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Your data matches 167 different statistics following compositions of up to 3 maps.
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Matching statistic: St000840
(load all 7 compositions to match this statistic)
(load all 7 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: St001136
St001136: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 1 = 0 + 1
[(1,2),(3,4)]
=> 1 = 0 + 1
[(1,3),(2,4)]
=> 1 = 0 + 1
[(1,4),(2,3)]
=> 2 = 1 + 1
[(1,2),(3,4),(5,6)]
=> 3 = 2 + 1
[(1,3),(2,4),(5,6)]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
[(1,5),(2,3),(4,6)]
=> 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> 2 = 1 + 1
[(1,5),(2,4),(3,6)]
=> 2 = 1 + 1
[(1,4),(2,5),(3,6)]
=> 1 = 0 + 1
[(1,3),(2,5),(4,6)]
=> 1 = 0 + 1
[(1,2),(3,5),(4,6)]
=> 1 = 0 + 1
[(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[(1,3),(2,6),(4,5)]
=> 1 = 0 + 1
[(1,4),(2,6),(3,5)]
=> 1 = 0 + 1
[(1,5),(2,6),(3,4)]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4)]
=> 3 = 2 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> 3 = 2 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> 2 = 1 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> 2 = 1 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> 2 = 1 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> 4 = 3 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> 4 = 3 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> 4 = 3 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> 3 = 2 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> 2 = 1 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> 2 = 1 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> 4 = 3 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> 4 = 3 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> 3 = 2 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> 3 = 2 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> 3 = 2 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> 3 = 2 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> 3 = 2 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> 3 = 2 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> 3 = 2 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> 4 = 3 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> 4 = 3 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> 4 = 3 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> 4 = 3 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> 3 = 2 + 1
Description
The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001227
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> 0
[(1,2),(3,4)]
=> [1,0,1,0]
=> 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
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 1
[(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]
=> 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
[(1,2),(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]
=> 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
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> 2
[(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]
=> 2
[(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)]
=> [1,1,1,0,0,1,0,0]
=> 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]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(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]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> 1
[(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]
=> 2
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> 3
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> 3
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> 1
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: St001291
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: St000010
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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]
=> 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[(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]
=> 2
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[(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]
=> 2
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[(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]
=> 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[(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]
=> 2
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[(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]
=> 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[(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]
=> 3
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[(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]
=> 1
Description
The length of the partition.
Matching statistic: St000141
(load all 9 compositions to match this statistic)
(load all 9 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
Description
The largest part of an integer partition.
Matching statistic: St000316
(load all 9 compositions to match this statistic)
(load all 9 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: St000054
(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
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: St000199
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [1,0]
=> [[1]]
=> 1 = 0 + 1
[(1,2),(3,4)]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2 = 1 + 1
[(1,3),(2,4)]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[(1,4),(2,3)]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 1 = 0 + 1
[(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 2 + 1
[(1,3),(2,4),(5,6)]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 3 = 2 + 1
[(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 3 = 2 + 1
[(1,5),(2,3),(4,6)]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[(1,6),(2,4),(3,5)]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 1 = 0 + 1
[(1,5),(2,4),(3,6)]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 1 = 0 + 1
[(1,4),(2,5),(3,6)]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 1 = 0 + 1
[(1,3),(2,5),(4,6)]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[(1,2),(3,5),(4,6)]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[(1,2),(3,6),(4,5)]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 2 = 1 + 1
[(1,3),(2,6),(4,5)]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 2 = 1 + 1
[(1,4),(2,6),(3,5)]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 1 = 0 + 1
[(1,5),(2,6),(3,4)]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 1 = 0 + 1
[(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 1 = 0 + 1
[(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,3),(2,4),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,4),(2,3),(5,6),(7,8)]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,5),(2,3),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,6),(2,3),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,7),(2,3),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,7),(2,4),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,6),(2,4),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,5),(2,4),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,4),(2,5),(3,6),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,2),(3,5),(4,6),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,2),(3,6),(4,5),(7,8)]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,3),(2,6),(4,5),(7,8)]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,4),(2,6),(3,5),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,5),(2,6),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,6),(2,5),(3,4),(7,8)]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,8),(2,5),(3,4),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,8),(2,6),(3,4),(5,7)]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2 = 1 + 1
[(1,7),(2,6),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2 = 1 + 1
[(1,6),(2,7),(3,4),(5,8)]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 2 = 1 + 1
[(1,5),(2,7),(3,4),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,4),(2,7),(3,5),(6,8)]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,3),(2,7),(4,5),(6,8)]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3 = 2 + 1
Description
The column of the unique '1' in the last row of the alternating sign matrix.
The following 157 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000200The row of the unique '1' in the last column of the alternating sign matrix. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St000019The cardinality of the support of a permutation. 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. St000133The "bounce" of a permutation. St000204The number of internal nodes of a binary tree. St000304The load of a permutation. St000356The number of occurrences of the pattern 13-2. St000378The diagonal inversion number of an integer partition. 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. St000653The last descent of a permutation. St000676The number of odd rises of a Dyck path. 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. St001480The number of simple summands of the module J^2/J^3. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. 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. St000028The number of stack-sorts needed to sort a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000240The number of indices that are not small excedances. St000443The number of long tunnels of a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. 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. 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. St000956The maximal displacement of a permutation. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000288The number of ones in a binary word. St000733The row containing the largest entry of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St001498The normalised height of a Nakayama algebra with magnitude 1. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000284The Plancherel distribution on integer partitions. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001394The genus of a permutation. St000233The number of nestings of a set partition. St000259The diameter of a connected graph. St000496The rcs statistic of a set partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000005The bounce statistic of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000891The number of distinct diagonal sums of a permutation matrix. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001571The Cartan determinant of the integer partition. St000422The energy of a graph, if it is integral. St000871The number of very big ascents of a permutation. St000971The smallest closer of a set partition. St000352The Elizalde-Pak rank of a permutation. St000647The number of big descents of a permutation. St000834The number of right outer peaks of a permutation. St001115The number of even descents of a permutation. St000662The staircase size of the code of a permutation. St000035The number of left outer peaks of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000703The number of deficiencies of a permutation. St000884The number of isolated descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000201The number of leaf nodes in a binary tree. St000670The reversal length of a permutation. St000497The lcb statistic of a set partition. St000919The number of maximal left branches of a binary tree. St000237The number of small exceedances. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001096The size of the overlap set of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001644The dimension of a graph. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001083The number of boxed occurrences of 132 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000007The number of saliances of the permutation. St001052The length of the exterior of a permutation. St000022The number of fixed points of a permutation. St001733The number of weak left to right maxima of a Dyck path. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000883The number of longest increasing subsequences of a permutation. St000842The breadth of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001330The hat guessing number of a graph. St000665The number of rafts of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000366The number of double descents of a permutation. St000568The hook number of a binary tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000731The number of double exceedences of a permutation. St000441The number of successions of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001403The number of vertical separators in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000214The number of adjacencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation.
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