Processing math: 100%

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001048
St001048: Perfect matchings ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 1
[(1,2),(3,4)]
=> 2
[(1,3),(2,4)]
=> 2
[(1,4),(2,3)]
=> 1
[(1,2),(3,4),(5,6)]
=> 2
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 3
[(1,6),(2,3),(4,5)]
=> 1
[(1,6),(2,4),(3,5)]
=> 1
[(1,5),(2,4),(3,6)]
=> 3
[(1,4),(2,5),(3,6)]
=> 3
[(1,3),(2,5),(4,6)]
=> 3
[(1,2),(3,5),(4,6)]
=> 3
[(1,2),(3,6),(4,5)]
=> 3
[(1,3),(2,6),(4,5)]
=> 3
[(1,4),(2,6),(3,5)]
=> 3
[(1,5),(2,6),(3,4)]
=> 3
[(1,6),(2,5),(3,4)]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> 3
[(1,3),(2,4),(5,6),(7,8)]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> 3
[(1,6),(2,3),(4,5),(7,8)]
=> 3
[(1,7),(2,3),(4,5),(6,8)]
=> 3
[(1,8),(2,3),(4,5),(6,7)]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> 3
[(1,6),(2,4),(3,5),(7,8)]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> 3
[(1,2),(3,5),(4,6),(7,8)]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> 3
[(1,4),(2,6),(3,5),(7,8)]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> 4
[(1,6),(2,7),(3,4),(5,8)]
=> 4
[(1,5),(2,7),(3,4),(6,8)]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> 4
[(1,3),(2,8),(4,5),(6,7)]
=> 4
[(1,4),(2,8),(3,5),(6,7)]
=> 4
Description
The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. The bijection between perfect matchings of {1,,2n} and trees with n+1 leaves is described in Example 5.2.6 of [1]. This statistic is the difference between [[St001045]] and n+1.
Matching statistic: St000374
Mp00058: Perfect matchings to permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000374: Permutations ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 27%
Values
[(1,2)]
=> [2,1] => [1,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2
[(1,4),(2,3)]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 2
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 3
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 3
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 3
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 3
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 3
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 3
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 3
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 3
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 3
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => ? = 3
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,3,4,1,2,8,5,9,7] => ? = 3
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,4,6,1,8,5,9,7] => ? = 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [2,3,8,5,6,1,4,9,7] => ? = 3
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [2,8,4,5,6,1,3,9,7] => ? = 3
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [8,6,4,5,1,2,3,9,7] => ? = 3
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [8,3,5,1,6,2,4,9,7] => ? = 3
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,8,1,5,6,3,4,9,7] => ? = 3
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ? = 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,3,4,1,6,8,2,9,7] => ? = 3
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [6,3,4,5,1,8,2,9,7] => ? = 3
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [8,3,4,5,6,1,2,9,7] => ? = 3
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,1,5] => ? = 4
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [9,3,4,8,6,7,1,2,5] => ? = 4
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,7,1,8,2,6] => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,1,7,8,2,6] => ? = 2
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,3,4,1,9,7,8,2,6] => ? = 2
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,4,1,5,9,7,8,3,6] => ? = 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,5,6,7,8,9,3] => ? = 4
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 4
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 4
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 4
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 4
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 4
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 4
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 4
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 4
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,5,6,7,8,9,3] => ? = 4
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,4,1,9,6,7,8,3,5] => ? = 4
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,1,6,7,8,2,5] => ? = 4
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,6,1,7,8,2,5] => ? = 4
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [9,3,7,5,6,1,8,2,4] => ? = 4
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,6),(5,7)]
=> [8,4,6,2,7,3,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,3),(4,6),(5,7)]
=> [8,3,2,6,7,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,7),(5,6)]
=> [8,4,7,2,6,5,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,7),(4,6)]
=> [8,5,7,6,2,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,7),(4,5)]
=> [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [10,8,9,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
Description
The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation π=[π1,,πn], this statistic counts the number of position j such that πj<j and there do not exist indices i,k with i<j<k and πi>πj>πk. See also [[St000213]] and [[St000119]].
Matching statistic: St000703
Mp00058: Perfect matchings to permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000703: Permutations ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 27%
Values
[(1,2)]
=> [2,1] => [1,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2
[(1,4),(2,3)]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 2
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 3
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 3
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 3
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 3
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 3
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 3
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 3
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 3
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 3
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => ? = 3
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,3,4,1,2,8,5,9,7] => ? = 3
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,4,6,1,8,5,9,7] => ? = 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [2,3,8,5,6,1,4,9,7] => ? = 3
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [2,8,4,5,6,1,3,9,7] => ? = 3
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [8,6,4,5,1,2,3,9,7] => ? = 3
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [8,3,5,1,6,2,4,9,7] => ? = 3
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,8,1,5,6,3,4,9,7] => ? = 3
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ? = 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,3,4,1,6,8,2,9,7] => ? = 3
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [6,3,4,5,1,8,2,9,7] => ? = 3
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [8,3,4,5,6,1,2,9,7] => ? = 3
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,1,5] => ? = 4
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [9,3,4,8,6,7,1,2,5] => ? = 4
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,7,1,8,2,6] => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,1,7,8,2,6] => ? = 2
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,3,4,1,9,7,8,2,6] => ? = 2
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,4,1,5,9,7,8,3,6] => ? = 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,5,6,7,8,9,3] => ? = 4
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 4
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 4
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 4
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 4
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 4
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 4
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 4
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 4
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,5,6,7,8,9,3] => ? = 4
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,4,1,9,6,7,8,3,5] => ? = 4
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,1,6,7,8,2,5] => ? = 4
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,6,1,7,8,2,5] => ? = 4
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [9,3,7,5,6,1,8,2,4] => ? = 4
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,6),(5,7)]
=> [8,4,6,2,7,3,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,3),(4,6),(5,7)]
=> [8,3,2,6,7,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,7),(5,6)]
=> [8,4,7,2,6,5,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,7),(4,6)]
=> [8,5,7,6,2,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,7),(4,5)]
=> [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [10,8,9,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
Description
The number of deficiencies of a permutation. This is defined as dec(σ)=#{i:σ(i)<i}. The number of exceedances is [[St000155]].
Matching statistic: St000862
Mp00058: Perfect matchings to permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000862: Permutations ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 27%
Values
[(1,2)]
=> [2,1] => [1,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 2
[(1,4),(2,3)]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 2
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 3
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 3
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 3
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 3
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 3
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 3
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 3
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 3
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 3
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => ? = 3
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,3,4,1,2,8,5,9,7] => ? = 3
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,4,6,1,8,5,9,7] => ? = 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [2,3,8,5,6,1,4,9,7] => ? = 3
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [2,8,4,5,6,1,3,9,7] => ? = 3
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [8,6,4,5,1,2,3,9,7] => ? = 3
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [8,3,5,1,6,2,4,9,7] => ? = 3
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,8,1,5,6,3,4,9,7] => ? = 3
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ? = 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,3,4,1,6,8,2,9,7] => ? = 3
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [6,3,4,5,1,8,2,9,7] => ? = 3
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [8,3,4,5,6,1,2,9,7] => ? = 3
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,1,5] => ? = 4
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [9,3,4,8,6,7,1,2,5] => ? = 4
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,7,1,8,2,6] => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,1,7,8,2,6] => ? = 2
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,3,4,1,9,7,8,2,6] => ? = 2
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,4,1,5,9,7,8,3,6] => ? = 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,5,6,7,8,9,3] => ? = 4
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 4
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 4
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 4
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 4
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 4
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 4
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 4
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 4
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,5,6,7,8,9,3] => ? = 4
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,4,1,9,6,7,8,3,5] => ? = 4
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,1,6,7,8,2,5] => ? = 4
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,6,1,7,8,2,5] => ? = 4
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [9,3,7,5,6,1,8,2,4] => ? = 4
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,6),(5,7)]
=> [8,4,6,2,7,3,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,3),(4,6),(5,7)]
=> [8,3,2,6,7,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,4),(3,7),(5,6)]
=> [8,4,7,2,6,5,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,5),(3,7),(4,6)]
=> [8,5,7,6,2,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,6),(3,7),(4,5)]
=> [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [10,8,9,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 1
Description
The number of parts of the shifted shape of a permutation. The diagram of a strict partition λ1<λ2<<λ of n is a tableau with rows, the i-th row being indented by i cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair (P,Q) of standard shifted Young tableaux of the same shape, where off-diagonal entries in Q may be circled. This statistic records the number of parts of the shifted shape.
Matching statistic: St000451
Mp00058: Perfect matchings to permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St000451: Permutations ⟶ ℤResult quality: 3% values known / values provided: 3%distinct values known / distinct values provided: 27%
Values
[(1,2)]
=> [2,1] => [1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[(1,2),(3,4)]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 3 = 2 + 1
[(1,3),(2,4)]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 3 = 2 + 1
[(1,4),(2,3)]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 2 + 1
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 2 + 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? = 3 + 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 1 + 1
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? = 3 + 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => 4 = 3 + 1
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? = 3 + 1
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => ? = 3 + 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 3 + 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? = 3 + 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? = 3 + 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? = 3 + 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => ? = 3 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,3,4,1,2,8,5,9,7] => ? = 3 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,4,6,1,8,5,9,7] => ? = 3 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [2,3,8,5,6,1,4,9,7] => ? = 3 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [2,8,4,5,6,1,3,9,7] => ? = 3 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [8,6,4,5,1,2,3,9,7] => ? = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [8,3,5,1,6,2,4,9,7] => ? = 3 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,8,1,5,6,3,4,9,7] => ? = 3 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ? = 3 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,3,4,1,6,8,2,9,7] => ? = 3 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [6,3,4,5,1,8,2,9,7] => ? = 3 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> [8,3,4,5,6,1,2,9,7] => ? = 3 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,6,8,1,9,7] => ? = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [2,3,4,5,9,7,8,1,6] => ? = 3 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,1,5] => ? = 4 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> [9,3,4,8,6,7,1,2,5] => ? = 4 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,7,1,8,2,6] => ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [9,3,4,5,1,7,8,2,6] => ? = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,3,4,1,9,7,8,2,6] => ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,4,1,5,9,7,8,3,6] => ? = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,5,6,7,8,9,3] => ? = 4 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 4 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 4 + 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 4 + 1
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 4 + 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 4 + 1
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [9,3,4,5,6,7,8,1,2] => ? = 4 + 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,3,4,5,6,7,1,9,2] => ? = 4 + 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [7,3,4,5,6,1,8,9,2] => ? = 4 + 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,9,2] => ? = 4 + 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,9,2] => ? = 4 + 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,5,6,7,8,9,3] => ? = 4 + 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,4,1,9,6,7,8,3,5] => ? = 4 + 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,1,6,7,8,2,5] => ? = 4 + 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [9,3,4,6,1,7,8,2,5] => ? = 4 + 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [9,3,7,5,6,1,8,2,4] => ? = 4 + 1
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,4),(3,6),(5,7)]
=> [8,4,6,2,7,3,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,3),(4,6),(5,7)]
=> [8,3,2,6,7,4,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,4),(3,7),(5,6)]
=> [8,4,7,2,6,5,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,5),(3,7),(4,6)]
=> [8,5,7,6,2,4,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,6),(3,7),(4,5)]
=> [8,6,7,5,4,2,3,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => 2 = 1 + 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,8),(3,9),(4,6),(5,7)]
=> [10,8,9,6,7,4,5,2,3,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 2 = 1 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).