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Your data matches 121 different statistics following compositions of up to 3 maps.
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Matching statistic: St000843
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
St000843: 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)]
=> 1
[(1,4),(2,3)]
=> 1
[(1,2),(3,4),(5,6)]
=> 3
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 1
[(1,6),(2,3),(4,5)]
=> 1
[(1,6),(2,4),(3,5)]
=> 1
[(1,5),(2,4),(3,6)]
=> 1
[(1,4),(2,5),(3,6)]
=> 1
[(1,3),(2,5),(4,6)]
=> 1
[(1,2),(3,5),(4,6)]
=> 2
[(1,2),(3,6),(4,5)]
=> 2
[(1,3),(2,6),(4,5)]
=> 1
[(1,4),(2,6),(3,5)]
=> 1
[(1,5),(2,6),(3,4)]
=> 1
[(1,6),(2,5),(3,4)]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> 4
[(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)]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> 1
[(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)]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> 3
[(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)]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> 1
[(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)]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> 1
Description
The decomposition number of a perfect matching.
This is the number of integers $i$ such that all elements in $\{1,\dots,i\}$ are matched among themselves.
Visually, it is the number of components of the arc diagram of the matching, where a component is a matching of a set of consecutive numbers $\{a,a+1,\dots,b\}$ such that there is no arc matching a number smaller than $a$ with a number larger than $b$.
E.g., $\{(1,6),(2,4),(3,5)\}$ is a hairpin under a single edge - crossing nested by a single arc. Thus, this matching has one component. However, $\{(1,2),(3,6),(4,5)\}$ is a single edge to the left of a ladder (a pair of nested edges), so it has two components.
Matching statistic: St000011
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
St000011: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> 2
[(1,3),(2,4)]
=> [1,1,0,0]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ? = 2
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? = 2
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ? = 2
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ? = 2
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? = 3
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ? = 4
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000007
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => 2
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ? => ? = 2
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ? => ? = 2
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ? => ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ? => ? = 4
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000025
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1,0]
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ?
=> ? = 2
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ?
=> ? = 4
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000056
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 2
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ? => ? = 2
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ? => ? = 2
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ? => ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ? => ? = 4
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
Matching statistic: St000084
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000084: Ordered trees ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [[]]
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [[],[]]
=> 2
[(1,3),(2,4)]
=> [1,1,0,0]
=> [[[]]]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [[[]]]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ?
=> ? = 2
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ?
=> ? = 4
Description
The number of subtrees.
Matching statistic: St000991
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 2
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ? => ? = 2
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ? => ? = 2
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ? => ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ? => ? = 4
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St001184
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1,0]
=> 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ?
=> ? = 2
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ?
=> ? = 4
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001461
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 2
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ? => ? = 2
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ? => ? = 2
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ? => ? = 2
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ? => ? = 2
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ? => ? = 2
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ? => ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ? => ? = 4
Description
The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$.
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
Matching statistic: St000234
(load all 66 compositions to match this statistic)
(load all 66 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 0 = 1 - 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 1 = 2 - 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2 = 3 - 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 2 - 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 2 - 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3 = 4 - 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 2 - 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0 = 1 - 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0 = 1 - 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0 = 1 - 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,8),(6,7),(9,12),(10,11)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,10),(6,7),(8,9),(11,12)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,9),(10,11)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,11),(5,10),(6,9),(7,8),(12,13)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,10),(6,9),(7,8),(11,12)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,12),(6,9),(7,8),(10,11)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ?
=> ? => ? = 2 - 1
[(1,4),(2,3),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,6),(4,5),(7,10),(8,9),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,8),(4,5),(6,7),(9,10),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,10),(2,9),(3,8),(4,7),(5,6),(11,14),(12,13)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,9),(3,8),(4,7),(5,6),(10,11),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,8),(4,7),(5,6),(9,10),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,10),(4,7),(5,6),(8,9),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,16),(4,15),(5,14),(6,7),(8,13),(9,12),(10,11)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,16),(4,15),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,10),(8,9),(11,12)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ?
=> ? => ? = 2 - 1
[(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2 - 1
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,8),(9,10),(15,16)]
=> ?
=> ? => ? = 2 - 1
[(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9),(15,16)]
=> ?
=> ? => ? = 2 - 1
[(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11),(15,16)]
=> ?
=> ? => ? = 2 - 1
[(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10),(15,16)]
=> ?
=> ? => ? = 2 - 1
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,7),(8,9),(15,16)]
=> ?
=> ? => ? = 2 - 1
[(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ?
=> ? => ? = 2 - 1
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 5 - 1
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5 - 1
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4 - 1
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 4 - 1
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5 - 1
[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4 - 1
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 4 - 1
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 3 - 1
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3 - 1
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> ?
=> ? => ? = 4 - 1
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
The following 111 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000237The number of small exceedances. St000439The position of the first down step of a Dyck path. St000546The number of global descents of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000010The length of the partition. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000288The number of ones in a binary word. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000382The first part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000740The last entry of a permutation. St000822The Hadwiger number of the graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000971The smallest closer of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001029The size of the core of a graph. St001050The number of terminal closers of a set partition. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000675The number of centered multitunnels of a Dyck path. St000738The first entry in the last row of a standard tableau. St000883The number of longest increasing subsequences of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000061The number of nodes on the left branch of a binary tree. St000717The number of ordinal summands of a poset. St000504The cardinality of the first block of a set partition. St000654The first descent of a permutation. St000906The length of the shortest maximal chain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St001812The biclique partition number of a graph. St000648The number of 2-excedences of a permutation. St001330The hat guessing number of a graph. St000924The number of topologically connected components of a perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000898The number of maximal entries in the last diagonal of the monotone triangle. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000291The number of descents of a binary word. St001889The size of the connectivity set of a signed permutation. St000292The number of ascents of a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001730The number of times the path corresponding to a binary word crosses the base line. St000306The bounce count of a Dyck path. St000386The number of factors DDU in a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000142The number of even parts of a partition. St000146The Andrews-Garvan crank of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000670The reversal length of a permutation. St001280The number of parts of an integer partition that are at least two. St000383The last part of an integer composition. St001333The cardinality of a minimal edge-isolating set of a graph. St000884The number of isolated descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000553The number of blocks of a graph. St001115The number of even descents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000534The number of 2-rises of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St001955The number of natural descents for set-valued two row standard Young tableaux. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
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