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Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St000840
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
St000840: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 0
[(1,2),(3,4)]
=> 1
[(1,3),(2,4)]
=> 0
[(1,4),(2,3)]
=> 0
[(1,2),(3,4),(5,6)]
=> 2
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 2
[(1,5),(2,3),(4,6)]
=> 1
[(1,6),(2,3),(4,5)]
=> 1
[(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)]
=> 1
[(1,2),(3,5),(4,6)]
=> 1
[(1,2),(3,6),(4,5)]
=> 1
[(1,3),(2,6),(4,5)]
=> 1
[(1,4),(2,6),(3,5)]
=> 0
[(1,5),(2,6),(3,4)]
=> 0
[(1,6),(2,5),(3,4)]
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> 3
[(1,3),(2,4),(5,6),(7,8)]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> 3
[(1,6),(2,3),(4,5),(7,8)]
=> 3
[(1,7),(2,3),(4,5),(6,8)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> 3
[(1,2),(3,5),(4,6),(7,8)]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> 3
[(1,4),(2,6),(3,5),(7,8)]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> 2
[(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)]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> 2
Description
The number of closers smaller than the largest opener in a perfect matching.
An opener (or left hand endpoint) of a perfect matching is a number that is matched with a larger number, which is then called a closer (or right hand endpoint).
Matching statistic: St001291
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
St001291: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> 1 = 0 + 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> 1 = 0 + 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> 1 = 0 + 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? = 4 + 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)]
=> ?
=> ? = 3 + 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)]
=> ?
=> ? = 5 + 1
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? = 4 + 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)]
=> ?
=> ? = 5 + 1
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> ?
=> ? = 4 + 1
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> ?
=> ? = 3 + 1
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> ?
=> ? = 2 + 1
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ? = 5 + 1
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> ?
=> ? = 4 + 1
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ? = 5 + 1
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> ?
=> ? = 4 + 1
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> ?
=> ? = 3 + 1
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ? = 5 + 1
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> ?
=> ? = 4 + 1
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ? = 5 + 1
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> ?
=> ? = 4 + 1
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> ?
=> ? = 3 + 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000141
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[(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] => 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(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] => 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[(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] => 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[(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] => 3
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[(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] => 3
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(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] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[(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] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(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)]
=> ?
=> ? => ? = 3
[(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)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(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)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000147
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> []
=> 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(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)]
=> ?
=> ?
=> ? = 3
[(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)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(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)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> ?
=> ?
=> ? = 3
Description
The largest part of an integer partition.
Matching statistic: St000316
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[(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] => 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(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] => 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[(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] => 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[(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] => 3
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[(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] => 3
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(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] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[(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] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(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)]
=> ?
=> ? => ? = 3
[(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)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(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)]
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> ?
=> ? => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St001227
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00113: Perfect matchings —reverse⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [(1,2)]
=> [1,0]
=> 0
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [1,0,1,0]
=> 1
[(1,3),(2,4)]
=> [(1,3),(2,4)]
=> [1,1,0,0]
=> 0
[(1,4),(2,3)]
=> [(1,4),(2,3)]
=> [1,1,0,0]
=> 0
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 2
[(1,3),(2,4),(5,6)]
=> [(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> 2
[(1,4),(2,3),(5,6)]
=> [(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> 2
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> 1
[(1,6),(2,3),(4,5)]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 1
[(1,6),(2,4),(3,5)]
=> [(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> 0
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> 0
[(1,4),(2,5),(3,6)]
=> [(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> 0
[(1,3),(2,5),(4,6)]
=> [(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> 1
[(1,2),(3,5),(4,6)]
=> [(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> 1
[(1,2),(3,6),(4,5)]
=> [(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 1
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> 1
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> 0
[(1,5),(2,6),(3,4)]
=> [(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> 0
[(1,6),(2,5),(3,4)]
=> [(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 3
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> 3
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> 3
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> 3
[(1,2),(3,5),(4,6),(7,8)]
=> [(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)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> 3
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,8),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> [(1,3),(2,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10),(11,12)]
=> ?
=> ? = 3
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> [(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)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> [(1,3),(2,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> [(1,4),(2,5),(3,7),(6,8),(9,10),(11,12)]
=> ?
=> ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> [(1,3),(2,6),(4,7),(5,8),(9,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> [(1,4),(2,6),(3,7),(5,8),(9,10),(11,12)]
=> ?
=> ? = 3
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10),(11,12)]
=> ?
=> ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> [(1,3),(2,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10),(11,12)]
=> ?
=> ? = 3
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ? = 5
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> [(1,3),(2,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ? = 4
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> [(1,4),(2,5),(3,7),(6,9),(8,10),(11,12)]
=> ?
=> ? = 3
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000054
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 1 = 0 + 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,2] => 1 = 0 + 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,2] => 1 = 0 + 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 2 + 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4 = 3 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 4 = 3 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 4 = 3 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 3 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 3 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 3 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4 = 3 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4 = 3 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 3 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4 + 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)]
=> ?
=> ? => ? = 3 + 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)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4 + 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)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> ?
=> ? => ? = 3 + 1
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> ?
=> ? => ? = 2 + 1
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3 + 1
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3 + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St001497
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1] => 1 = 0 + 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,2] => 2 = 1 + 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3 = 2 + 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2,1,3] => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4 = 3 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4 = 3 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4 = 3 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 3 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 3 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4 = 3 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4 = 3 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 3 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4 = 3 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4 = 3 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4 = 3 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4 = 3 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4 = 3 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4 + 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)]
=> ?
=> ? => ? = 3 + 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)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4 + 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)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> ?
=> ? => ? = 3 + 1
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> ?
=> ? => ? = 2 + 1
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3 + 1
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ? => ? = 5 + 1
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> ?
=> ? => ? = 4 + 1
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> ?
=> ? => ? = 3 + 1
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St000010
Mp00113: Perfect matchings —reverse⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [(1,2)]
=> [1,0]
=> []
=> 0
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> [1,0,1,0]
=> [1]
=> 1
[(1,3),(2,4)]
=> [(1,3),(2,4)]
=> [1,1,0,0]
=> []
=> 0
[(1,4),(2,3)]
=> [(1,4),(2,3)]
=> [1,1,0,0]
=> []
=> 0
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[(1,3),(2,4),(5,6)]
=> [(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[(1,4),(2,3),(5,6)]
=> [(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,6),(2,3),(4,5)]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,6),(2,4),(3,5)]
=> [(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,5),(2,4),(3,6)]
=> [(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,4),(2,5),(3,6)]
=> [(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,3),(2,5),(4,6)]
=> [(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,2),(3,5),(4,6)]
=> [(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[(1,2),(3,6),(4,5)]
=> [(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,5),(2,6),(3,4)]
=> [(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,6),(2,5),(3,4)]
=> [(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> []
=> 0
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,7),(6,8)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,8),(2,3),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,3),(2,8),(4,6),(5,7)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,2),(3,8),(4,6),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,2),(3,6),(4,8),(5,7)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,2),(3,6),(4,7),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,2),(3,7),(4,6),(5,8)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,2),(3,7),(4,8),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,3),(2,8),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,8),(2,4),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,8),(3,7),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,4),(2,7),(3,8),(5,6)]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,3),(2,7),(4,8),(5,6)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,3),(2,7),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,7),(2,3),(4,6),(5,8)]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> [(1,3),(2,4),(5,6),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,6),(7,9),(8,11),(10,12)]
=> [(1,3),(2,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,6),(7,10),(8,11),(9,12)]
=> [(1,4),(2,5),(3,6),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,7),(6,8),(9,10),(11,12)]
=> [(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)]
=> [(1,3),(2,4),(5,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> [(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> [(1,3),(2,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,7),(6,10),(8,11),(9,12)]
=> [(1,4),(2,5),(3,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,8),(6,9),(7,10),(11,12)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> [(1,3),(2,6),(4,7),(5,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> [(1,4),(2,6),(3,7),(5,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> [(1,5),(2,6),(3,7),(4,8),(9,10),(11,12)]
=> ?
=> ?
=> ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> [(1,3),(2,4),(5,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> [(1,3),(2,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> [(1,4),(2,5),(3,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 3
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> [(1,3),(2,4),(5,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> [(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 5
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> [(1,3),(2,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 4
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> [(1,4),(2,5),(3,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? = 3
Description
The length of the partition.
Matching statistic: St000019
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [1,0]
=> [1] => 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[(1,4),(2,3)]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 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]
=> [3,4,1,2] => 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]
=> [3,1,4,2] => 3
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 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]
=> [2,4,1,3] => 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]
=> [3,1,4,2] => 3
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 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]
=> [2,1,3,4] => 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]
=> [2,1,3,4] => 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]
=> [2,3,1,4] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12)]
=> ?
=> ?
=> ? => ? = 4
[(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)]
=> ?
=> ?
=> ? => ? = 3
[(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)]
=> ?
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,7),(6,9),(8,11),(10,12)]
=> ?
=> ?
=> ? => ? = 4
[(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)]
=> ?
=> ?
=> ? => ? = 5
[(1,2),(3,4),(5,8),(6,9),(7,11),(10,12)]
=> ?
=> ?
=> ? => ? = 4
[(1,2),(3,4),(5,8),(6,10),(7,11),(9,12)]
=> ?
=> ?
=> ? => ? = 3
[(1,2),(3,4),(5,9),(6,10),(7,11),(8,12)]
=> ?
=> ?
=> ? => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10),(11,12)]
=> ?
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,6),(7,8),(9,11),(10,12)]
=> ?
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,6),(7,9),(8,10),(11,12)]
=> ?
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,6),(7,9),(8,11),(10,12)]
=> ?
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,6),(7,10),(8,11),(9,12)]
=> ?
=> ?
=> ? => ? = 3
[(1,2),(3,5),(4,7),(6,8),(9,10),(11,12)]
=> ?
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,7),(6,8),(9,11),(10,12)]
=> ?
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,7),(6,9),(8,10),(11,12)]
=> ?
=> ?
=> ? => ? = 5
[(1,2),(3,5),(4,7),(6,9),(8,11),(10,12)]
=> ?
=> ?
=> ? => ? = 4
[(1,2),(3,5),(4,7),(6,10),(8,11),(9,12)]
=> ?
=> ?
=> ? => ? = 3
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000024The number of double up and double down steps of a Dyck path. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000204The number of internal nodes of a binary tree. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000240The number of indices that are not small excedances. St000443The number of long tunnels of a Dyck path. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000730The maximal arc length of a set partition. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000444The length of the maximal rise of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000676The number of odd rises of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000005The bounce statistic of a Dyck path. St001571The Cartan determinant of the integer partition. St000052The number of valleys of a Dyck path not on the x-axis.
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