- FindStat

Your data matches 137 different statistics following compositions of up to 3 maps.
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Matching statistic: St001043

Mapping

St001043: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. The bijection between perfect matchings of

$\{1,\dots,2n\}$ and trees with

$n+1$ leaves is described in Example 5.2.6 of [1].

Matching statistic: St000630

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%

Values

[(1,2)]
 => [1,0]
 => []
 => ? => ? = 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1]
 => 1 => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => []
 => ? => ? ∊ {1,1}
[(1,4),(2,3)]
 => [1,1,0,0]
 => []
 => ? => ? ∊ {1,1}
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [2,1]
 => 01 => 2
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => 0 => 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2]
 => 0 => 1
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1]
 => 1 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1]
 => 1 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,2,2}
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,2,2}
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,2,2}
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1]
 => 1 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1]
 => 11 => 1
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1]
 => 11 => 1
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1]
 => 1 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,2,2}
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,2,2}
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,2,2}
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10 => 2
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10 => 2
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 11 => 1
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 11 => 1
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 01 => 2
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 01 => 2
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 11 => 1
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 111 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 111 => 1
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 11 => 1
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1 => 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 01 => 2
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 011 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 011 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 01 => 2
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,5),(2,8),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 0 => 1
[(1,6),(2,8),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(1,7),(2,8),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(1,8),(2,7),(3,4),(5,6)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(1,8),(2,7),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,5),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,4),(2,8),(3,6),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 => 1
[(1,3),(2,8),(4,6),(5,7)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 11 => 1
[(1,2),(3,8),(4,6),(5,7)]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 111 => 1
[(1,2),(3,7),(4,6),(5,8)]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 111 => 1
[(1,3),(2,7),(4,6),(5,8)]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 11 => 1
[(1,5),(2,7),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,5),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,5),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,6),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,6),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,7),(4,8)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,6),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,5),(3,8),(4,7)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,5),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,5),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,6),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,8),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,7),(3,8),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,5),(2,8),(3,7),(4,6)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,7),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,6),(4,5)]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,10),(2,9),(3,8),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,9),(2,10),(3,8),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,10),(3,9),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,10),(3,9),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,10),(3,9),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,9),(3,10),(4,7),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,9),(3,10),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,9),(3,10),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,9),(2,8),(3,10),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,10),(2,8),(3,9),(4,6),(5,7)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,10),(2,7),(3,9),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,9),(2,7),(3,10),(4,6),(5,8)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,8),(2,7),(3,10),(4,6),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,8),(3,10),(4,6),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,8),(3,10),(4,7),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,6),(2,7),(3,10),(4,8),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[(1,7),(2,6),(3,10),(4,8),(5,9)]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}

Description

The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word

$w=a_1,\dots,a_n$ is any list of factors

$p_1,\dots,p_k$ such that

$w=p_1\dots p_k$ and each

$p_i$ is a palindrome, i.e. coincides with itself read backwards.

Matching statistic: St001199
(load all 8 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%

Values

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

Description

The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001597

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%

Values

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

Description

The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.

Matching statistic: St001568
(load all 3 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%

Values

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

Description

The smallest positive integer that does not appear twice in the partition.

Matching statistic: St000755
(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
St000755: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. Consider the recurrence

$f(n)=\sum_{p\in\lambda} f(n-p).$ This statistic returns the number of distinct real roots of the associated characteristic polynomial. For example, the partition

$(2,1)$ corresponds to the recurrence

$f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial

$x^2-x-1$ , which has two real roots.

Matching statistic: St000781
(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
St000781: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001934
(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
St001934: Integer partitions ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of monotone factorisations of genus zero of a permutation of given cycle type. A monotone factorisation of genus zero of a permutation

$\pi\in\mathfrak S_n$ with

$\ell$ cycles, including fixed points, is a tuple of

$r = n - \ell$ transpositions

$(a_1, b_1),\dots,(a_r, b_r)$ with

$b_1 \leq \dots \leq b_r$ and

$a_i < b_i$ for all

$i$ , whose product, in this order, is

$\pi$ . For example, the cycle

$(2,3,1)$ has the two factorizations

$(2,3)(1,3)$ and

$(1,2)(2,3)$ .

Matching statistic: St001432

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%

Values

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

Description

The order dimension of the partition. Given a partition

$\lambda$ , let

$I(\lambda)$ be the principal order ideal in the Young lattice generated by

$\lambda$ . The order dimension of a partition is defined as the order dimension of the poset

$I(\lambda)$ .

Matching statistic: St001780

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%

Values

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

Description

The order of promotion on the set of standard tableaux of given shape.

The following 127 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001330The hat guessing number of a graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000920The logarithmic height of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000891The number of distinct diagonal sums of a permutation matrix. St001394The genus of a permutation. St000661The number of rises of length 3 of a Dyck path. St000534The number of 2-rises of a permutation. St000731The number of double exceedences of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000842The breadth of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000117The number of centered tunnels of a Dyck path. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000617The number of global maxima of a Dyck path. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000649The number of 3-excedences of a permutation. St000254The nesting number of a set partition. St000648The number of 2-excedences of a permutation. St000232The number of crossings of a set partition. St000237The number of small exceedances. St000563The number of overlapping pairs of blocks of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000253The crossing number of a set partition. St000065The number of entries equal to -1 in an alternating sign matrix. St000895The number of ones on the main diagonal of an alternating sign matrix. St000233The number of nestings of a set partition. St001434The number of negative sum pairs of a signed permutation. St001947The number of ties in a parking function. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000535The rank-width of a graph. St000223The number of nestings in the permutation. St000370The genus of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001353The number of prime nodes in the modular decomposition of a graph. St000366The number of double descents of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000546The number of global descents of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000352The Elizalde-Pak rank of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000678The number of up steps after the last double rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001115The number of even descents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000359The number of occurrences of the pattern 23-1. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000007The number of saliances of the permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001732The number of peaks visible from the left. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000356The number of occurrences of the pattern 13-2. St001172The number of 1-rises at odd height of a Dyck path. St000408The number of occurrences of the pattern 4231 in a permutation. St000002The number of occurrences of the pattern 123 in a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000187The determinant of an alternating sign matrix. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000069The number of maximal elements of a poset. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000516The number of stretching pairs of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000365The number of double ascents of a permutation. St000664The number of right ropes of a permutation. St000449The number of pairs of vertices of a graph with distance 4. St000962The 3-shifted major index of a permutation. St000028The number of stack-sorts needed to sort a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001728The number of invisible descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St001513The number of nested exceedences of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000650The number of 3-rises of a permutation.