 1 + 3
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a 321 pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
Matching statistic: St001085
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,1,2] => 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => 0
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => 0
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => 1
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 1
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,2,3,1,5,6] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 1
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,4,5,6] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,4,1,2] => 0
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,2,1,3,4] => ? = 0
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,2,1,3,4] => ? = 0
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [7,8,4,2,3,1,5,6] => ? = 0
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [7,8,4,2,3,1,5,6] => ? = 0
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,2,3,1,7,8] => ? = 1
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,2,3,1,7,8] => ? = 1
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [7,8,3,2,1,4,5,6] => ? = 1
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [7,8,3,2,1,4,5,6] => ? = 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [7,8,3,2,1,4,5,6] => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [7,8,4,2,3,1,5,6] => ? = 0
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,8,4,3,5,6,1,2] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,8,4,3,5,6,1,2] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [7,8,4,2,3,1,5,6] => ? = 0
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [7,8,3,2,1,4,5,6] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [7,8,3,2,1,4,5,6] => ? = 1
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [7,8,3,2,1,4,5,6] => ? = 1
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,2,3,1,7,8] => ? = 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,1,2] => ? = 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,1,2] => ? = 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,2,3,1,7,8] => ? = 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,2,3,6,1,7,8] => ? = 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,8,1,2] => ? = 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,8,1,2] => ? = 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,2,3,6,1,7,8] => ? = 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [5,4,2,3,6,1,7,8] => ? = 1
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,8,1,2] => ? = 1
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,1,2] => ? = 1
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [6,4,5,2,3,1,7,8] => ? = 1
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,1,5,7,8] => ? = 1
[(1,6),(2,4),(3,7),(5,8)]
=> [4,6,7,2,8,1,3,5] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,7),(2,4),(3,6),(5,8)]
=> [4,6,7,2,8,3,1,5] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,8),(2,4),(3,6),(5,7)]
=> [4,6,7,2,8,3,5,1] => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,1,6,7,8] => ? = 1
[(1,4),(2,3),(5,7),(6,8)]
=> [3,4,2,1,7,8,5,6] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,2,1,3,4] => 1
[(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,2,1,3,4] => 1
[(1,3),(2,4),(5,8),(6,7)]
=> [3,4,1,2,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,2,1,3,4] => 1
[(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,2,1,3,4] => 1
[(1,5),(2,6),(3,8),(4,7)]
=> [5,6,7,8,1,2,4,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,6),(2,5),(3,8),(4,7)]
=> [5,6,7,8,2,1,4,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,7),(2,5),(3,8),(4,6)]
=> [5,6,7,8,2,4,1,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,8),(2,5),(3,7),(4,6)]
=> [5,6,7,8,2,4,3,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,8),(2,6),(3,7),(4,5)]
=> [5,6,7,8,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,7),(2,6),(3,8),(4,5)]
=> [5,6,7,8,4,2,1,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,6),(2,7),(3,8),(4,5)]
=> [5,6,7,8,4,1,2,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,5),(2,7),(3,8),(4,6)]
=> [5,6,7,8,1,4,2,3] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,5),(2,8),(3,7),(4,6)]
=> [5,6,7,8,1,4,3,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,6),(2,8),(3,7),(4,5)]
=> [5,6,7,8,4,1,3,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,7),(2,8),(3,6),(4,5)]
=> [5,6,7,8,4,3,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,1,5,6,7,8] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [9,10,7,8,5,6,3,4,1,2] => 0
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [6,7,8,9,10,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,4,3,2,1,6,7,8,9,10] => 1
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [6,7,8,9,10,1,4,5,3,2] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,4,3,2,1,6,7,8,9,10] => 1
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [6,7,8,9,10,1,4,5,2,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,4,3,2,1,6,7,8,9,10] => 1
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [6,7,8,9,10,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,4,3,2,1,6,7,8,9,10] => 1
[(1,9),(2,8),(3,7),(4,6),(5,10)]
=> [6,7,8,9,10,4,3,2,1,5] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,4,3,2,1,6,7,8,9,10] => 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern 213, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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