- FindStat

Your data matches 68 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001195
(load all 29 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => 0
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => 0
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => 0
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => 1
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => 1
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => 1
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => 1
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => 1
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => 1
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => 1
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => 1
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => 0
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => 0
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => 0
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => 0
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 1
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => 0
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => 0
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => 0
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => 0
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 1
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => 1
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => 1
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 1
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 1
[(1,6),(2,8),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,8),(2,7),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => 1

Description

The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ .

Matching statistic: St000326

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [2,1]
 => 1010 => 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => 100 => 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => 100 => 1
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1]
 => 10 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1]
 => 10 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,1,1,1}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,1,1,1}
[(1,4),(2,5),(3,6)]
 => [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]
 => 10 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1]
 => 110 => 1
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1]
 => 110 => 1
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1]
 => 10 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,1,1,1}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,1,1,1}
[(1,6),(2,5),(3,4)]
 => [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]
 => [3,2,1]
 => 101010 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 1
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 1
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 1
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 1
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 1
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 1
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => 1
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => 1
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[(1,6),(2,8),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[(1,8),(2,7),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[(1,8),(2,7),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[(1,3),(2,8),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 1
[(1,2),(3,8),(4,6),(5,7)]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 1
[(1,2),(3,7),(4,6),(5,8)]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 1
[(1,3),(2,7),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 1
[(1,4),(2,7),(3,6),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,10),(4,7),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,6),(3,10),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,6),(3,9),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St001722

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [2,1]
 => 01 => 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => 0 => 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => 0 => 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]
 => []
 => ? => ? ∊ {0,0,0,1,1,1}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,1,1,1}
[(1,4),(2,5),(3,6)]
 => [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]
 => 1 => 1
[(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 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,1,1,1}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,1,1,1}
[(1,6),(2,5),(3,4)]
 => [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]
 => [3,2,1]
 => 101 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10 => 1
[(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 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 01 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(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 => 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 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]
 => 0 => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 01 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 011 => 1
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 011 => 1
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 01 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 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]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(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
[(1,4),(2,7),(3,6),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,10),(4,7),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,6),(3,10),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,6),(3,9),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

Description

The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence

$01$ , or a trailing

$0$ . A peak is a subsequence

$10$ or a trailing

$1$ . Let

$P$ be the lattice on binary words of length

$n$ , where the covering elements of a word are obtained by replacing a valley with a peak. An interval

$[w_1, w_2]$ in

$P$ is small if

$w_2$ is obtained from

$w_1$ by replacing some valleys with peaks. This statistic counts the number of chains

$w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word

$0110$ :

$0110 < 1011 < 1101 < 1110 < 1111$ and

$0110 < 1010 < 1101 < 1110 < 1111.$

Matching statistic: St000260
(load all 3 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,0,1,1}
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,1,1}
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0,0,1,1}
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,1,1}
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,1,1}
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,6),(2,8),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,8),(2,7),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,8),(2,7),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,3),(2,8),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,2),(3,8),(4,6),(5,7)]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,7),(4,6),(5,8)]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,7),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,4),(2,7),(3,6),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,4),(2,6),(3,7),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,3),(2,6),(4,7),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,2),(3,6),(4,7),(5,8)]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,7),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,7),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[(1,4),(2,5),(3,7),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,5),(2,4),(3,7),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,6),(2,4),(3,7),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[(1,4),(2,3),(5,7),(6,8)]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,7),(6,8)]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,7),(6,8)]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,8),(6,7)]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,8),(6,7)]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,8),(6,7)]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,8),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,8),(5,7)]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,7),(4,8),(5,6)]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,8),(4,7),(5,6)]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St001487

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 70%●distinct values known / distinct values provided: 50%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 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]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1,1,1}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1,1,1}
[(1,4),(2,5),(3,6)]
 => [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]
 => [[1],[]]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[(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]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1,1,1}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1,1,1}
[(1,6),(2,5),(3,4)]
 => [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]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? ∊ {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}
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 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]
 => [[2,1],[]]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[(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]
 => [[3,1,1],[]]
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 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]
 => [[2],[]]
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 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]
 => [[2],[]]
 => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => 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]
 => [[2,1],[]]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 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]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[(1,3),(2,8),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[(1,2),(3,8),(4,6),(5,7)]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[(1,2),(3,7),(4,6),(5,8)]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[(1,3),(2,7),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[(1,4),(2,7),(3,6),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,7),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[(1,5),(2,6),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[4,3,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

Description

The number of inner corners of a skew partition.

Matching statistic: St001490

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 70%●distinct values known / distinct values provided: 50%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => [[2],[]]
 => 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]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1,1,1}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1,1,1}
[(1,4),(2,5),(3,6)]
 => [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]
 => [[1],[]]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[(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]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1,1,1}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,1,1,1}
[(1,6),(2,5),(3,4)]
 => [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]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? ∊ {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}
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[3,2],[]]
 => 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]
 => [[2,1],[]]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[(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]
 => [[3,1,1],[]]
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[3,1,1],[]]
 => 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]
 => [[2],[]]
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 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]
 => [[2],[]]
 => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[2,1],[]]
 => 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]
 => [[2,1],[]]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 1
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[2],[]]
 => 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]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[(1,3),(2,8),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[(1,2),(3,8),(4,6),(5,7)]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[(1,2),(3,7),(4,6),(5,8)]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[(1,3),(2,7),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1,1],[]]
 => 1
[(1,4),(2,7),(3,6),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,7),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1],[]]
 => 1
[(1,5),(2,6),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => [[],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[4,3,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[4,3,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[4,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[4,2],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[4,3,1,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[4,3,1],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[4,3],[]]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}

Description

The number of connected components of a skew partition.

Matching statistic: St001199
(load all 3 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,5),(2,6),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,6),(2,5),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,7),(2,5),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,8),(2,5),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,8),(2,6),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,7),(2,6),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,6),(2,7),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,5),(2,7),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,5),(2,8),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,6),(2,8),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,7),(2,8),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,8),(2,7),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,8),(2,10),(3,5),(4,6),(7,9)]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,7),(2,10),(3,5),(4,6),(8,9)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,6),(2,10),(3,5),(4,7),(8,9)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,5),(2,10),(3,6),(4,7),(8,9)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,5),(2,9),(3,6),(4,7),(8,10)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,6),(2,9),(3,5),(4,7),(8,10)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,7),(2,9),(3,5),(4,6),(8,10)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,8),(2,9),(3,5),(4,6),(7,10)]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,9),(2,8),(3,5),(4,6),(7,10)]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,10),(2,8),(3,5),(4,6),(7,9)]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,10),(2,7),(3,5),(4,6),(8,9)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,9),(2,7),(3,5),(4,6),(8,10)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,8),(2,7),(3,5),(4,6),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,7),(2,8),(3,5),(4,6),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,6),(2,8),(3,5),(4,7),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,5),(2,8),(3,6),(4,7),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,5),(2,7),(3,6),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,6),(2,7),(3,5),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,7),(2,6),(3,5),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,8),(2,6),(3,5),(4,7),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,9),(2,6),(3,5),(4,7),(8,10)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,10),(2,6),(3,5),(4,7),(8,9)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,10),(2,5),(3,6),(4,7),(8,9)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[(1,9),(2,5),(3,6),(4,7),(8,10)]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1

Description

The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St000781
(load all 2 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 50%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 1
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,2),(3,8),(4,6),(5,7)]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,7),(4,6),(5,8)]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,7),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,4),(2,7),(3,6),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,7),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,8),(2,3),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,7),(2,3),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,6),(2,3),(4,7),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,4),(2,3),(5,7),(6,8)]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[(1,3),(2,4),(5,7),(6,8)]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[(1,3),(2,4),(5,8),(6,7)]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[(1,4),(2,3),(5,8),(6,7)]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[(1,6),(2,3),(4,8),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,7),(2,3),(4,8),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,8),(2,3),(4,7),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,3),(2,6),(4,8),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,3),(2,7),(4,8),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,3),(2,8),(4,7),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1

Description

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

Matching statistic: St001901

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 50%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => 1
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => 1
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,2),(3,8),(4,6),(5,7)]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,7),(4,6),(5,8)]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,7),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,4),(2,7),(3,6),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,7),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,8),(2,3),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,7),(2,3),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,6),(2,3),(4,7),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,4),(2,3),(5,7),(6,8)]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[(1,3),(2,4),(5,7),(6,8)]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[(1,3),(2,4),(5,8),(6,7)]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[(1,4),(2,3),(5,8),(6,7)]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => [1]
 => 1
[(1,6),(2,3),(4,8),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,7),(2,3),(4,8),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,8),(2,3),(4,7),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,3),(2,6),(4,8),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,3),(2,7),(4,8),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,3),(2,8),(4,7),(5,6)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1

Description

The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.

Matching statistic: St001123

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%

Values

[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => [2]
 => 1
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,8),(2,7),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,8),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 0
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 0
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => [1,1]
 => 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 0
[(1,2),(3,7),(4,5),(6,8),(9,10)]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => 1
[(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => [3]
 => 0
[(1,4),(2,8),(3,5),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => [2]
 => 1
[(1,8),(2,7),(3,5),(4,6),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,7),(2,8),(3,5),(4,6),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,6),(2,8),(3,5),(4,7),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,5),(2,8),(3,6),(4,7),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,3),(2,8),(4,6),(5,7),(9,10)]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 1
[(1,3),(2,7),(4,6),(5,8),(9,10)]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 1
[(1,5),(2,7),(3,6),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,6),(2,7),(3,5),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,7),(2,6),(3,5),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,8),(2,6),(3,5),(4,7),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,8),(2,5),(3,6),(4,7),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,7),(2,5),(3,6),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,6),(2,5),(3,7),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,5),(2,6),(3,7),(4,8),(9,10)]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => [2]
 => 1
[(1,3),(2,6),(4,7),(5,8),(9,10)]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => [2,1]
 => 1
[(1,2),(3,5),(4,7),(6,8),(9,10)]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => [2]
 => 1

Description

The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{21^{n-2}}$ , for

$\lambda\vdash n$ .

The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001568The smallest positive integer that does not appear twice in the partition. St000862The number of parts of the shifted shape of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St000701The protection number of a binary tree. St000660The number of rises of length at least 3 of a Dyck path. St001624The breadth of a lattice. St000007The number of saliances of the permutation. St000834The number of right outer peaks of a permutation. St000035The number of left outer peaks of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St000153The number of adjacent cycles of a permutation. St000451The length of the longest pattern of the form k 1 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000891The number of distinct diagonal sums of a permutation matrix. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000402Half the size of the symmetry class of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000742The number of big ascents of a permutation after prepending zero. St000215The number of adjacencies of a permutation, zero appended. St000883The number of longest increasing subsequences of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000884The number of isolated descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000675The number of centered multitunnels of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000359The number of occurrences of the pattern 23-1. St000648The number of 2-excedences of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000187The determinant of an alternating sign matrix. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000069The number of maximal elements of a poset. St000486The number of cycles of length at least 3 of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000054The first entry of the permutation. St000546The number of global descents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000022The number of fixed points of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000245The number of ascents of a permutation.