Processing math: 42%

Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001710
St001710: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 4
[1,1,1]
=> 4
[4]
=> 2
[3,1]
=> 1
[2,2]
=> 10
[2,1,1]
=> 10
[1,1,1,1]
=> 10
[5]
=> 1
[4,1]
=> 2
[3,2]
=> 2
[3,1,1]
=> 2
[2,2,1]
=> 26
[2,1,1,1]
=> 26
[1,1,1,1,1]
=> 26
[6]
=> 4
[5,1]
=> 1
[4,2]
=> 4
[4,1,1]
=> 4
[3,3]
=> 4
[3,2,1]
=> 4
[3,1,1,1]
=> 4
[2,2,2]
=> 76
[2,2,1,1]
=> 76
[2,1,1,1,1]
=> 76
[1,1,1,1,1,1]
=> 76
[7]
=> 1
[6,1]
=> 4
[5,2]
=> 2
[5,1,1]
=> 2
[4,3]
=> 2
[4,2,1]
=> 8
[4,1,1,1]
=> 8
[3,3,1]
=> 4
[3,2,2]
=> 10
[3,2,1,1]
=> 10
[3,1,1,1,1]
=> 10
[2,2,2,1]
=> 232
[2,2,1,1,1]
=> 232
[2,1,1,1,1,1]
=> 232
[1,1,1,1,1,1,1]
=> 232
[8]
=> 4
[7,1]
=> 1
[6,2]
=> 8
[6,1,1]
=> 8
[5,3]
=> 1
[5,2,1]
=> 4
Description
The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. Let α be any permutation of cycle type λ. This statistic is the number of permutations π such that απα1=π1.
Matching statistic: St000455
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00160: Permutations graph of inversionsGraphs
St000455: Graphs ⟶ ℤResult quality: 14% values known / values provided: 14%distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2} - 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {2,2} - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {4,4} - 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {4,4} - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,10,10,10} - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,10,10,10} - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,10,10,10} - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,10,10,10} - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {1,2,2,26,26,26} - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,26,26,26} - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,26,26,26} - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,2,26,26,26} - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,2,2,26,26,26} - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {1,2,2,26,26,26} - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {4,4,4,4,4,4,76,76,76,76} - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ? ∊ {1,4,4,8,8,10,10,10,232,232,232,232} - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(8,9)],10)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => ([(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => ([(0,7),(1,6),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => ([(0,1),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(6,7)],8)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 2 - 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} - 1
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000028
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000028: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {4,4}
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {4,4}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? ∊ {1,10,10,10}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {1,10,10,10}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? ∊ {1,10,10,10}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,10,10,10}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? ∊ {1,2,2,2,26,26,26}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? ∊ {1,2,2,2,26,26,26}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {1,2,2,2,26,26,26}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {1,2,2,2,26,26,26}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {1,2,2,2,26,26,26}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,26,26,26}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? ∊ {1,2,2,2,26,26,26}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1,16,15] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [6,8,9,10,11,5,12,7,4,3,2,1,14,13] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [5,6,8,9,4,3,10,7,2,1,12,11] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [4,7,8,3,9,10,6,5,2,1,12,11] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,7,8,10,11,6,5,12,9,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,6,9,10,5,11,12,8,7,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,8,10,11,12,13,7,14,9,6,5,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1,18,17] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> [7,9,10,11,12,13,6,14,8,5,4,3,2,1,16,15] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> [6,7,9,10,11,5,4,12,8,3,2,1,14,13] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> [5,8,9,10,4,11,12,7,6,3,2,1,14,13] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]
=> [5,6,7,9,4,3,2,10,8,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
=> [4,6,8,3,9,5,10,7,2,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [3,7,2,8,9,10,6,5,4,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> [5,6,7,8,4,3,2,1,11,12,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
Description
The number of stack-sorts needed to sort a permutation. A permutation is (West) t-stack sortable if it is sortable using t stacks in series. Let Wt(n,k) be the number of permutations of size n with k descents which are t-stack sortable. Then the polynomials Wn,t(x)=nk=0Wt(n,k)xk are symmetric and unimodal. We have Wn,1(x)=An(x), the Eulerian polynomials. One can show that Wn,1(x) and Wn,2(x) are real-rooted. Precisely the permutations that avoid the pattern 231 have statistic at most 1, see [3]. These are counted by \frac{1}{n+1}\binom{2n}{n} ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern 2341 and the barred pattern 3\bar 5241 have statistic at most 2, see [4]. These are counted by \frac{2(3n)!}{(n+1)!(2n+1)!} ([[OEIS:A000139]]).
Matching statistic: St000141
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000141: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {4,4}
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {4,4}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? ∊ {1,10,10,10}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {1,10,10,10}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? ∊ {1,10,10,10}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,10,10,10}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? ∊ {1,2,2,2,26,26,26}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? ∊ {1,2,2,2,26,26,26}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {1,2,2,2,26,26,26}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {1,2,2,2,26,26,26}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {1,2,2,2,26,26,26}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,26,26,26}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? ∊ {1,2,2,2,26,26,26}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1,16,15] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [6,8,9,10,11,5,12,7,4,3,2,1,14,13] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [5,6,8,9,4,3,10,7,2,1,12,11] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [4,7,8,3,9,10,6,5,2,1,12,11] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,7,8,10,11,6,5,12,9,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,6,9,10,5,11,12,8,7,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,8,10,11,12,13,7,14,9,6,5,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1,18,17] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> [7,9,10,11,12,13,6,14,8,5,4,3,2,1,16,15] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> [6,7,9,10,11,5,4,12,8,3,2,1,14,13] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> [5,8,9,10,4,11,12,7,6,3,2,1,14,13] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]
=> [5,6,7,9,4,3,2,10,8,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
=> [4,6,8,3,9,5,10,7,2,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [3,7,2,8,9,10,6,5,4,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> [5,6,7,8,4,3,2,1,11,12,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
Description
The maximum drop size of a permutation. The maximum drop size of a permutation \pi of [n]=\{1,2,\ldots, n\} is defined to be the maximum value of i-\pi(i).
Matching statistic: St000662
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000662: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 2
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {4,4}
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {4,4}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? ∊ {1,10,10,10}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {1,10,10,10}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? ∊ {1,10,10,10}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,10,10,10}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? ∊ {1,2,2,2,26,26,26}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? ∊ {1,2,2,2,26,26,26}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {1,2,2,2,26,26,26}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {1,2,2,2,26,26,26}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {1,2,2,2,26,26,26}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,26,26,26}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? ∊ {1,2,2,2,26,26,26}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1,16,15] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [6,8,9,10,11,5,12,7,4,3,2,1,14,13] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [5,6,8,9,4,3,10,7,2,1,12,11] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [4,7,8,3,9,10,6,5,2,1,12,11] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,7,8,10,11,6,5,12,9,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,6,9,10,5,11,12,8,7,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,8,10,11,12,13,7,14,9,6,5,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1,18,17] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> [7,9,10,11,12,13,6,14,8,5,4,3,2,1,16,15] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> [6,7,9,10,11,5,4,12,8,3,2,1,14,13] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> [5,8,9,10,4,11,12,7,6,3,2,1,14,13] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]
=> [5,6,7,9,4,3,2,10,8,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
=> [4,6,8,3,9,5,10,7,2,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [3,7,2,8,9,10,6,5,4,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> [5,6,7,8,4,3,2,1,11,12,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764}
Description
The staircase size of the code of a permutation. The code c(\pi) of a permutation \pi of length n is given by the sequence (c_1,\ldots,c_{n}) with c_i = |\{j > i : \pi(j) < \pi(i)\}|. This is a bijection between permutations and all sequences (c_1,\ldots,c_n) with 0 \leq c_i \leq n-i. The staircase size of the code is the maximal k such that there exists a subsequence (c_{i_k},\ldots,c_{i_1}) of c(\pi) with c_{i_j} \geq j. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000891
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
Mp00283: Perfect matchings non-nesting-exceedence permutationPermutations
St000891: Permutations ⟶ ℤResult quality: 9% values known / values provided: 9%distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {4,4} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {4,4} + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? ∊ {1,10,10,10} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {1,10,10,10} + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 3 = 2 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? ∊ {1,10,10,10} + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,10,10,10} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? ∊ {1,2,2,2,26,26,26} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? ∊ {1,2,2,2,26,26,26} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {1,2,2,2,26,26,26} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {1,2,2,2,26,26,26} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {1,2,2,2,26,26,26} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,2,2,2,26,26,26} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? ∊ {1,2,2,2,26,26,26} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? ∊ {4,4,4,4,4,4,76,76,76,76} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1,16,15] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [6,8,9,10,11,5,12,7,4,3,2,1,14,13] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [5,6,8,9,4,3,10,7,2,1,12,11] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [4,7,8,3,9,10,6,5,2,1,12,11] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,7,8,10,11,6,5,12,9,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,6,9,10,5,11,12,8,7,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,8,10,11,12,13,7,14,9,6,5,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? ∊ {1,2,2,2,4,4,8,8,10,10,10,232,232,232,232} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18)]
=> [9,10,11,12,13,14,15,16,8,7,6,5,4,3,2,1,18,17] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> [7,9,10,11,12,13,6,14,8,5,4,3,2,1,16,15] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> [6,7,9,10,11,5,4,12,8,3,2,1,14,13] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> [5,8,9,10,4,11,12,7,6,3,2,1,14,13] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9),(11,12)]
=> [5,6,7,9,4,3,2,10,8,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8),(11,12)]
=> [4,6,8,3,9,5,10,7,2,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,12)]
=> [3,7,2,8,9,10,6,5,4,1,12,11] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> [5,6,7,8,4,3,2,1,11,12,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> [2,1,7,8,9,11,6,5,4,12,10,3] => ? ∊ {1,1,2,4,4,4,8,8,8,8,12,20,20,20,26,26,26,764,764,764,764,764} + 1
Description
The number of distinct diagonal sums of a permutation matrix. For example, the sums of the diagonals of the matrix \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right) are (1,0,1,0,2,0), so the statistic is 3.