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Your data matches 83 different statistics following compositions of up to 3 maps.
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Matching statistic: St001045
St001045: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 1
[(1,2),(3,4)]
=> 1
[(1,3),(2,4)]
=> 1
[(1,4),(2,3)]
=> 2
[(1,2),(3,4),(5,6)]
=> 2
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 1
[(1,6),(2,3),(4,5)]
=> 3
[(1,6),(2,4),(3,5)]
=> 3
[(1,5),(2,4),(3,6)]
=> 1
[(1,4),(2,5),(3,6)]
=> 1
[(1,3),(2,5),(4,6)]
=> 1
[(1,2),(3,5),(4,6)]
=> 1
[(1,2),(3,6),(4,5)]
=> 1
[(1,3),(2,6),(4,5)]
=> 1
[(1,4),(2,6),(3,5)]
=> 1
[(1,5),(2,6),(3,4)]
=> 1
[(1,6),(2,5),(3,4)]
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> 2
[(1,3),(2,4),(5,6),(7,8)]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> 4
[(1,8),(2,4),(3,5),(6,7)]
=> 4
[(1,7),(2,4),(3,5),(6,8)]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> 2
[(1,3),(2,6),(4,5),(7,8)]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> 4
[(1,8),(2,6),(3,4),(5,7)]
=> 4
[(1,7),(2,6),(3,4),(5,8)]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> 3
[(1,4),(2,7),(3,5),(6,8)]
=> 3
[(1,3),(2,7),(4,5),(6,8)]
=> 3
[(1,2),(3,7),(4,5),(6,8)]
=> 3
[(1,2),(3,8),(4,5),(6,7)]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> 1
Description
The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of {1,…,2n} and trees with n+1 leaves is described in Example 5.2.6 of [1].
The number of trees having precisely j leaves in the subtree not containing 1, computed in [2], is
\binom{n}{j}(2j-3)!!(2n-2j-1)!!
Matching statistic: St001132
St001132: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 1
[(1,2),(3,4)]
=> 1
[(1,3),(2,4)]
=> 1
[(1,4),(2,3)]
=> 2
[(1,2),(3,4),(5,6)]
=> 1
[(1,3),(2,4),(5,6)]
=> 1
[(1,4),(2,3),(5,6)]
=> 1
[(1,5),(2,3),(4,6)]
=> 2
[(1,6),(2,3),(4,5)]
=> 3
[(1,6),(2,4),(3,5)]
=> 3
[(1,5),(2,4),(3,6)]
=> 2
[(1,4),(2,5),(3,6)]
=> 1
[(1,3),(2,5),(4,6)]
=> 1
[(1,2),(3,5),(4,6)]
=> 1
[(1,2),(3,6),(4,5)]
=> 1
[(1,3),(2,6),(4,5)]
=> 1
[(1,4),(2,6),(3,5)]
=> 1
[(1,5),(2,6),(3,4)]
=> 2
[(1,6),(2,5),(3,4)]
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> 4
[(1,8),(2,4),(3,5),(6,7)]
=> 4
[(1,7),(2,4),(3,5),(6,8)]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> 4
[(1,8),(2,6),(3,4),(5,7)]
=> 4
[(1,7),(2,6),(3,4),(5,8)]
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> 1
Description
The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of \{1,\dots,2n\} and trees with n+1 leaves is described in Example 5.2.6 of [1].
Matching statistic: St000667
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> []
=> ? ∊ {1,1,2}
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St001389
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 80%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 80%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> []
=> ? ∊ {1,1,2}
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 5
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 5
Description
The number of partitions of the same length below the given integer partition.
For a partition \lambda_1 \geq \dots \lambda_k > 0, this number is
\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.
Matching statistic: St001432
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> []
=> ? ∊ {1,1,2}
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 3
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
Description
The order dimension of the partition.
Given a partition \lambda, let I(\lambda) be the principal order ideal in the Young lattice generated by \lambda. The order dimension of a partition is defined as the order dimension of the poset I(\lambda).
Matching statistic: St001527
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001527: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001527: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> []
=> ? ∊ {1,1,2}
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 5
Description
The cyclic permutation representation number of an integer partition.
This is the size of the largest cyclic group C such that the fake degree is the character of a permutation representation of C.
Matching statistic: St001571
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 60%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> []
=> ? ∊ {1,1,2}
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1,2}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
Description
The Cartan determinant of the integer partition.
Let p=[p_1,...,p_r] be a given integer partition with highest part t. Let A=K[x]/(x^t) be the finite dimensional algebra over the field K and M the direct sum of the indecomposable A-modules of vector space dimension p_i for each i. Then the Cartan determinant of p is the Cartan determinant of the endomorphism algebra of M over A.
Explicitly, this is the determinant of the matrix \left(\min(\bar p_i, \bar p_j)\right)_{i,j}, where \bar p is the set of distinct parts of the partition.
Matching statistic: St001199
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 80%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 80%
Values
[(1,2)]
=> [1,0]
=> [1] => [1,0]
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => [1,0,1,0]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => [1,1,0,0]
=> ? ∊ {1,2}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => [1,1,0,0]
=> ? ∊ {1,2}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,3,3}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,3,3}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,3,3}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,3,3}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,3,3}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,3,3}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [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]
=> [2,3,4,1] => [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]
=> [3,2,4,1] => [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]
=> [3,2,4,1] => [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]
=> [3,2,1,4] => [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]
=> [3,2,1,4] => [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]
=> [3,2,1,4] => [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]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [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]
=> [3,2,1,4] => [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]
=> [3,2,1,4] => [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]
=> [3,2,4,1] => [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]
=> [3,2,4,1] => [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]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [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]
=> [3,2,4,1] => [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]
=> [2,3,4,1] => [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]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [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]
=> [3,2,4,1] => [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]
=> [3,2,4,1] => [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]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,6),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[(1,4),(2,5),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[(1,5),(2,4),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[(1,6),(2,4),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,4),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,4),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[(1,8),(2,4),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,4),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,4),(3,8),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,6),(3,8),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,7),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
Description
The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
Matching statistic: St001232
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> []
=> []
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {1,2}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {1,2}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,2,3,3,3}
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,2,3,3,3}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,2,3,3,3}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,2,3,3,3}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,2,3,3,3}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,2,3,3,3}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,2,3,3,3}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [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]
=> [3]
=> [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]
=> [3]
=> [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]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [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]
=> [3]
=> [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]
=> [3]
=> [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]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,6),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,5),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,5),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,5),(2,4),(3,7),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[(1,6),(2,4),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,7),(2,4),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,8),(2,4),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[(1,5),(2,3),(4,7),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,3),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,5),(4,8),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000207
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {1,1,2}
[(1,3),(2,4)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,1,2}
[(1,4),(2,3)]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,1,2}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,3,3,3}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4}
[(1,3),(2,6),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,6),(2,3),(4,7),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,7),(6,8)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[(1,6),(2,3),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,7),(2,3),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,6),(4,8),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 5
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 5
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 4
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 4
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 5
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 5
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given \lambda count how many ''integer compositions'' w (weight) there are, such that
P_{\lambda,w} is integral, i.e., w such that the Gelfand-Tsetlin polytope P_{\lambda,w} has all vertices in integer lattice points.
The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000456The monochromatic index of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000260The radius of a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000534The number of 2-rises of a permutation. St000451The length of the longest pattern of the form k 1 2. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000356The number of occurrences of the pattern 13-2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000007The number of saliances of the permutation. St000264The girth of a graph, which is not a tree. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000352The Elizalde-Pak rank of a permutation. St000883The number of longest increasing subsequences of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001096The size of the overlap set of a permutation. St000648The number of 2-excedences of a permutation. St000842The breadth of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000237The number of small exceedances. St000366The number of double descents of a permutation. St000731The number of double exceedences of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001552The number of inversions between excedances and fixed points of a permutation. St000022The number of fixed points of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001052The length of the exterior of a permutation.
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