- FindStat

Your data matches 97 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000164

Mapping

St000164: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of short pairs. A short pair is a matching pair of the form

$(i,i+1)$ .

Matching statistic: St000442

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 75%●distinct values known / distinct values provided: 67%

Values

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

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St001431

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 68%●distinct values known / distinct values provided: 67%

Values

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

Description

Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I. See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.

Matching statistic: St001632
(load all 3 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

Matching statistic: St000137
(load all 3 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%

Values

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

Description

The Grundy value of an integer partition. Consider the two-player game on an integer partition. In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition. The first player that cannot move lose. This happens exactly when the empty partition is reached. The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1]. This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.

Matching statistic: St001122
(load all 2 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%

Values

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

Description

The multiplicity of the sign 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

$g_{\lambda,\lambda}^{1^n}$ , for

$\lambda\vdash n$ . It equals

$1$ if and only if

$\lambda$ is self-conjugate.

Matching statistic: St001176
(load all 2 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 67%

Values

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

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St001384
(load all 2 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.

Matching statistic: St001525
(load all 2 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of symmetric hooks on the diagonal of a partition.

Matching statistic: St001593
(load all 2 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001593: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 59%●distinct values known / distinct values provided: 50%

Values

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

Description

This is the number of standard Young tableaux of the given shifted shape. For an integer partition

$\lambda = (\lambda_1,\dots,\lambda_k)$ , the shifted diagram is obtained by moving the

$i$ -th row in the diagram

$i-1$ boxes to the right, i.e.,

$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$ In particular, this statistic is zero if and only if

$\lambda_{i+1} = \lambda_i$ for some

$i$ .

The following 87 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001527The cyclic permutation representation number of an integer partition. St001541The Gini index of an integer partition. St001571The Cartan determinant of the integer partition. St001587Half of the largest even part of an integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001657The number of twos in an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000153The number of adjacent cycles of a permutation. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001651The Frankl number of a lattice. St000382The first part of an integer composition. St000214The number of adjacencies of a permutation. St000260The radius of a connected graph. St000383The last part of an integer composition. St000454The largest eigenvalue of a graph if it is integral. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000925The number of topologically connected components of a set partition. St000546The number of global descents of a permutation. St001115The number of even descents of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000932The number of occurrences of the pattern UDU in a Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001061The number of indices that are both descents and recoils of a permutation.