Your data matches 125 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001476
(load all 3 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
St001476: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => 1
[1,2] => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => 1
[1,2,3] => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => 1
[2,1,3] => ([(1,2)],3)
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,2,3,4] => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => 1
[1,3,2,4] => ([(2,3)],4)
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => ([(2,3)],4)
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
Description
The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1).
Matching statistic: St000770
(load all 4 compositions to match this statistic)
Mapping
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 52%●distinct values known / distinct values provided: 20%
Values
[1] => [] => []
 => ?
 => ? = 1
[1,2] => [1] => [1]
 => []
 => ? ∊ {1,1}
[2,1] => [1] => [1]
 => []
 => ? ∊ {1,1}
[1,2,3] => [1,2] => [1,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[1,3,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[2,1,3] => [2,1] => [2]
 => []
 => ? ∊ {1,1,1,1,1,1}
[2,3,1] => [2,1] => [2]
 => []
 => ? ∊ {1,1,1,1,1,1}
[3,1,2] => [1,2] => [1,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[3,2,1] => [2,1] => [2]
 => []
 => ? ∊ {1,1,1,1,1,1}
[1,2,3,4] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
Description
The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St001964
(load all 27 compositions to match this statistic)
Mapping
Mp00277: Permutations —catalanization⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 49%●distinct values known / distinct values provided: 27%
Values
[1] => [1] => [1] => ([],1)
 => 0 = 1 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
 => 0 = 1 - 1
[2,1] => [2,1] => [2,1] => ([],2)
 => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,1,2] => ([(1,2)],3)
 => ? ∊ {1,1,1} - 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[2,3,1] => [2,3,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {1,1,1} - 1
[3,1,2] => [2,3,1] => [2,3,1] => ([(1,2)],3)
 => ? ∊ {1,1,1} - 1
[3,2,1] => [3,2,1] => [3,2,1] => ([],3)
 => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,1,2,2} - 1
[1,4,2,3] => [1,3,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,1,2,2} - 1
[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2} - 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 0 = 1 - 1
[2,3,1,4] => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
[2,3,4,1] => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 0 = 1 - 1
[2,4,1,3] => [4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 1 = 2 - 1
[2,4,3,1] => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2} - 1
[3,1,2,4] => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 0 = 1 - 1
[3,1,4,2] => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 0 = 1 - 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,4,1] => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[3,4,2,1] => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2} - 1
[4,1,2,3] => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 0 = 1 - 1
[4,1,3,2] => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2} - 1
[4,2,1,3] => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[4,2,3,1] => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2} - 1
[4,3,1,2] => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2} - 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 0 = 1 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 0 = 1 - 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 1 = 2 - 1
[1,3,5,4,2] => [1,3,5,4,2] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[1,4,2,3,5] => [1,3,4,2,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 0 = 1 - 1
[1,4,2,5,3] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,5,2] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 0 = 1 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[1,4,5,3,2] => [1,4,5,3,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[1,5,2,3,4] => [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 0 = 1 - 1
[1,5,2,4,3] => [1,3,5,4,2] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[1,5,3,2,4] => [1,4,3,5,2] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 0 = 1 - 1
[1,5,3,4,2] => [1,4,5,3,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[1,5,4,2,3] => [1,4,5,3,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1 = 2 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => 1 = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,1,4,5,3] => [2,1,4,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 1 - 1
[2,1,5,3,4] => [2,1,4,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 1 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1 = 2 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => 1 = 2 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 0 = 1 - 1
[2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 0 = 1 - 1
[2,3,5,1,4] => [2,5,4,1,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
[2,3,5,4,1] => [2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 0 = 1 - 1
[2,4,5,1,3] => [3,5,4,1,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[2,4,5,3,1] => [2,4,5,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[2,5,3,4,1] => [2,4,5,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[2,5,4,3,1] => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[3,2,5,1,4] => [5,2,4,1,3] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[3,4,1,5,2] => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[3,4,5,1,2] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[3,5,1,4,2] => [5,3,2,4,1] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[3,5,2,4,1] => [5,4,2,3,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[3,5,4,2,1] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[4,1,5,2,3] => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[4,1,5,3,2] => [2,4,5,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[4,2,5,1,3] => [5,2,4,3,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[4,3,2,5,1] => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[4,3,5,1,2] => [5,3,4,2,1] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[4,5,1,3,2] => [5,3,4,2,1] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[4,5,3,2,1] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,1,3,4,2] => [2,4,5,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,1,4,2,3] => [2,4,5,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,1,4,3,2] => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,2,1,4,3] => [3,2,5,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,2,4,1,3] => [5,4,2,3,1] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,2,4,3,1] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,3,2,1,4] => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,3,4,2,1] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,4,1,3,2] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,4,2,3,1] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[5,4,3,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5} - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,6,1,2,3,5] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16} - 1
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,6,5,2,3,4] => ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
 => ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,14,16,16,16,16,16,16,16,16,16,16,16,16,16} - 1
Description
The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001199
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 27% ●values known / values provided: 36%●distinct values known / distinct values provided: 27%
Values
[1] => => [] => ?
 => ? = 1
[1,2] => 1 => [1] => [1,0]
 => ? ∊ {1,1}
[2,1] => 0 => [1] => [1,0]
 => ? ∊ {1,1}
[1,2,3] => 11 => [2] => [1,1,0,0]
 => ? ∊ {1,1,1,1}
[1,3,2] => 10 => [1,1] => [1,0,1,0]
 => 1
[2,1,3] => 01 => [1,1] => [1,0,1,0]
 => 1
[2,3,1] => 00 => [2] => [1,1,0,0]
 => ? ∊ {1,1,1,1}
[3,1,2] => 00 => [2] => [1,1,0,0]
 => ? ∊ {1,1,1,1}
[3,2,1] => 00 => [2] => [1,1,0,0]
 => ? ∊ {1,1,1,1}
[1,2,3,4] => 111 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,4,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 1
[1,3,2,4] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[1,3,4,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[1,4,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[1,4,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3,4] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,4,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 2
[2,3,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[2,3,4,1] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,4,1,3] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,4,3,1] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,2,4] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[3,1,4,2] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,2,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
 => 1
[3,2,4,1] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,4,1,2] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,4,2,1] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,1,2,3] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,1,3,2] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,2,1,3] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,2,3,1] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,3,1,2] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,3,2,1] => 000 => [3] => [1,1,1,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,3,4,5] => 1111 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,3,5,4] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,2,4,3,5] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,2,4,5,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,2,5,3,4] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,2,5,4,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4,5] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[1,3,4,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,3,4,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,5,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,5,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,4,2,5,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,4,3,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,5,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,5,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,2,3,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,2,4,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,3,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,3,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,4,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,4,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,3,4,5] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,3,5,4] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,1,4,3,5] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
[2,1,4,5,3] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[2,1,5,3,4] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[2,1,5,4,3] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[2,3,1,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,3,4,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,3,4,5,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,5,1,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,5,4,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,1,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,4,1,5,3] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,3,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,4,3,5,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,5,1,3] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,5,3,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,1,3,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,1,4,3] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,3,1,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,3,4,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,4,1,3] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,4,3,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,1,2,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,1,2,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[3,1,4,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,4,5,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,1,5,2,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,1,5,4,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,2,1,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,4,5,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,2,5,1,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,2,5,4,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,1,5,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,2,5,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,5,1,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,5,2,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,5,1,2,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,5,1,4,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,5,2,1,4] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,5,2,4,1] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,5,4,1,2] => 0000 => [4] => [1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000771
(load all 4 compositions to match this statistic)
Mapping
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 36%●distinct values known / distinct values provided: 27%
Values
[1] => [1] => [1] => ([],1)
 => 1
[1,2] => [1,2] => [1,2] => ([],2)
 => ? = 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => ? ∊ {1,1,1,1}
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1,1,1}
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,1,1,1}
[2,3,1] => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1,1,1}
[3,1,2] => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => [3,2,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[2,4,1,3] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,4,3,1] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,2,1,4] => [3,2,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[3,4,1,2] => [2,4,1,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,4,2,1] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2}
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,3,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,3] => [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
[4,2,3,1] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => [4,3,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,2,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,2,4,3] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => [1,5,3,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,2,3] => [1,5,4,2,3] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,5,3,4] => [2,1,5,3,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,1,3,4] => [2,5,1,3,4] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,5,1,4,3] => [2,5,1,4,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[3,1,5,2,4] => [3,1,5,2,4] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[3,2,5,1,4] => [3,2,5,1,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,5,1,2,4] => [3,5,1,2,4] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,5,1,4,2] => [2,5,1,4,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,5,2,1,4] => [3,5,2,1,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[3,5,2,4,1] => [2,5,1,4,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,5,4,1,2] => [2,5,4,1,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[4,1,5,2,3] => [3,1,5,2,4] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[4,2,5,1,3] => [3,2,5,1,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,3,5,1,2] => [3,2,5,1,4] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,5,1,2,3] => [3,5,1,2,4] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,5,1,3,2] => [2,5,1,4,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[4,5,2,1,3] => [3,5,2,1,4] => [4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[4,5,2,3,1] => [2,5,1,4,3] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[4,5,3,1,2] => [2,5,4,1,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,2,3,4] => [5,1,2,3,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[5,1,2,4,3] => [5,1,2,4,3] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[5,1,3,2,4] => [5,1,3,2,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[5,1,3,4,2] => [5,1,2,4,3] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[5,1,4,2,3] => [5,1,4,2,3] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,1,4,3,2] => [5,1,4,3,2] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[5,2,1,3,4] => [5,2,1,3,4] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[5,2,1,4,3] => [5,2,1,4,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[5,2,3,1,4] => [5,1,3,2,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[5,2,3,4,1] => [5,1,2,4,3] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[5,2,4,1,3] => [5,2,4,1,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[5,2,4,3,1] => [5,1,4,3,2] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[5,3,1,2,4] => [5,3,1,2,4] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[5,3,1,4,2] => [5,2,1,4,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[5,3,2,1,4] => [5,3,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[5,3,2,4,1] => [5,2,1,4,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
[5,3,4,1,2] => [5,2,4,1,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[5,3,4,2,1] => [5,1,4,3,2] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[5,4,1,2,3] => [5,4,1,2,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[5,4,1,3,2] => [5,4,1,3,2] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St001630
(load all 13 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 13% ●values known / values provided: 35%●distinct values known / distinct values provided: 13%
Values
[1] => ([],1)
 => ([],1)
 => ? = 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1}
[2,1] => ([],2)
 => ([],1)
 => ? ∊ {1,1}
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1}
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1}
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1}
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1}
[3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1}
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[4,3,2,1] => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2}
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 14 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 13% ●values known / values provided: 35%●distinct values known / distinct values provided: 13%
Values
[1] => ([],1)
 => ([],1)
 => ? = 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1}
[2,1] => ([],2)
 => ([],1)
 => ? ∊ {1,1}
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1}
[2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1}
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1}
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1}
[3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {1,1,1,1,1}
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[4,3,2,1] => ([],4)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,2),(2,1)],3)
 => 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001605
(load all 9 compositions to match this statistic)
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 35%●distinct values known / distinct values provided: 13%
Values
[1] => [1]
 => []
 => []
 => ? = 1
[1,2] => [1,1]
 => [1]
 => [1]
 => ? ∊ {1,1}
[2,1] => [2]
 => []
 => []
 => ? ∊ {1,1}
[1,2,3] => [1,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1}
[1,3,2] => [2,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[2,1,3] => [2,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[2,3,1] => [3]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1}
[3,1,2] => [3]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1}
[3,2,1] => [2,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,4,2] => [3,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,2,3] => [3,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,4,3] => [2,2]
 => [2]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,1,4] => [3,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,4,1] => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,1,3] => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,3,1] => [3,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,4] => [3,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,4,2] => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,4,1] => [3,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,1,2] => [2,2]
 => [2]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,2,1] => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,3] => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,3,2] => [3,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,1,3] => [3,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1,2] => [4]
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,2,1] => [2,2]
 => [2]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => [4,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,5,2,4] => [4,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => [4,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[1,4,5,3,2] => [4,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,2,3,4] => [4,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[2,1,4,5,3] => [3,2]
 => [2]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,5,3,4] => [3,2]
 => [2]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,5,4,3] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[2,3,1,4,5] => [3,1,1]
 => [1,1]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,1,5,4] => [3,2]
 => [2]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[3,2,1,5,4] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[3,4,1,2,5] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[3,5,1,4,2] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[4,2,3,1,5] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[4,2,5,1,3] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[4,3,2,1,5] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[4,5,3,1,2] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[5,2,3,4,1] => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[5,2,4,3,1] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[5,3,2,4,1] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[5,4,3,2,1] => [2,2,1]
 => [2,1]
 => [3]
 => 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [3,2]
 => 2
[1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,2,3,5,6,4] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,3,6,4,5] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,3,6,5,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 2
[1,2,4,5,3,6] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,5,3,4,6] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,5,4,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,5,6,3,4] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 2
[1,2,6,3,5,4] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,6,4,3,5] => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1
[1,2,6,4,5,3] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,2,6,5,4,3] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 2
[1,3,2,4,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 1
[1,3,2,4,6,5] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 2
[1,3,2,5,4,6] => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 2
[1,3,2,5,6,4] => [3,2,1]
 => [2,1]
 => [3]
 => 1
[1,3,2,6,4,5] => [3,2,1]
 => [2,1]
 => [3]
 => 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition. Two colourings are considered equal, if they are obtained by an action of the cyclic group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001603
(load all 12 compositions to match this statistic)
Mapping
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => []
 => ? = 1
[1,2] => [1,2] => [1,2] => []
 => ? ∊ {1,1}
[2,1] => [2,1] => [-2,1] => [2]
 => ? ∊ {1,1}
[1,2,3] => [1,2,3] => [1,2,3] => []
 => ? ∊ {1,1,1}
[1,3,2] => [1,3,2] => [-3,1,2] => [3]
 => 1
[2,1,3] => [2,1,3] => [-2,1,3] => [2]
 => ? ∊ {1,1,1}
[2,3,1] => [2,3,1] => [-3,-2,1] => [2,1]
 => 1
[3,1,2] => [3,1,2] => [3,1,2] => []
 => ? ∊ {1,1,1}
[3,2,1] => [3,2,1] => [2,-3,1] => [3]
 => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => [4]
 => 1
[1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => [3]
 => 1
[1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => []
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,2,3] => [1,4,2,3] => [4,1,2,3] => []
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => [3,1]
 => 1
[2,3,1,4] => [2,3,1,4] => [-3,-2,1,4] => [2,1]
 => 1
[2,3,4,1] => [2,3,4,1] => [-4,-3,-2,1] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[2,4,1,3] => [2,4,1,3] => [2,-4,1,3] => [4]
 => 1
[2,4,3,1] => [2,4,3,1] => [3,-4,-2,1] => []
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => []
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[3,1,4,2] => [3,1,4,2] => [4,-3,1,2] => [4]
 => 1
[3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => [3]
 => 1
[3,2,4,1] => [3,2,4,1] => [2,-4,-3,1] => [3,1]
 => 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => []
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[3,4,2,1] => [3,4,2,1] => [2,4,-3,1] => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[4,1,2,3] => [4,1,2,3] => [1,-4,2,3] => [3]
 => 1
[4,1,3,2] => [4,1,3,2] => [-3,-4,1,2] => [2,2]
 => 2
[4,2,1,3] => [4,2,1,3] => [-2,-4,1,3] => []
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => [4]
 => 1
[4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => [4]
 => 1
[4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => []
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,3,4] => [5]
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [-4,1,2,3,5] => [4]
 => 1
[1,2,4,5,3] => [1,2,4,5,3] => [-5,-4,1,2,3] => [3,2]
 => 2
[1,2,5,3,4] => [1,2,5,3,4] => [5,1,2,3,4] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => [1,2,5,4,3] => [4,-5,1,2,3] => [5]
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [-3,1,2,4,5] => [3]
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [-5,-3,1,2,4] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,2,5] => [1,3,4,2,5] => [-4,-3,1,2,5] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => [1,3,4,5,2] => [-5,-4,-3,1,2] => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,5,2,4] => [1,3,5,2,4] => [3,-5,1,2,4] => [3]
 => 1
[1,3,5,4,2] => [1,3,5,4,2] => [4,-5,-3,1,2] => [2,1]
 => 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1,2,3,5] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => [1,4,2,5,3] => [5,-4,1,2,3] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,2,5] => [1,4,3,2,5] => [3,-4,1,2,5] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,5,2] => [1,4,3,5,2] => [3,-5,-4,1,2] => [3,2]
 => 2
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,3,2] => [1,4,5,3,2] => [3,5,-4,1,2] => [3]
 => 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,-5,2,3,4] => [4]
 => 1
[1,5,2,4,3] => [1,5,2,4,3] => [-4,-5,1,2,3] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => [1,5,3,2,4] => [-3,-5,1,2,4] => [3,2]
 => 2
[1,5,3,4,2] => [1,5,3,4,2] => [3,4,-5,1,2] => [5]
 => 1
[1,5,4,2,3] => [1,5,4,2,3] => [-5,4,1,2,3] => [3]
 => 1
[1,5,4,3,2] => [1,5,4,3,2] => [-4,3,-5,1,2] => [3,2]
 => 2
[2,1,3,4,5] => [2,1,3,4,5] => [-2,1,3,4,5] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,3,5,4] => [2,1,3,5,4] => [-5,-2,1,3,4] => [4,1]
 => 1
[2,1,4,3,5] => [2,1,4,3,5] => [-4,-2,1,3,5] => [3,1]
 => 1
[2,1,4,5,3] => [2,1,4,5,3] => [-5,-4,-2,1,3] => [5]
 => 1
[2,1,5,3,4] => [2,1,5,3,4] => [2,-5,1,3,4] => [5]
 => 1
[2,1,5,4,3] => [2,1,5,4,3] => [4,-5,-2,1,3] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,1,4,5] => [2,3,1,4,5] => [-3,-2,1,4,5] => [2,1]
 => 1
[2,3,1,5,4] => [2,3,1,5,4] => [-5,-3,-2,1,4] => [3]
 => 1
[2,3,4,1,5] => [2,3,4,1,5] => [-4,-3,-2,1,5] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,4,5,1] => [2,3,4,5,1] => [-5,-4,-3,-2,1] => [2,1]
 => 1
[2,3,5,1,4] => [2,3,5,1,4] => [3,-5,-2,1,4] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,5,4,1] => [2,3,5,4,1] => [4,-5,-3,-2,1] => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,1,3,5] => [2,4,1,3,5] => [2,-4,1,3,5] => [4]
 => 1
[2,4,1,5,3] => [2,4,1,5,3] => [2,-5,-4,1,3] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,3,1,5] => [2,4,3,1,5] => [3,-4,-2,1,5] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,3,5,1] => [2,4,3,5,1] => [3,-5,-4,-2,1] => [5]
 => 1
[2,4,5,1,3] => [2,4,5,1,3] => [2,5,-4,1,3] => [5]
 => 1
[2,4,5,3,1] => [2,4,5,3,1] => [3,5,-4,-2,1] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,1,3,4] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,1,4,3] => [2,5,1,4,3] => [2,4,-5,1,3] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,5,3,1,4] => [2,5,3,1,4] => [2,3,-5,1,4] => [5]
 => 1
[2,5,3,4,1] => [2,5,3,4,1] => [3,4,-5,-2,1] => [3,2]
 => 2
[2,5,4,1,3] => [2,5,4,1,3] => [-4,2,-5,1,3] => [2,2]
 => 2
[2,5,4,3,1] => [2,5,4,3,1] => [-4,3,-5,-2,1] => [5]
 => 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,1,2,5,4] => [3,1,2,5,4] => [5,-3,1,2,4] => [5]
 => 1
[3,1,4,2,5] => [3,1,4,2,5] => [4,-3,1,2,5] => [4]
 => 1
[3,1,4,5,2] => [3,1,4,5,2] => [5,-4,-3,1,2] => [4,1]
 => 1
[3,1,5,2,4] => [3,1,5,2,4] => [3,5,1,2,4] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,1,5,4,2] => [3,1,5,4,2] => [-4,-5,-3,1,2] => [2,2,1]
 => 4
[3,2,1,4,5] => [3,2,1,4,5] => [2,-3,1,4,5] => [3]
 => 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,-5,-3,1,4] => [4,1]
 => 1
[3,2,4,1,5] => [3,2,4,1,5] => [2,-4,-3,1,5] => [3,1]
 => 1
[3,2,5,1,4] => [3,2,5,1,4] => [2,5,-3,1,4] => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,2,5,4,1] => [3,2,5,4,1] => [2,4,-5,-3,1] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,1,5,2] => [3,4,1,5,2] => [4,5,-3,1,2] => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,2,1,5] => [3,4,2,1,5] => [2,4,-3,1,5] => [1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,2,5,1] => [3,4,2,5,1] => [2,5,-4,-3,1] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[3,5,2,1,4] => [3,5,2,1,4] => [-3,2,-5,1,4] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[4,1,2,5,3] => [4,1,2,5,3] => [1,-5,-4,2,3] => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[4,1,3,5,2] => [4,1,3,5,2] => [-3,-5,-4,1,2] => [2]
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000772
(load all 2 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 32%●distinct values known / distinct values provided: 20%
Values
[1] => ([],1)
 => ([],0)
 => ([],0)
 => ? = 1
[1,2] => ([],2)
 => ([],0)
 => ([],0)
 => ? = 1
[2,1] => ([(0,1)],2)
 => ([],1)
 => ([],1)
 => 1
[1,2,3] => ([],3)
 => ([],0)
 => ([],0)
 => ? ∊ {1,1,1,1}
[1,3,2] => ([(1,2)],3)
 => ([],1)
 => ([],1)
 => 1
[2,1,3] => ([(1,2)],3)
 => ([],1)
 => ([],1)
 => 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1}
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1}
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1}
[1,2,3,4] => ([],4)
 => ([],0)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,4,3] => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[1,3,2,4] => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,4] => ([(2,3)],4)
 => ([],1)
 => ([],1)
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([],2)
 => ([(0,1)],2)
 => 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,4,5] => ([],5)
 => ([],0)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,3,5,4] => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,2,4,5] => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,1,3,4,5] => ([(3,4)],5)
 => ([],1)
 => ([],1)
 => 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,1)],2)
 => 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
 => 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
 => 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 1
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => 2
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 1
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
 => 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
The following 115 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000225Difference between largest and smallest parts in a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000456The monochromatic index of a connected graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001877Number of indecomposable injective modules with projective dimension 2. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000284The Plancherel distribution on integer partitions. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000260The radius of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000422The energy of a graph, if it is integral. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001845The number of join irreducibles minus the rank of a lattice. St000618The number of self-evacuating tableaux of given shape. St000706The product of the factorials of the multiplicities of an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000068The number of minimal elements in a poset. St000454The largest eigenvalue of a graph if it is integral. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001621The number of atoms of a lattice. St001867The number of alignments of type EN of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001862The number of crossings of a signed permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001427The number of descents of a signed permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000264The girth of a graph, which is not a tree. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000322The skewness of a graph. St000449The number of pairs of vertices of a graph with distance 4. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001578The minimal number of edges to add or remove to make a graph a line graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St001765The number of connected components of the friends and strangers graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001570The minimal number of edges to add to make a graph Hamiltonian.