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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000164
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000164: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 1
[(1,2),(3,4)]
=> 2
[(1,3),(2,4)]
=> 0
[(1,4),(2,3)]
=> 1
[(1,2),(3,4),(5,6)]
=> 3
[(1,3),(2,4),(5,6)]
=> 1
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 1
[(1,6),(2,3),(4,5)]
=> 2
[(1,6),(2,4),(3,5)]
=> 0
[(1,5),(2,4),(3,6)]
=> 0
[(1,4),(2,5),(3,6)]
=> 0
[(1,3),(2,5),(4,6)]
=> 0
[(1,2),(3,5),(4,6)]
=> 1
[(1,2),(3,6),(4,5)]
=> 2
[(1,3),(2,6),(4,5)]
=> 1
[(1,4),(2,6),(3,5)]
=> 0
[(1,5),(2,6),(3,4)]
=> 1
[(1,6),(2,5),(3,4)]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> 4
[(1,3),(2,4),(5,6),(7,8)]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> 3
[(1,7),(2,3),(4,5),(6,8)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> 3
[(1,8),(2,4),(3,5),(6,7)]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> 0
[(1,6),(2,4),(3,5),(7,8)]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> 1
[(1,2),(3,5),(4,6),(7,8)]
=> 2
[(1,2),(3,6),(4,5),(7,8)]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> 0
[(1,3),(2,7),(4,5),(6,8)]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> 3
[(1,3),(2,8),(4,5),(6,7)]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> 1
Description
The number of short pairs.
A short pair is a matching pair of the form (i,i+1).
Matching statistic: St000153
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
St000153: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => 0
[(1,4),(2,3)]
=> [4,3,2,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => 0
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 2
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => 0
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => 2
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => 2
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => 0
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => 2
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => 2
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => 0
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 3
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => ? = 3
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ? = 3
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [7,3,2,5,4,8,1,6,10,9] => ? = 3
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [9,3,2,5,4,7,6,10,1,8] => ? = 3
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [10,4,5,2,3,7,6,9,8,1] => ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [9,4,5,2,3,7,6,10,1,8] => ? = 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [8,4,5,2,3,7,6,1,10,9] => ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [7,4,5,2,3,8,1,6,10,9] => ? = 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [6,4,5,2,3,1,8,7,10,9] => ? = 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [5,4,6,2,1,3,8,7,10,9] => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => ? = 2
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => ? = 3
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ? = 3
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,6,5,1,3,2,8,7,10,9] => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [5,6,4,3,1,2,8,7,10,9] => ? = 3
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [7,5,4,3,2,8,1,6,10,9] => ? = 2
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [9,5,4,3,2,7,6,10,1,8] => ? = 2
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [10,6,4,3,7,2,5,9,8,1] => ? = 2
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [9,6,4,3,7,2,5,10,1,8] => ? = 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [8,6,4,3,7,2,5,1,10,9] => ? = 2
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [7,6,4,3,8,2,1,5,10,9] => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [6,7,4,3,8,1,2,5,10,9] => ? = 2
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [5,7,4,3,1,8,2,6,10,9] => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,7,5,1,3,8,2,6,10,9] => ? = 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,7,1,5,4,8,2,6,10,9] => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,7,5,4,8,3,6,10,9] => ? = 3
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,8,1,5,4,7,6,2,10,9] => ? = 3
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,8,5,1,3,7,6,2,10,9] => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [5,8,4,3,1,7,6,2,10,9] => ? = 3
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [6,8,4,3,7,1,5,2,10,9] => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [7,8,4,3,6,5,1,2,10,9] => ? = 3
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [9,7,4,3,6,5,2,10,1,8] => ? = 2
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [10,8,4,3,6,5,9,2,7,1] => ? = 2
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [9,8,4,3,6,5,10,2,1,7] => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [8,9,4,3,6,5,10,1,2,7] => ? = 2
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [7,9,4,3,6,5,1,10,2,8] => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [6,9,4,3,7,1,5,10,2,8] => ? = 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [5,9,4,3,1,7,6,10,2,8] => ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,9,5,1,3,7,6,10,2,8] => ? = 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,9,1,5,4,7,6,10,2,8] => ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,9,5,4,7,6,10,3,8] => ? = 3
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,10,1,5,4,7,6,9,8,2] => ? = 3
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,10,5,1,3,7,6,9,8,2] => ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [5,10,4,3,1,7,6,9,8,2] => ? = 3
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [6,10,4,3,7,1,5,9,8,2] => ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [7,10,4,3,6,5,1,9,8,2] => ? = 3
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [8,10,4,3,6,5,9,1,7,2] => ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [9,10,4,3,6,5,8,7,1,2] => ? = 3
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [10,9,5,6,3,4,8,7,2,1] => ? = 1
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000237
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
St000237: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => 0
[(1,4),(2,3)]
=> [4,3,2,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => 0
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 2
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => 0
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => 2
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => 2
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => 0
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => 2
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => 2
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => 0
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 3
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => ? = 3
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ? = 3
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [7,3,2,5,4,8,1,6,10,9] => ? = 3
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [9,3,2,5,4,7,6,10,1,8] => ? = 3
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [10,4,5,2,3,7,6,9,8,1] => ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [9,4,5,2,3,7,6,10,1,8] => ? = 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [8,4,5,2,3,7,6,1,10,9] => ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [7,4,5,2,3,8,1,6,10,9] => ? = 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [6,4,5,2,3,1,8,7,10,9] => ? = 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [5,4,6,2,1,3,8,7,10,9] => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => ? = 2
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => ? = 3
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ? = 3
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,6,5,1,3,2,8,7,10,9] => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [5,6,4,3,1,2,8,7,10,9] => ? = 3
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [7,5,4,3,2,8,1,6,10,9] => ? = 2
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [9,5,4,3,2,7,6,10,1,8] => ? = 2
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [10,6,4,3,7,2,5,9,8,1] => ? = 2
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [9,6,4,3,7,2,5,10,1,8] => ? = 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [8,6,4,3,7,2,5,1,10,9] => ? = 2
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [7,6,4,3,8,2,1,5,10,9] => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [6,7,4,3,8,1,2,5,10,9] => ? = 2
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [5,7,4,3,1,8,2,6,10,9] => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,7,5,1,3,8,2,6,10,9] => ? = 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,7,1,5,4,8,2,6,10,9] => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,7,5,4,8,3,6,10,9] => ? = 3
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,8,1,5,4,7,6,2,10,9] => ? = 3
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,8,5,1,3,7,6,2,10,9] => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [5,8,4,3,1,7,6,2,10,9] => ? = 3
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [6,8,4,3,7,1,5,2,10,9] => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [7,8,4,3,6,5,1,2,10,9] => ? = 3
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [9,7,4,3,6,5,2,10,1,8] => ? = 2
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [10,8,4,3,6,5,9,2,7,1] => ? = 2
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [9,8,4,3,6,5,10,2,1,7] => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [8,9,4,3,6,5,10,1,2,7] => ? = 2
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [7,9,4,3,6,5,1,10,2,8] => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [6,9,4,3,7,1,5,10,2,8] => ? = 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [5,9,4,3,1,7,6,10,2,8] => ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,9,5,1,3,7,6,10,2,8] => ? = 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,9,1,5,4,7,6,10,2,8] => ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,9,5,4,7,6,10,3,8] => ? = 3
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,10,1,5,4,7,6,9,8,2] => ? = 3
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,10,5,1,3,7,6,9,8,2] => ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [5,10,4,3,1,7,6,9,8,2] => ? = 3
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [6,10,4,3,7,1,5,9,8,2] => ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [7,10,4,3,6,5,1,9,8,2] => ? = 3
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [8,10,4,3,6,5,9,1,7,2] => ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [9,10,4,3,6,5,8,7,1,2] => ? = 3
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [10,9,5,6,3,4,8,7,2,1] => ? = 1
Description
The number of small exceedances.
This is the number of indices i such that πi=i+1.
Matching statistic: St001465
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
St001465: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => 0
[(1,4),(2,3)]
=> [4,3,2,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => 0
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 2
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => 0
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => 2
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => 2
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => 0
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => 2
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => 2
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => 0
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => 3
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => ? = 3
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ? = 3
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [7,3,2,5,4,8,1,6,10,9] => ? = 3
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [9,3,2,5,4,7,6,10,1,8] => ? = 3
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [10,4,5,2,3,7,6,9,8,1] => ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [9,4,5,2,3,7,6,10,1,8] => ? = 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [8,4,5,2,3,7,6,1,10,9] => ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [7,4,5,2,3,8,1,6,10,9] => ? = 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [6,4,5,2,3,1,8,7,10,9] => ? = 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [5,4,6,2,1,3,8,7,10,9] => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => ? = 2
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => ? = 3
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ? = 3
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,6,5,1,3,2,8,7,10,9] => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [5,6,4,3,1,2,8,7,10,9] => ? = 3
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [7,5,4,3,2,8,1,6,10,9] => ? = 2
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [9,5,4,3,2,7,6,10,1,8] => ? = 2
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [10,6,4,3,7,2,5,9,8,1] => ? = 2
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [9,6,4,3,7,2,5,10,1,8] => ? = 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [8,6,4,3,7,2,5,1,10,9] => ? = 2
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [7,6,4,3,8,2,1,5,10,9] => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [6,7,4,3,8,1,2,5,10,9] => ? = 2
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [5,7,4,3,1,8,2,6,10,9] => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,7,5,1,3,8,2,6,10,9] => ? = 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,7,1,5,4,8,2,6,10,9] => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,7,5,4,8,3,6,10,9] => ? = 3
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,8,1,5,4,7,6,2,10,9] => ? = 3
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,8,5,1,3,7,6,2,10,9] => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [5,8,4,3,1,7,6,2,10,9] => ? = 3
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [6,8,4,3,7,1,5,2,10,9] => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [7,8,4,3,6,5,1,2,10,9] => ? = 3
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [9,7,4,3,6,5,2,10,1,8] => ? = 2
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [10,8,4,3,6,5,9,2,7,1] => ? = 2
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [9,8,4,3,6,5,10,2,1,7] => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [8,9,4,3,6,5,10,1,2,7] => ? = 2
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [7,9,4,3,6,5,1,10,2,8] => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [6,9,4,3,7,1,5,10,2,8] => ? = 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [5,9,4,3,1,7,6,10,2,8] => ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> [4,9,5,1,3,7,6,10,2,8] => ? = 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> [3,9,1,5,4,7,6,10,2,8] => ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> [2,1,9,5,4,7,6,10,3,8] => ? = 3
[(1,3),(2,10),(4,5),(6,7),(8,9)]
=> [3,10,1,5,4,7,6,9,8,2] => ? = 3
[(1,4),(2,10),(3,5),(6,7),(8,9)]
=> [4,10,5,1,3,7,6,9,8,2] => ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
=> [5,10,4,3,1,7,6,9,8,2] => ? = 3
[(1,6),(2,10),(3,4),(5,7),(8,9)]
=> [6,10,4,3,7,1,5,9,8,2] => ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
=> [7,10,4,3,6,5,1,9,8,2] => ? = 3
[(1,8),(2,10),(3,4),(5,6),(7,9)]
=> [8,10,4,3,6,5,9,1,7,2] => ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
=> [9,10,4,3,6,5,8,7,1,2] => ? = 3
[(1,10),(2,9),(3,5),(4,6),(7,8)]
=> [10,9,5,6,3,4,8,7,2,1] => ? = 1
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000382
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 86%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 86%
Values
[(1,2)]
=> {{1,2}}
=> {{1,2}}
=> [2] => 2 = 1 + 1
[(1,2),(3,4)]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [3,1] => 3 = 2 + 1
[(1,3),(2,4)]
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> [1,3] => 1 = 0 + 1
[(1,4),(2,3)]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [2,2] => 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> {{1,2,4,6},{3},{5}}
=> [4,1,1] => 4 = 3 + 1
[(1,3),(2,4),(5,6)]
=> {{1,3},{2,4},{5,6}}
=> {{1,6},{2,3,4},{5}}
=> [2,3,1] => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> {{1,3,6},{2,4},{5}}
=> [3,2,1] => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> {{1,5},{2,3},{4,6}}
=> {{1,3},{2,6},{4,5}}
=> [2,2,2] => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> {{1,3,5},{2},{4,6}}
=> [3,1,2] => 3 = 2 + 1
[(1,6),(2,4),(3,5)]
=> {{1,6},{2,4},{3,5}}
=> {{1},{2,4,5},{3,6}}
=> [1,3,2] => 1 = 0 + 1
[(1,5),(2,4),(3,6)]
=> {{1,5},{2,4},{3,6}}
=> {{1},{2,4},{3,5,6}}
=> [1,2,3] => 1 = 0 + 1
[(1,4),(2,5),(3,6)]
=> {{1,4},{2,5},{3,6}}
=> {{1},{2},{3,4,5,6}}
=> [1,1,4] => 1 = 0 + 1
[(1,3),(2,5),(4,6)]
=> {{1,3},{2,5},{4,6}}
=> {{1},{2,3,6},{4,5}}
=> [1,3,2] => 1 = 0 + 1
[(1,2),(3,5),(4,6)]
=> {{1,2},{3,5},{4,6}}
=> {{1,2},{3,5,6},{4}}
=> [2,3,1] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> {{1,2,5},{3,6},{4}}
=> [3,2,1] => 3 = 2 + 1
[(1,3),(2,6),(4,5)]
=> {{1,3},{2,6},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> [2,2,2] => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> {{1,4},{2,6},{3,5}}
=> {{1},{2,5},{3,4,6}}
=> [1,2,3] => 1 = 0 + 1
[(1,5),(2,6),(3,4)]
=> {{1,5},{2,6},{3,4}}
=> {{1,4},{2},{3,5,6}}
=> [2,1,3] => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> {{1,4},{2,5},{3,6}}
=> [2,2,2] => 2 = 1 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> [5,1,1,1] => 5 = 4 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> {{1,3},{2,4},{5,6},{7,8}}
=> {{1,6,8},{2,3,4},{5},{7}}
=> [3,3,1,1] => 3 = 2 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> {{1,3,6,8},{2,4},{5},{7}}
=> [4,2,1,1] => 4 = 3 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> {{1,5},{2,3},{4,6},{7,8}}
=> {{1,3,8},{2,6},{4,5},{7}}
=> [3,2,2,1] => 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> {{1,3,5,8},{2},{4,6},{7}}
=> [4,1,2,1] => 4 = 3 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> {{1,7},{2,3},{4,5},{6,8}}
=> {{1,3,5},{2,8},{4},{6,7}}
=> [3,2,1,2] => 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> {{1,3,5,7},{2},{4},{6,8}}
=> [4,1,1,2] => 4 = 3 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> {{1,8},{2,4},{3,5},{6,7}}
=> {{1,7},{2,4,5},{3},{6,8}}
=> [2,3,1,2] => 2 = 1 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> {{1,7},{2,4},{3,5},{6,8}}
=> {{1},{2,4,5,8},{3},{6,7}}
=> [1,4,1,2] => 1 = 0 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> {{1,6},{2,4},{3,5},{7,8}}
=> {{1,8},{2,4,5},{3,6},{7}}
=> [2,3,2,1] => 2 = 1 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> {{1,5},{2,4},{3,6},{7,8}}
=> {{1,8},{2,4},{3,5,6},{7}}
=> [2,2,3,1] => 2 = 1 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> {{1,4},{2,5},{3,6},{7,8}}
=> {{1,8},{2},{3,4,5,6},{7}}
=> [2,1,4,1] => 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> {{1,3},{2,5},{4,6},{7,8}}
=> {{1,8},{2,3,6},{4,5},{7}}
=> [2,3,2,1] => 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> {{1,2},{3,5},{4,6},{7,8}}
=> {{1,2,8},{3,5,6},{4},{7}}
=> [3,3,1,1] => 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> {{1,2,5,8},{3,6},{4},{7}}
=> [4,2,1,1] => 4 = 3 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> {{1,3},{2,6},{4,5},{7,8}}
=> {{1,5,8},{2,3},{4,6},{7}}
=> [3,2,2,1] => 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> {{1,4},{2,6},{3,5},{7,8}}
=> {{1,8},{2,5},{3,4,6},{7}}
=> [2,2,3,1] => 2 = 1 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> {{1,5},{2,6},{3,4},{7,8}}
=> {{1,4,8},{2},{3,5,6},{7}}
=> [3,1,3,1] => 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> {{1,4,8},{2,5},{3,6},{7}}
=> [3,2,2,1] => 3 = 2 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> {{1,7},{2,5},{3,4},{6,8}}
=> {{1,4},{2,5,8},{3},{6,7}}
=> [2,3,1,2] => 2 = 1 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> {{1,4,7},{2,5},{3},{6,8}}
=> [3,2,1,2] => 3 = 2 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> {{1,8},{2,6},{3,4},{5,7}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> [2,2,2,2] => 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> {{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2},{3,6,8},{5,7}}
=> [2,1,3,2] => 2 = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> {{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2},{3,8},{5,6,7}}
=> [2,1,2,3] => 2 = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> {{1,5},{2,7},{3,4},{6,8}}
=> {{1,4},{2,8},{3,5},{6,7}}
=> [2,2,2,2] => 2 = 1 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> {{1,4},{2,7},{3,5},{6,8}}
=> {{1},{2,5,8},{3,4},{6,7}}
=> [1,3,2,2] => 1 = 0 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> {{1,3},{2,7},{4,5},{6,8}}
=> {{1,5},{2,3,8},{4},{6,7}}
=> [2,3,1,2] => 2 = 1 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> {{1,2},{3,7},{4,5},{6,8}}
=> {{1,2,5},{3,8},{4,7},{6}}
=> [3,2,2,1] => 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> {{1,2,5,7},{3},{4,8},{6}}
=> [4,1,2,1] => 4 = 3 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> {{1,3},{2,8},{4,5},{6,7}}
=> {{1,5,7},{2,3},{4},{6,8}}
=> [3,2,1,2] => 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> {{1,4},{2,8},{3,5},{6,7}}
=> {{1,7},{2,5},{3,4},{6,8}}
=> [2,2,2,2] => 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> {{1,3},{2,4},{5,6},{7,8},{9,10}}
=> {{1,6,8,10},{2,3,4},{5},{7},{9}}
=> ? => ? = 3 + 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> {{1,5},{2,3},{4,6},{7,8},{9,10}}
=> ?
=> ? => ? = 3 + 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> {{1,6},{2,3},{4,5},{7,8},{9,10}}
=> {{1,3,5,8,10},{2},{4,6},{7},{9}}
=> [5,1,2,1,1] => ? = 4 + 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> {{1,7},{2,3},{4,5},{6,8},{9,10}}
=> ?
=> ? => ? = 3 + 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> {{1,8},{2,3},{4,5},{6,7},{9,10}}
=> {{1,3,5,7,10},{2},{4},{6,8},{9}}
=> [5,1,1,2,1] => ? = 4 + 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> {{1,9},{2,3},{4,5},{6,7},{8,10}}
=> ?
=> ? => ? = 3 + 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> {{1,3,5,7,9},{2},{4},{6},{8,10}}
=> [5,1,1,1,2] => ? = 4 + 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> {{1,10},{2,4},{3,5},{6,7},{8,9}}
=> ?
=> ? => ? = 2 + 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> {{1,9},{2,4},{3,5},{6,7},{8,10}}
=> ?
=> ? => ? = 1 + 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> {{1,8},{2,4},{3,5},{6,7},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> {{1,7},{2,4},{3,5},{6,8},{9,10}}
=> ?
=> ? => ? = 1 + 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> {{1,6},{2,4},{3,5},{7,8},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> {{1,5},{2,4},{3,6},{7,8},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> {{1,4},{2,5},{3,6},{7,8},{9,10}}
=> {{1,8,10},{2},{3,4,5,6},{7},{9}}
=> ? => ? = 2 + 1
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> {{1,3},{2,5},{4,6},{7,8},{9,10}}
=> {{1,8,10},{2,3,6},{4,5},{7},{9}}
=> ? => ? = 2 + 1
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> {{1,2},{3,5},{4,6},{7,8},{9,10}}
=> {{1,2,8,10},{3,5,6},{4},{7},{9}}
=> ? => ? = 3 + 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> {{1,3},{2,6},{4,5},{7,8},{9,10}}
=> ?
=> ? => ? = 3 + 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> {{1,4},{2,6},{3,5},{7,8},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> {{1,5},{2,6},{3,4},{7,8},{9,10}}
=> ?
=> ? => ? = 3 + 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> {{1,7},{2,5},{3,4},{6,8},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> {{1,8},{2,5},{3,4},{6,7},{9,10}}
=> {{1,4,7,10},{2,5},{3},{6,8},{9}}
=> [4,2,1,2,1] => ? = 3 + 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> {{1,9},{2,5},{3,4},{6,7},{8,10}}
=> ?
=> ? => ? = 2 + 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> {{1,10},{2,5},{3,4},{6,7},{8,9}}
=> {{1,4,7,9},{2,5},{3},{6},{8,10}}
=> [4,2,1,1,2] => ? = 3 + 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> {{1,10},{2,6},{3,4},{5,7},{8,9}}
=> ?
=> ? => ? = 2 + 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> {{1,9},{2,6},{3,4},{5,7},{8,10}}
=> ?
=> ? => ? = 1 + 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> {{1,8},{2,6},{3,4},{5,7},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> {{1,7},{2,6},{3,4},{5,8},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> {{1,6},{2,7},{3,4},{5,8},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> {{1,5},{2,7},{3,4},{6,8},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> {{1,4},{2,7},{3,5},{6,8},{9,10}}
=> ?
=> ? => ? = 1 + 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> {{1,3},{2,7},{4,5},{6,8},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> {{1,2},{3,7},{4,5},{6,8},{9,10}}
=> ?
=> ? => ? = 3 + 1
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> {{1,2},{3,8},{4,5},{6,7},{9,10}}
=> {{1,2,5,7,10},{3},{4,8},{6},{9}}
=> [5,1,2,1,1] => ? = 4 + 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> {{1,3},{2,8},{4,5},{6,7},{9,10}}
=> ?
=> ? => ? = 3 + 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> {{1,4},{2,8},{3,5},{6,7},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> {{1,5},{2,8},{3,4},{6,7},{9,10}}
=> ?
=> ? => ? = 3 + 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> {{1,6},{2,8},{3,4},{5,7},{9,10}}
=> ?
=> ? => ? = 2 + 1
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> {{1,7},{2,8},{3,4},{5,6},{9,10}}
=> ?
=> ? => ? = 3 + 1
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> {{1,8},{2,7},{3,4},{5,6},{9,10}}
=> {{1,4,6,10},{2},{3,7},{5,8},{9}}
=> [4,1,2,2,1] => ? = 3 + 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> {{1,9},{2,7},{3,4},{5,6},{8,10}}
=> ?
=> ? => ? = 2 + 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> {{1,10},{2,7},{3,4},{5,6},{8,9}}
=> {{1,4,6,9},{2},{3,7},{5},{8,10}}
=> [4,1,2,1,2] => ? = 3 + 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> {{1,10},{2,8},{3,4},{5,6},{7,9}}
=> ?
=> ? => ? = 2 + 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> {{1,9},{2,8},{3,4},{5,6},{7,10}}
=> ?
=> ? => ? = 2 + 1
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> {{1,8},{2,9},{3,4},{5,6},{7,10}}
=> ?
=> ? => ? = 2 + 1
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> {{1,7},{2,9},{3,4},{5,6},{8,10}}
=> ?
=> ? => ? = 2 + 1
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> {{1,6},{2,9},{3,4},{5,7},{8,10}}
=> ?
=> ? => ? = 1 + 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> {{1,5},{2,9},{3,4},{6,7},{8,10}}
=> ?
=> ? => ? = 2 + 1
[(1,4),(2,9),(3,5),(6,7),(8,10)]
=> {{1,4},{2,9},{3,5},{6,7},{8,10}}
=> ?
=> ? => ? = 1 + 1
[(1,3),(2,9),(4,5),(6,7),(8,10)]
=> {{1,3},{2,9},{4,5},{6,7},{8,10}}
=> ?
=> ? => ? = 2 + 1
[(1,2),(3,9),(4,5),(6,7),(8,10)]
=> {{1,2},{3,9},{4,5},{6,7},{8,10}}
=> ?
=> ? => ? = 3 + 1
Description
The first part of an integer composition.
Matching statistic: St000214
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [2,1] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => 0
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,2,5,1,6,3] => 0
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => 2
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,5,3,6,2] => 0
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,3,5,1,6,2] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => 2
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => 2
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => 0
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => 2
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => 2
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => 0
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => 3
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => 1
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => [3,1,4,2,6,5,8,7,10,9] => ? = 3
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [3,2,4,1,6,5,8,7,10,9] => ? = 4
[(1,5),(2,3),(4,6),(7,8),(9,10)]
=> [5,3,2,6,1,4,8,7,10,9] => ? => ? = 3
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [3,2,5,4,6,1,8,7,10,9] => ? = 4
[(1,7),(2,3),(4,5),(6,8),(9,10)]
=> [7,3,2,5,4,8,1,6,10,9] => ? => ? = 3
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [3,2,5,4,7,6,8,1,10,9] => ? = 4
[(1,9),(2,3),(4,5),(6,7),(8,10)]
=> [9,3,2,5,4,7,6,10,1,8] => ? => ? = 3
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [3,2,5,4,7,6,9,8,10,1] => ? = 4
[(1,10),(2,4),(3,5),(6,7),(8,9)]
=> [10,4,5,2,3,7,6,9,8,1] => ? => ? = 2
[(1,9),(2,4),(3,5),(6,7),(8,10)]
=> [9,4,5,2,3,7,6,10,1,8] => ? => ? = 1
[(1,8),(2,4),(3,5),(6,7),(9,10)]
=> [8,4,5,2,3,7,6,1,10,9] => ? => ? = 2
[(1,7),(2,4),(3,5),(6,8),(9,10)]
=> [7,4,5,2,3,8,1,6,10,9] => ? => ? = 1
[(1,6),(2,4),(3,5),(7,8),(9,10)]
=> [6,4,5,2,3,1,8,7,10,9] => [4,2,5,3,6,1,8,7,10,9] => ? = 2
[(1,5),(2,4),(3,6),(7,8),(9,10)]
=> [5,4,6,2,1,3,8,7,10,9] => ? => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => [4,1,5,2,6,3,8,7,10,9] => ? = 2
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => [3,1,5,2,6,4,8,7,10,9] => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => [2,1,5,3,6,4,8,7,10,9] => ? = 3
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,5,4,6,3,8,7,10,9] => ? = 4
[(1,3),(2,6),(4,5),(7,8),(9,10)]
=> [3,6,1,5,4,2,8,7,10,9] => ? => ? = 3
[(1,4),(2,6),(3,5),(7,8),(9,10)]
=> [4,6,5,1,3,2,8,7,10,9] => ? => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
=> [5,6,4,3,1,2,8,7,10,9] => ? => ? = 3
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [4,3,5,2,6,1,8,7,10,9] => ? = 3
[(1,7),(2,5),(3,4),(6,8),(9,10)]
=> [7,5,4,3,2,8,1,6,10,9] => ? => ? = 2
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [4,3,5,2,7,6,8,1,10,9] => ? = 3
[(1,9),(2,5),(3,4),(6,7),(8,10)]
=> [9,5,4,3,2,7,6,10,1,8] => ? => ? = 2
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [4,3,5,2,7,6,9,8,10,1] => ? = 3
[(1,10),(2,6),(3,4),(5,7),(8,9)]
=> [10,6,4,3,7,2,5,9,8,1] => ? => ? = 2
[(1,9),(2,6),(3,4),(5,7),(8,10)]
=> [9,6,4,3,7,2,5,10,1,8] => ? => ? = 1
[(1,8),(2,6),(3,4),(5,7),(9,10)]
=> [8,6,4,3,7,2,5,1,10,9] => ? => ? = 2
[(1,7),(2,6),(3,4),(5,8),(9,10)]
=> [7,6,4,3,8,2,1,5,10,9] => ? => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
=> [6,7,4,3,8,1,2,5,10,9] => ? => ? = 2
[(1,5),(2,7),(3,4),(6,8),(9,10)]
=> [5,7,4,3,1,8,2,6,10,9] => ? => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
=> [4,7,5,1,3,8,2,6,10,9] => ? => ? = 1
[(1,3),(2,7),(4,5),(6,8),(9,10)]
=> [3,7,1,5,4,8,2,6,10,9] => ? => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
=> [2,1,7,5,4,8,3,6,10,9] => ? => ? = 3
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,5,4,7,6,8,3,10,9] => ? = 4
[(1,3),(2,8),(4,5),(6,7),(9,10)]
=> [3,8,1,5,4,7,6,2,10,9] => ? => ? = 3
[(1,4),(2,8),(3,5),(6,7),(9,10)]
=> [4,8,5,1,3,7,6,2,10,9] => ? => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
=> [5,8,4,3,1,7,6,2,10,9] => ? => ? = 3
[(1,6),(2,8),(3,4),(5,7),(9,10)]
=> [6,8,4,3,7,1,5,2,10,9] => ? => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
=> [7,8,4,3,6,5,1,2,10,9] => ? => ? = 3
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [4,3,6,5,7,2,8,1,10,9] => ? = 3
[(1,9),(2,7),(3,4),(5,6),(8,10)]
=> [9,7,4,3,6,5,2,10,1,8] => ? => ? = 2
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [4,3,6,5,7,2,9,8,10,1] => ? = 3
[(1,10),(2,8),(3,4),(5,6),(7,9)]
=> [10,8,4,3,6,5,9,2,7,1] => ? => ? = 2
[(1,9),(2,8),(3,4),(5,6),(7,10)]
=> [9,8,4,3,6,5,10,2,1,7] => ? => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
=> [8,9,4,3,6,5,10,1,2,7] => ? => ? = 2
[(1,7),(2,9),(3,4),(5,6),(8,10)]
=> [7,9,4,3,6,5,1,10,2,8] => ? => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
=> [6,9,4,3,7,1,5,10,2,8] => ? => ? = 1
[(1,5),(2,9),(3,4),(6,7),(8,10)]
=> [5,9,4,3,1,7,6,10,2,8] => ? => ? = 2
Description
The number of adjacencies of a permutation.
An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000441
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 86%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 86%
Values
[(1,2)]
=> [2,1] => [2,1] => [1,2] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 0
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => 3
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => [5,6,2,4,1,3] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => [5,6,1,4,2,3] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => [4,6,1,5,2,3] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => [1,6,4,5,2,3] => 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => [1,6,3,5,2,4] => 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,2,5,1,6,3] => [3,6,1,5,2,4] => 0
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [3,6,2,5,1,4] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => [4,6,3,5,1,2] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => [3,6,4,5,1,2] => 2
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,6,2] => [2,6,4,5,1,3] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,5,3,6,2] => [2,6,3,5,1,4] => 0
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,3,5,1,6,2] => [2,6,1,5,3,4] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => [1,6,2,5,3,4] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => [7,8,5,6,2,4,1,3] => ? = 2
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => [7,8,5,6,1,4,2,3] => ? = 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => [7,8,4,6,1,5,2,3] => ? = 2
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => [7,8,1,6,4,5,2,3] => ? = 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => [6,8,1,7,4,5,2,3] => ? = 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => [1,8,6,7,4,5,2,3] => ? = 3
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => [1,8,6,7,3,5,2,4] => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => [6,8,1,7,3,5,2,4] => ? = 0
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => [7,8,1,6,3,5,2,4] => ? = 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => [7,8,3,6,1,5,2,4] => ? = 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => [7,8,3,6,2,5,1,4] => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => [7,8,4,6,2,5,1,3] => ? = 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => [7,8,4,6,3,5,1,2] => ? = 2
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => [7,8,3,6,4,5,1,2] => ? = 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => [7,8,2,6,4,5,1,3] => ? = 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => [7,8,2,6,3,5,1,4] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => [7,8,2,6,1,5,3,4] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => [7,8,1,6,2,5,3,4] => ? = 2
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => [6,8,1,7,2,5,3,4] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => [1,8,6,7,2,5,3,4] => ? = 2
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => [1,8,5,7,2,6,3,4] => ? = 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => [5,8,1,7,2,6,3,4] => ? = 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => [5,8,2,7,1,6,3,4] => ? = 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => [6,8,2,7,1,5,3,4] => ? = 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => [6,8,2,7,3,5,1,4] => ? = 0
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => [6,8,2,7,4,5,1,3] => ? = 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => [6,8,3,7,4,5,1,2] => ? = 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => [3,8,6,7,4,5,1,2] => ? = 3
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => [2,8,6,7,4,5,1,3] => ? = 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => [2,8,6,7,3,5,1,4] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => [2,8,6,7,1,5,3,4] => ? = 2
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => [2,8,5,7,1,6,3,4] => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => [2,8,1,7,5,6,3,4] => 2
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => [1,8,2,7,5,6,3,4] => ? = 2
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [1,8,2,7,4,6,3,5] => ? = 0
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [2,8,1,7,4,6,3,5] => ? = 0
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => [2,8,4,7,1,6,3,5] => ? = 0
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [5,1,6,3,7,4,8,2] => [2,8,4,7,3,6,1,5] => ? = 0
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [4,1,6,3,7,5,8,2] => [2,8,5,7,3,6,1,4] => ? = 0
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [3,1,6,4,7,5,8,2] => [2,8,5,7,4,6,1,3] => ? = 0
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [2,1,6,4,7,5,8,3] => [3,8,5,7,4,6,1,2] => ? = 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [2,1,6,4,7,3,8,5] => [5,8,3,7,4,6,1,2] => ? = 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [3,1,6,4,7,2,8,5] => [5,8,2,7,4,6,1,3] => ? = 0
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [4,1,6,3,7,2,8,5] => [5,8,2,7,3,6,1,4] => ? = 0
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [5,1,6,3,7,2,8,4] => [4,8,2,7,3,6,1,5] => ? = 0
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [5,3,6,1,7,2,8,4] => [4,8,2,7,1,6,3,5] => ? = 0
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [4,8,1,7,2,6,3,5] => ? = 0
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [1,8,4,7,2,6,3,5] => ? = 0
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => [1,8,4,7,3,6,2,5] => ? = 0
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [5,2,6,3,7,1,8,4] => [4,8,1,7,3,6,2,5] => ? = 0
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [5,2,6,1,7,3,8,4] => [4,8,3,7,1,6,2,5] => ? = 0
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [4,8,3,7,2,6,1,5] => ? = 0
[(1,6),(2,3),(4,8),(5,7)]
=> [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => [4,8,5,7,1,6,2,3] => 1
[(1,5),(2,4),(3,8),(6,7)]
=> [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => [3,8,6,7,1,5,2,4] => 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => 5
Description
The number of successions of a permutation.
A succession of a permutation π is an index i such that π(i)+1=π(i+1). Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000665
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 86%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 86%
Values
[(1,2)]
=> [2,1] => [2,1] => [1,2] => 1
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 0
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => [1,4,2,3] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => 3
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => [5,6,2,4,1,3] => 1
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => [5,6,1,4,2,3] => 2
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => [4,6,1,5,2,3] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => [1,6,4,5,2,3] => 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => [1,6,3,5,2,4] => 0
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,2,5,1,6,3] => [3,6,1,5,2,4] => 0
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [3,6,2,5,1,4] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => [4,6,3,5,1,2] => 1
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => [3,6,4,5,1,2] => 2
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,6,2] => [2,6,4,5,1,3] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,5,3,6,2] => [2,6,3,5,1,4] => 0
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,3,5,1,6,2] => [2,6,1,5,3,4] => 1
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => [1,6,2,5,3,4] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => [7,8,5,6,2,4,1,3] => ? = 2
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => [7,8,5,6,1,4,2,3] => ? = 3
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => [7,8,4,6,1,5,2,3] => ? = 2
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => [7,8,1,6,4,5,2,3] => ? = 3
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => [6,8,1,7,4,5,2,3] => ? = 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => [1,8,6,7,4,5,2,3] => ? = 3
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => [1,8,6,7,3,5,2,4] => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => [6,8,1,7,3,5,2,4] => ? = 0
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => [7,8,1,6,3,5,2,4] => ? = 1
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => [7,8,3,6,1,5,2,4] => ? = 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => [7,8,3,6,2,5,1,4] => ? = 1
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => [7,8,4,6,2,5,1,3] => ? = 1
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => [7,8,4,6,3,5,1,2] => ? = 2
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => [7,8,3,6,4,5,1,2] => ? = 3
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => [7,8,2,6,4,5,1,3] => ? = 2
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => [7,8,2,6,3,5,1,4] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => [7,8,2,6,1,5,3,4] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => [7,8,1,6,2,5,3,4] => ? = 2
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => [6,8,1,7,2,5,3,4] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => [1,8,6,7,2,5,3,4] => ? = 2
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => [1,8,5,7,2,6,3,4] => ? = 1
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => [5,8,1,7,2,6,3,4] => ? = 1
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => [5,8,2,7,1,6,3,4] => ? = 1
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => [6,8,2,7,1,5,3,4] => ? = 1
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => [6,8,2,7,3,5,1,4] => ? = 0
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => [6,8,2,7,4,5,1,3] => ? = 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => [6,8,3,7,4,5,1,2] => ? = 2
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => [3,8,6,7,4,5,1,2] => ? = 3
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => [2,8,6,7,4,5,1,3] => ? = 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => [2,8,6,7,3,5,1,4] => ? = 1
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => [2,8,6,7,1,5,3,4] => ? = 2
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => [2,8,5,7,1,6,3,4] => ? = 1
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => [2,8,1,7,5,6,3,4] => 2
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => [1,8,2,7,5,6,3,4] => ? = 2
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [1,8,2,7,4,6,3,5] => ? = 0
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [2,8,1,7,4,6,3,5] => ? = 0
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => [2,8,4,7,1,6,3,5] => ? = 0
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [5,1,6,3,7,4,8,2] => [2,8,4,7,3,6,1,5] => ? = 0
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [4,1,6,3,7,5,8,2] => [2,8,5,7,3,6,1,4] => ? = 0
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [3,1,6,4,7,5,8,2] => [2,8,5,7,4,6,1,3] => ? = 0
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [2,1,6,4,7,5,8,3] => [3,8,5,7,4,6,1,2] => ? = 1
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [2,1,6,4,7,3,8,5] => [5,8,3,7,4,6,1,2] => ? = 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [3,1,6,4,7,2,8,5] => [5,8,2,7,4,6,1,3] => ? = 0
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [4,1,6,3,7,2,8,5] => [5,8,2,7,3,6,1,4] => ? = 0
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [5,1,6,3,7,2,8,4] => [4,8,2,7,3,6,1,5] => ? = 0
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [5,3,6,1,7,2,8,4] => [4,8,2,7,1,6,3,5] => ? = 0
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [4,8,1,7,2,6,3,5] => ? = 0
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [1,8,4,7,2,6,3,5] => ? = 0
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => [1,8,4,7,3,6,2,5] => ? = 0
[(1,7),(2,5),(3,6),(4,8)]
=> [7,5,6,8,2,3,1,4] => [5,2,6,3,7,1,8,4] => [4,8,1,7,3,6,2,5] => ? = 0
[(1,6),(2,5),(3,7),(4,8)]
=> [6,5,7,8,2,1,3,4] => [5,2,6,1,7,3,8,4] => [4,8,3,7,1,6,2,5] => ? = 0
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [4,8,3,7,2,6,1,5] => ? = 0
[(1,6),(2,3),(4,8),(5,7)]
=> [6,3,2,8,7,1,5,4] => [3,2,6,1,7,5,8,4] => [4,8,5,7,1,6,2,3] => 1
[(1,5),(2,4),(3,8),(6,7)]
=> [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => [3,8,6,7,1,5,2,4] => 1
[(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,8,2,7,3,6,4,5] => 1
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => 5
Description
The number of rafts of a permutation.
Let π be a permutation of length n. A small ascent of π is an index i such that π(i+1)=π(i)+1, see [[St000441]], and a raft of π is a non-empty maximal sequence of consecutive small ascents.
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