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Your data matches 90 different statistics following compositions of up to 3 maps.
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Matching statistic: St001831
St001831: 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)]
=> 0
[(1,2),(3,4),(5,6)]
=> 1
[(1,3),(2,4),(5,6)]
=> 1
[(1,4),(2,3),(5,6)]
=> 0
[(1,5),(2,3),(4,6)]
=> 0
[(1,6),(2,3),(4,5)]
=> 0
[(1,6),(2,4),(3,5)]
=> 0
[(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)]
=> 0
[(1,3),(2,6),(4,5)]
=> 0
[(1,4),(2,6),(3,5)]
=> 1
[(1,5),(2,6),(3,4)]
=> 1
[(1,6),(2,5),(3,4)]
=> 0
[(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)]
=> 0
[(1,5),(2,3),(4,6),(7,8)]
=> 0
[(1,6),(2,3),(4,5),(7,8)]
=> 0
[(1,7),(2,3),(4,5),(6,8)]
=> 0
[(1,8),(2,3),(4,5),(6,7)]
=> 0
[(1,8),(2,4),(3,5),(6,7)]
=> 0
[(1,7),(2,4),(3,5),(6,8)]
=> 0
[(1,6),(2,4),(3,5),(7,8)]
=> 0
[(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)]
=> 0
[(1,3),(2,6),(4,5),(7,8)]
=> 0
[(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)]
=> 0
[(1,7),(2,5),(3,4),(6,8)]
=> 0
[(1,8),(2,5),(3,4),(6,7)]
=> 0
[(1,8),(2,6),(3,4),(5,7)]
=> 0
[(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)]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> 0
[(1,2),(3,7),(4,5),(6,8)]
=> 0
[(1,2),(3,8),(4,5),(6,7)]
=> 0
[(1,3),(2,8),(4,5),(6,7)]
=> 0
[(1,4),(2,8),(3,5),(6,7)]
=> 0
Description
The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching.
Given a perfect matching, we obtain a formal sum of non-crossing perfect matchings by replacing recursively every matching $M$ that has a crossing $(a, c), (b, d)$ with $a < b < c < d$ with the sum of the two matchings $(M\setminus \{(a,c), (b,d)\})\cup \{(a,b), (c,d)\}$ and $(M\setminus \{(a,c), (b,d)\})\cup \{(a,d), (b,c)\}$.
This statistic is the coefficient of the unique non-nesting perfect matching in the formal sum.
Computer experiments indicate that this statistic has the same distribution as [[StatisticsDatabase/St000383oMp00128oMp00215oMp00092]].
Matching statistic: St001940
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[(1,2)]
=> [1,0]
=> [1,0]
=> []
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? = 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[(1,4),(2,7),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,4),(2,6),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,7),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,6),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,8),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,8),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,7),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,6),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,6),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,6),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,6),(3,8),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,6),(3,8),(4,7),(5,10)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,6),(3,9),(4,7),(5,10)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,9),(4,8),(5,10)]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St000149
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> [1]
=> 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 0
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 0
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 0
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 0
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 0
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 0
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 0
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 0
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St001276
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[(1,2)]
=> [1,0]
=> []
=> []
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> [1,0]
=> 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 0
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 0
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 0
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 0
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 0
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 0
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 0
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(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]
=> 0
[(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]
=> 0
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 0
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 0
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 0
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 0
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 0
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra.
Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional.
The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St001283
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001283: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> [1]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 0
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 0
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 0
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 0
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 0
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 0
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers.
A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
Matching statistic: St001284
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001284: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> [1]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 0
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 0
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 0
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 0
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 1
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 0
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 0
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 0
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 0
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of finite groups that are realised by the given partition over the complex numbers.
A finite group $G$ is 'realised' by the partition $(a_1,...,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers.
Matching statistic: St001413
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[(1,2)]
=> [1,0]
=> []
=> ? => ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> 1 => 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ? => ? ∊ {1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ? => ? ∊ {1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> 01 => 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 0 => 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 0 => 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 11 => 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 11 => 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101 => 0
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10 => 0
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10 => 0
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 111 => 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 111 => 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 011 => 0
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 011 => 0
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 11 => 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 11 => 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
Half the length of the longest even length palindromic prefix of a binary word.
Matching statistic: St001484
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> []
=> ?
=> ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> []
=> 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ?
=> ? ∊ {1,1}
[(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]
=> []
=> 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 0
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 0
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 0
[(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]
=> []
=> 0
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,1,1,1}
[(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]
=> []
=> 0
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[(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]
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 0
[(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]
=> []
=> 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[(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]
=> 0
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 0
[(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]
=> []
=> 0
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[(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]
=> 0
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 0
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St001524
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[(1,2)]
=> [1,0]
=> []
=> ? => ? = 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> 1 => 0
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> ? => ? ∊ {1,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> ? => ? ∊ {1,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> 01 => 0
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 0 => 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 0 => 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 11 => 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 11 => 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> ? => ? ∊ {1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101 => 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10 => 0
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 10 => 0
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 111 => 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 111 => 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 11 => 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1 => 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 011 => 0
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 011 => 0
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 11 => 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 1
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 1
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 11 => 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The degree of symmetry of a binary word.
For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.
Matching statistic: St001010
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001010: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001010: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●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
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {0,1}
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {0,1}
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(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]
=> 0
[(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]
=> 0
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,5),(2,8),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 0
[(1,6),(2,8),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[(1,7),(2,8),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[(1,8),(2,7),(3,4),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[(1,8),(2,7),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,7),(2,8),(3,5),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,6),(2,8),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,5),(2,8),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,4),(2,8),(3,6),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> 1
[(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[(1,5),(2,7),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,6),(2,7),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,7),(2,6),(3,5),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,8),(2,6),(3,5),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,8),(2,5),(3,6),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,7),(2,5),(3,6),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,6),(2,5),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,5),(2,6),(3,7),(4,8)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,5),(2,6),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,6),(2,5),(3,8),(4,7)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,7),(2,5),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,8),(2,5),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,8),(2,6),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,7),(2,6),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,6),(2,7),(3,8),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,5),(2,7),(3,8),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,5),(2,8),(3,7),(4,6)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,6),(2,8),(3,7),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,7),(2,8),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,8),(2,7),(3,6),(4,5)]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2}
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,8),(4,6),(5,7),(9,10)]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,7),(4,6),(5,8),(9,10)]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,6),(4,7),(5,8),(9,10)]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,5),(4,7),(6,8),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,4),(5,7),(6,8),(9,10)]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,5),(4,8),(6,7),(9,10)]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,6),(4,8),(5,7),(9,10)]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,7),(4,8),(5,6),(9,10)]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path.
The following 80 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001587Half of the largest even part of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001657The number of twos in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000667The greatest common divisor of the parts of the partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St001651The Frankl number of a lattice. St000383The last part of an integer composition. St000617The number of global maxima of a Dyck path. St000237The number of small exceedances. St000546The number of global descents of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000674The number of hills of a Dyck path. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000665The number of rafts of 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. St001075The minimal size of a block of a set partition. St000022The number of fixed points of a permutation. St000989The number of final rises of a permutation.
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