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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001418
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St000782
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 0
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> ? = 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> ? = 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> ? = 3
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> ? = 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> ? = 3
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> ? = 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> ? = 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> ? = 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> ? = 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> ? = 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
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