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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000087
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000087: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000087: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,1] => [2] => [1] => ([],1)
=> 1
[1,1,1] => [3] => [1] => ([],1)
=> 1
[1,1,2] => [3] => [1] => ([],1)
=> 1
[1,2,1] => [3] => [1] => ([],1)
=> 1
[2,1,1] => [3] => [1] => ([],1)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[3,1,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[2,1,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[2,2,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[1,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,2] => [4] => [1] => ([],1)
=> 1
[1,1,2,1] => [4] => [1] => ([],1)
=> 1
[1,2,1,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,3] => [4] => [1] => ([],1)
=> 1
[1,1,3,1] => [4] => [1] => ([],1)
=> 1
[1,3,1,1] => [4] => [1] => ([],1)
=> 1
[3,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[4,1,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,2,2] => [4] => [1] => ([],1)
=> 1
[1,2,1,2] => [4] => [1] => ([],1)
=> 1
[1,2,2,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,2] => [4] => [1] => ([],1)
=> 1
[2,1,2,1] => [4] => [1] => ([],1)
=> 1
[2,2,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,2,3] => [4] => [1] => ([],1)
=> 1
[1,1,3,2] => [4] => [1] => ([],1)
=> 1
[1,2,1,3] => [4] => [1] => ([],1)
=> 1
[1,2,3,1] => [4] => [1] => ([],1)
=> 1
[1,3,1,2] => [4] => [1] => ([],1)
=> 1
[1,3,2,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,3] => [4] => [1] => ([],1)
=> 1
[2,1,3,1] => [4] => [1] => ([],1)
=> 1
[2,3,1,1] => [4] => [1] => ([],1)
=> 1
[3,1,1,2] => [4] => [1] => ([],1)
=> 1
[3,1,2,1] => [4] => [1] => ([],1)
=> 1
[3,2,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,2,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,4,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,2,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[2,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
Description
The number of induced subgraphs.
A subgraph $H \subseteq G$ is induced if $E(H)$ consists of all edges in $E(G)$ that connect the vertices of $H$.
Matching statistic: St000867
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000867: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000867: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,1] => [2] => [1] => [1]
=> 1
[1,1,1] => [3] => [1] => [1]
=> 1
[1,1,2] => [3] => [1] => [1]
=> 1
[1,2,1] => [3] => [1] => [1]
=> 1
[2,1,1] => [3] => [1] => [1]
=> 1
[1,1,3] => [2,1] => [1,1] => [1,1]
=> 2
[1,3,1] => [2,1] => [1,1] => [1,1]
=> 2
[3,1,1] => [2,1] => [1,1] => [1,1]
=> 2
[1,2,2] => [1,2] => [1,1] => [1,1]
=> 2
[2,1,2] => [1,2] => [1,1] => [1,1]
=> 2
[2,2,1] => [1,2] => [1,1] => [1,1]
=> 2
[1,1,1,1] => [4] => [1] => [1]
=> 1
[1,1,1,2] => [4] => [1] => [1]
=> 1
[1,1,2,1] => [4] => [1] => [1]
=> 1
[1,2,1,1] => [4] => [1] => [1]
=> 1
[2,1,1,1] => [4] => [1] => [1]
=> 1
[1,1,1,3] => [4] => [1] => [1]
=> 1
[1,1,3,1] => [4] => [1] => [1]
=> 1
[1,3,1,1] => [4] => [1] => [1]
=> 1
[3,1,1,1] => [4] => [1] => [1]
=> 1
[1,1,1,4] => [3,1] => [1,1] => [1,1]
=> 2
[1,1,4,1] => [3,1] => [1,1] => [1,1]
=> 2
[1,4,1,1] => [3,1] => [1,1] => [1,1]
=> 2
[4,1,1,1] => [3,1] => [1,1] => [1,1]
=> 2
[1,1,2,2] => [4] => [1] => [1]
=> 1
[1,2,1,2] => [4] => [1] => [1]
=> 1
[1,2,2,1] => [4] => [1] => [1]
=> 1
[2,1,1,2] => [4] => [1] => [1]
=> 1
[2,1,2,1] => [4] => [1] => [1]
=> 1
[2,2,1,1] => [4] => [1] => [1]
=> 1
[1,1,2,3] => [4] => [1] => [1]
=> 1
[1,1,3,2] => [4] => [1] => [1]
=> 1
[1,2,1,3] => [4] => [1] => [1]
=> 1
[1,2,3,1] => [4] => [1] => [1]
=> 1
[1,3,1,2] => [4] => [1] => [1]
=> 1
[1,3,2,1] => [4] => [1] => [1]
=> 1
[2,1,1,3] => [4] => [1] => [1]
=> 1
[2,1,3,1] => [4] => [1] => [1]
=> 1
[2,3,1,1] => [4] => [1] => [1]
=> 1
[3,1,1,2] => [4] => [1] => [1]
=> 1
[3,1,2,1] => [4] => [1] => [1]
=> 1
[3,2,1,1] => [4] => [1] => [1]
=> 1
[1,1,2,4] => [3,1] => [1,1] => [1,1]
=> 2
[1,1,4,2] => [3,1] => [1,1] => [1,1]
=> 2
[1,2,1,4] => [3,1] => [1,1] => [1,1]
=> 2
[1,2,4,1] => [3,1] => [1,1] => [1,1]
=> 2
[1,4,1,2] => [3,1] => [1,1] => [1,1]
=> 2
[1,4,2,1] => [3,1] => [1,1] => [1,1]
=> 2
[2,1,1,4] => [3,1] => [1,1] => [1,1]
=> 2
Description
The sum of the hook lengths in the first row of an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition.
Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.
Matching statistic: St000926
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000926: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000926: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,1] => [2] => [1] => ([],1)
=> 1
[1,1,1] => [3] => [1] => ([],1)
=> 1
[1,1,2] => [3] => [1] => ([],1)
=> 1
[1,2,1] => [3] => [1] => ([],1)
=> 1
[2,1,1] => [3] => [1] => ([],1)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[3,1,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[2,1,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[2,2,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[1,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,2] => [4] => [1] => ([],1)
=> 1
[1,1,2,1] => [4] => [1] => ([],1)
=> 1
[1,2,1,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,3] => [4] => [1] => ([],1)
=> 1
[1,1,3,1] => [4] => [1] => ([],1)
=> 1
[1,3,1,1] => [4] => [1] => ([],1)
=> 1
[3,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[4,1,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,2,2] => [4] => [1] => ([],1)
=> 1
[1,2,1,2] => [4] => [1] => ([],1)
=> 1
[1,2,2,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,2] => [4] => [1] => ([],1)
=> 1
[2,1,2,1] => [4] => [1] => ([],1)
=> 1
[2,2,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,2,3] => [4] => [1] => ([],1)
=> 1
[1,1,3,2] => [4] => [1] => ([],1)
=> 1
[1,2,1,3] => [4] => [1] => ([],1)
=> 1
[1,2,3,1] => [4] => [1] => ([],1)
=> 1
[1,3,1,2] => [4] => [1] => ([],1)
=> 1
[1,3,2,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,3] => [4] => [1] => ([],1)
=> 1
[2,1,3,1] => [4] => [1] => ([],1)
=> 1
[2,3,1,1] => [4] => [1] => ([],1)
=> 1
[3,1,1,2] => [4] => [1] => ([],1)
=> 1
[3,1,2,1] => [4] => [1] => ([],1)
=> 1
[3,2,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,2,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,4,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,2,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[2,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
Description
The clique-coclique number of a graph.
This is the product of the size of a maximal clique [[St000097]] and the size of a maximal independent set [[St000093]].
Matching statistic: St001228
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1] => [1,0]
=> 1
[1,1,1] => [3] => [1] => [1,0]
=> 1
[1,1,2] => [3] => [1] => [1,0]
=> 1
[1,2,1] => [3] => [1] => [1,0]
=> 1
[2,1,1] => [3] => [1] => [1,0]
=> 1
[1,1,3] => [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,3,1] => [2,1] => [1,1] => [1,0,1,0]
=> 2
[3,1,1] => [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,2,2] => [1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1,2] => [1,2] => [1,1] => [1,0,1,0]
=> 2
[2,2,1] => [1,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,1,2] => [4] => [1] => [1,0]
=> 1
[1,1,2,1] => [4] => [1] => [1,0]
=> 1
[1,2,1,1] => [4] => [1] => [1,0]
=> 1
[2,1,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,1,3] => [4] => [1] => [1,0]
=> 1
[1,1,3,1] => [4] => [1] => [1,0]
=> 1
[1,3,1,1] => [4] => [1] => [1,0]
=> 1
[3,1,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,4,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,4,1,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[4,1,1,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,2,2] => [4] => [1] => [1,0]
=> 1
[1,2,1,2] => [4] => [1] => [1,0]
=> 1
[1,2,2,1] => [4] => [1] => [1,0]
=> 1
[2,1,1,2] => [4] => [1] => [1,0]
=> 1
[2,1,2,1] => [4] => [1] => [1,0]
=> 1
[2,2,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,2,3] => [4] => [1] => [1,0]
=> 1
[1,1,3,2] => [4] => [1] => [1,0]
=> 1
[1,2,1,3] => [4] => [1] => [1,0]
=> 1
[1,2,3,1] => [4] => [1] => [1,0]
=> 1
[1,3,1,2] => [4] => [1] => [1,0]
=> 1
[1,3,2,1] => [4] => [1] => [1,0]
=> 1
[2,1,1,3] => [4] => [1] => [1,0]
=> 1
[2,1,3,1] => [4] => [1] => [1,0]
=> 1
[2,3,1,1] => [4] => [1] => [1,0]
=> 1
[3,1,1,2] => [4] => [1] => [1,0]
=> 1
[3,1,2,1] => [4] => [1] => [1,0]
=> 1
[3,2,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,2,4] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,4,2] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,2,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,2,4,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,4,1,2] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,4,2,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[2,1,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 2
Description
The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001254
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1] => [1,0]
=> 1
[1,1,1] => [3] => [1] => [1,0]
=> 1
[1,1,2] => [3] => [1] => [1,0]
=> 1
[1,2,1] => [3] => [1] => [1,0]
=> 1
[2,1,1] => [3] => [1] => [1,0]
=> 1
[1,1,3] => [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,3,1] => [2,1] => [1,1] => [1,0,1,0]
=> 2
[3,1,1] => [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,2,2] => [1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1,2] => [1,2] => [1,1] => [1,0,1,0]
=> 2
[2,2,1] => [1,2] => [1,1] => [1,0,1,0]
=> 2
[1,1,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,1,2] => [4] => [1] => [1,0]
=> 1
[1,1,2,1] => [4] => [1] => [1,0]
=> 1
[1,2,1,1] => [4] => [1] => [1,0]
=> 1
[2,1,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,1,3] => [4] => [1] => [1,0]
=> 1
[1,1,3,1] => [4] => [1] => [1,0]
=> 1
[1,3,1,1] => [4] => [1] => [1,0]
=> 1
[3,1,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,4,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,4,1,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[4,1,1,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,2,2] => [4] => [1] => [1,0]
=> 1
[1,2,1,2] => [4] => [1] => [1,0]
=> 1
[1,2,2,1] => [4] => [1] => [1,0]
=> 1
[2,1,1,2] => [4] => [1] => [1,0]
=> 1
[2,1,2,1] => [4] => [1] => [1,0]
=> 1
[2,2,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,2,3] => [4] => [1] => [1,0]
=> 1
[1,1,3,2] => [4] => [1] => [1,0]
=> 1
[1,2,1,3] => [4] => [1] => [1,0]
=> 1
[1,2,3,1] => [4] => [1] => [1,0]
=> 1
[1,3,1,2] => [4] => [1] => [1,0]
=> 1
[1,3,2,1] => [4] => [1] => [1,0]
=> 1
[2,1,1,3] => [4] => [1] => [1,0]
=> 1
[2,1,3,1] => [4] => [1] => [1,0]
=> 1
[2,3,1,1] => [4] => [1] => [1,0]
=> 1
[3,1,1,2] => [4] => [1] => [1,0]
=> 1
[3,1,2,1] => [4] => [1] => [1,0]
=> 1
[3,2,1,1] => [4] => [1] => [1,0]
=> 1
[1,1,2,4] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,4,2] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,2,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,2,4,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,4,1,2] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[1,4,2,1] => [3,1] => [1,1] => [1,0,1,0]
=> 2
[2,1,1,4] => [3,1] => [1,1] => [1,0,1,0]
=> 2
Description
The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,1] => [2] => [1] => ([],1)
=> 1
[1,1,1] => [3] => [1] => ([],1)
=> 1
[1,1,2] => [3] => [1] => ([],1)
=> 1
[1,2,1] => [3] => [1] => ([],1)
=> 1
[2,1,1] => [3] => [1] => ([],1)
=> 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[3,1,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[2,1,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[2,2,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 2
[1,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,2] => [4] => [1] => ([],1)
=> 1
[1,1,2,1] => [4] => [1] => ([],1)
=> 1
[1,2,1,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,3] => [4] => [1] => ([],1)
=> 1
[1,1,3,1] => [4] => [1] => ([],1)
=> 1
[1,3,1,1] => [4] => [1] => ([],1)
=> 1
[3,1,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[4,1,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,2,2] => [4] => [1] => ([],1)
=> 1
[1,2,1,2] => [4] => [1] => ([],1)
=> 1
[1,2,2,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,2] => [4] => [1] => ([],1)
=> 1
[2,1,2,1] => [4] => [1] => ([],1)
=> 1
[2,2,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,2,3] => [4] => [1] => ([],1)
=> 1
[1,1,3,2] => [4] => [1] => ([],1)
=> 1
[1,2,1,3] => [4] => [1] => ([],1)
=> 1
[1,2,3,1] => [4] => [1] => ([],1)
=> 1
[1,3,1,2] => [4] => [1] => ([],1)
=> 1
[1,3,2,1] => [4] => [1] => ([],1)
=> 1
[2,1,1,3] => [4] => [1] => ([],1)
=> 1
[2,1,3,1] => [4] => [1] => ([],1)
=> 1
[2,3,1,1] => [4] => [1] => ([],1)
=> 1
[3,1,1,2] => [4] => [1] => ([],1)
=> 1
[3,1,2,1] => [4] => [1] => ([],1)
=> 1
[3,2,1,1] => [4] => [1] => ([],1)
=> 1
[1,1,2,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,1,4,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,2,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[1,4,2,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
[2,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 2
Description
The pebbling number of a connected graph.
Matching statistic: St000108
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 2 = 1 + 1
[1,1] => [2] => [1] => [1]
=> 2 = 1 + 1
[1,1,1] => [3] => [1] => [1]
=> 2 = 1 + 1
[1,1,2] => [3] => [1] => [1]
=> 2 = 1 + 1
[1,2,1] => [3] => [1] => [1]
=> 2 = 1 + 1
[2,1,1] => [3] => [1] => [1]
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,3,1] => [2,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[3,1,1] => [2,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,2,2] => [1,2] => [1,1] => [1,1]
=> 3 = 2 + 1
[2,1,2] => [1,2] => [1,1] => [1,1]
=> 3 = 2 + 1
[2,2,1] => [1,2] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,1,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,1,3] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,3,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,3,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[3,1,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,1,4,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,4,1,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[4,1,1,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,1,2,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,2,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,2,3] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,3,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,1,3] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,3,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,3,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,3,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,1,3] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,3,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,3,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[3,1,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[3,1,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[3,2,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,2,4] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,1,4,2] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,2,1,4] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,2,4,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,4,1,2] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,4,2,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[2,1,1,4] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
Description
The number of partitions contained in the given partition.
Matching statistic: St000300
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000300: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000300: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 2 = 1 + 1
[1,1] => [2] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,1] => [3] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,2] => [3] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,1] => [3] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,1] => [3] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[3,1,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[2,1,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[2,2,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,1,3] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,3,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,3,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[3,1,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,4,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[4,1,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,2,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,2,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,2,3] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,3,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,1,3] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,3,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,3,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,3,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,1,3] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,3,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,3,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[3,1,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[3,1,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[3,2,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,2,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,4,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,2,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,2,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,4,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,4,2,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[2,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
Description
The number of independent sets of vertices of a graph.
An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent.
This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers.
The size of the largest independent set, also called independence number of $G$, is [[St000093]]
Matching statistic: St000301
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000301: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000301: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 2 = 1 + 1
[1,1] => [2] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,1] => [3] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,2] => [3] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,1] => [3] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,1] => [3] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,3,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[3,1,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[2,1,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[2,2,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,1,3] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,3,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,3,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[3,1,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,4,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[4,1,1,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,2,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,2,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,2,3] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,3,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,1,3] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,2,3,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,3,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,3,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,1,3] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,1,3,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[2,3,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[3,1,1,2] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[3,1,2,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[3,2,1,1] => [4] => [1] => ([],1)
=> 2 = 1 + 1
[1,1,2,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,1,4,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,2,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,2,4,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,4,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[1,4,2,1] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
[2,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 3 = 2 + 1
Description
The number of facets of the stable set polytope of a graph.
The stable set polytope of a graph $G$ is the convex hull of the characteristic vectors of stable (or independent) sets of vertices of $G$ inside $\mathbb{R}^{V(G)}$.
Matching statistic: St000532
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 2 = 1 + 1
[1,1] => [2] => [1] => [1]
=> 2 = 1 + 1
[1,1,1] => [3] => [1] => [1]
=> 2 = 1 + 1
[1,1,2] => [3] => [1] => [1]
=> 2 = 1 + 1
[1,2,1] => [3] => [1] => [1]
=> 2 = 1 + 1
[2,1,1] => [3] => [1] => [1]
=> 2 = 1 + 1
[1,1,3] => [2,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,3,1] => [2,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[3,1,1] => [2,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,2,2] => [1,2] => [1,1] => [1,1]
=> 3 = 2 + 1
[2,1,2] => [1,2] => [1,1] => [1,1]
=> 3 = 2 + 1
[2,2,1] => [1,2] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,1,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,1,3] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,3,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,3,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[3,1,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,1,4] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,1,4,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,4,1,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[4,1,1,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,1,2,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,2,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,2,3] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,3,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,1,3] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,2,3,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,3,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,3,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,1,3] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,1,3,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[2,3,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[3,1,1,2] => [4] => [1] => [1]
=> 2 = 1 + 1
[3,1,2,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[3,2,1,1] => [4] => [1] => [1]
=> 2 = 1 + 1
[1,1,2,4] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,1,4,2] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,2,1,4] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,2,4,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,4,1,2] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[1,4,2,1] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
[2,1,1,4] => [3,1] => [1,1] => [1,1]
=> 3 = 2 + 1
Description
The total number of rook placements on a Ferrers board.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001658The total number of rook placements on a Ferrers board. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000699The toughness times the least common multiple of 1,. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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