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]].
Mp00150: Perfect matchings to Dyck pathDyck paths
Mp00228: Dyck paths reflect parallelogram polyominoDyck paths
Mp00027: Dyck paths to partitionInteger 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 pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00313: Integer partitions Glaisher-Franklin inverseInteger 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 pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck 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]] .
Mp00150: Perfect matchings to Dyck pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00313: Integer partitions Glaisher-Franklin inverseInteger 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 $(5,4,3,3,1)$.
Mp00150: Perfect matchings to Dyck pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00313: Integer partitions Glaisher-Franklin inverseInteger 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 pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00317: Integer partitions odd partsBinary 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 pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger 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 pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00317: Integer partitions odd partsBinary 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 pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck 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.