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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000741
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2] => ([],2)
=> 1
[2,1] => [1,2] => [2] => ([],2)
=> 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [1,2,3] => [3] => ([],3)
=> 1
[2,1,3] => [1,2,3] => [3] => ([],3)
=> 1
[2,3,1] => [1,2,3] => [3] => ([],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
=> 1
[1,4,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
=> 1
[2,4,1,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,5,2,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,5,3,2,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,5,3,4,2] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,5,4,2,3] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,5,4,3,2] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[2,1,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[2,1,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[2,1,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[2,1,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[2,3,1,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[2,3,1,5,4] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[2,3,4,1,5] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
=> 1
[2,4,1,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001499
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 100%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,1,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,4,1,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[4,2,1,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,5,3,4,2] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,4,2,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,4,3,2] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,3,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,4,3,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,4,5,3] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,3,1,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,3,1,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,3,4,1,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,3,4,5,1] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,4,1,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,4,1,5,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St001199
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 90%●distinct values known / distinct values provided: 80%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 90%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> ? = 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> ? = 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> ? = 1
[1,3,2] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> ? = 1
[2,1,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> ? = 1
[2,3,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
[1,2,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
[1,3,2,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
[1,3,4,2] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
[1,4,2,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
[2,1,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
[2,3,1,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
[2,3,4,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1
[2,4,1,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[4,2,1,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,5,2,4,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,5,3,4,2] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,4,2,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,5,4,3,2] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,1,3,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,1,4,3,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,1,4,5,3] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,3,1,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,3,4,1,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,3,4,5,1] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,4,1,5,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,4,3,1,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,4,3,5,1] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,4,5,1,3] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,4,5,3,1] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[2,5,1,4,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[2,5,3,4,1] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,5,4,1,3] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[2,5,4,3,1] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[3,5,1,2,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,5,1,4,2] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,5,2,1,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,5,2,4,1] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[4,1,2,3,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[4,1,5,3,2] => [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[4,2,1,3,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[4,2,5,3,1] => [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[4,5,1,2,3] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[4,5,1,3,2] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[4,5,2,1,3] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[4,5,2,3,1] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[4,5,3,1,2] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[4,5,3,2,1] => [1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[5,1,2,4,3] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000454
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [.,.]
=> ([],1)
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,5,2,3,4] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,1,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,1,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,1,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,3,1,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,3,4,1,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1
[3,5,1,2,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,1,4,2,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,1,6,2,4] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,1,6,4,2] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,2,1,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,2,4,1,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,2,6,1,4] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,5,2,6,4,1] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,6,1,2,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,6,1,4,5,2] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,6,1,5,2,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,6,1,5,4,2] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,6,2,1,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,6,2,4,5,1] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,6,2,5,1,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,6,2,5,4,1] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,1,5,3,2,6] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,2,5,3,1,6] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,1,2,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,2,1,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,3,1,2,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,3,2,1,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,6,1,2,3] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,6,1,3,2] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,6,2,1,3] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,6,2,3,1] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,5,3,1,2] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[4,6,5,3,2,1] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,3,6,4,2] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,4,2,3,6] => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,1,6,4,3,2] => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,2,3,6,4,1] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,2,4,1,3,6] => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,2,6,4,3,1] => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[5,3,1,6,4,2] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001557
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 60%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => [1] => ? = 0 - 1
[1,2] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [3,4,2,1] => [1,3,2,4] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 1 = 2 - 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,5,3,2,4] => [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 1 = 2 - 1
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,1,5,4] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,4,1,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [1,2,3,4,5] => 0 = 1 - 1
[2,4,1,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [2,3,5,1,4,6] => [1,2,3,5,4,6] => ? = 1 - 1
[1,4,2,5,6,3] => [1,2,4,5,6,3] => [2,3,6,1,4,5] => [1,2,3,6,5,4] => ? = 1 - 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [3,4,6,2,5,1] => [1,3,6,2,4,5] => ? = 2 - 1
[1,4,2,6,5,3] => [1,2,4,6,3,5] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,3,2,5,6] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [2,3,5,1,4,6] => [1,2,3,5,4,6] => ? = 1 - 1
[1,4,3,5,6,2] => [1,2,4,5,6,3] => [2,3,6,1,4,5] => [1,2,3,6,5,4] => ? = 1 - 1
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [3,4,6,2,5,1] => [1,3,6,2,4,5] => ? = 2 - 1
[1,4,3,6,5,2] => [1,2,4,6,3,5] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,5,2,6,3] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,5,3,6,2] => [1,2,4,3,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,5,6,2,3] => [1,2,4,6,3,5] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,5,6,3,2] => [1,2,4,6,3,5] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,6,2,3,5] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => [1,3,5,6,2,4] => ? = 2 - 1
[1,4,6,2,5,3] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => [1,3,5,6,2,4] => ? = 2 - 1
[1,4,6,3,2,5] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => [1,3,5,6,2,4] => ? = 2 - 1
[1,4,6,3,5,2] => [1,2,4,3,6,5] => [3,4,5,2,6,1] => [1,3,5,6,2,4] => ? = 2 - 1
[1,4,6,5,2,3] => [1,2,4,5,3,6] => [2,3,5,1,4,6] => [1,2,3,5,4,6] => ? = 1 - 1
[1,4,6,5,3,2] => [1,2,4,5,3,6] => [2,3,5,1,4,6] => [1,2,3,5,4,6] => ? = 1 - 1
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [3,4,5,2,1,6] => [1,3,5,2,4,6] => ? = 2 - 1
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [3,4,6,5,1,2] => [1,3,6,2,4,5] => ? = 2 - 1
[1,5,2,4,3,6] => [1,2,5,3,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,5,2,4,6,3] => [1,2,5,6,3,4] => [2,3,1,6,4,5] => [1,2,3,4,6,5] => ? = 1 - 1
[1,5,2,6,3,4] => [1,2,5,3,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ? = 1 - 1
[1,5,2,6,4,3] => [1,2,5,4,6,3] => [3,4,6,2,1,5] => [1,3,6,5,2,4] => ? = 2 - 1
[1,5,3,2,4,6] => [1,2,5,4,3,6] => [3,4,5,2,1,6] => [1,3,5,2,4,6] => ? = 2 - 1
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [3,4,6,5,1,2] => [1,3,6,2,4,5] => ? = 2 - 1
[1,5,3,4,2,6] => [1,2,5,3,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ? = 1 - 1
Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001960
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 60%
Mp00064: Permutations —reverse⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => [1] => ? = 0 - 1
[1,2] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [3,1,2,4] => 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 0 = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 0 = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0 = 1 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0 = 1 - 1
[1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => [4,3,1,2,5] => 1 = 2 - 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0 = 1 - 1
[1,5,3,2,4] => [1,2,5,4,3] => [3,4,5,2,1] => [4,3,1,2,5] => 1 = 2 - 1
[1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0 = 1 - 1
[1,5,4,2,3] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0 = 1 - 1
[1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => [3,4,1,2,5] => 0 = 1 - 1
[2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0 = 1 - 1
[2,4,1,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [3,5,1,2,4] => 0 = 1 - 1
[2,4,1,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 0 = 1 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [3,5,6,1,2,4] => ? = 1 - 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [3,5,6,1,2,4] => ? = 1 - 1
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => [3,6,1,2,4,5] => ? = 1 - 1
[1,4,2,5,6,3] => [1,2,4,5,6,3] => [3,6,5,4,2,1] => [3,1,2,4,5,6] => ? = 1 - 1
[1,4,2,6,3,5] => [1,2,4,6,5,3] => [3,5,6,4,2,1] => [5,3,1,2,4,6] => ? = 2 - 1
[1,4,2,6,5,3] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => [3,5,1,2,4,6] => ? = 1 - 1
[1,4,3,2,5,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [3,5,6,1,2,4] => ? = 1 - 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [3,5,6,1,2,4] => ? = 1 - 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => [3,6,1,2,4,5] => ? = 1 - 1
[1,4,3,5,6,2] => [1,2,4,5,6,3] => [3,6,5,4,2,1] => [3,1,2,4,5,6] => ? = 1 - 1
[1,4,3,6,2,5] => [1,2,4,6,5,3] => [3,5,6,4,2,1] => [5,3,1,2,4,6] => ? = 2 - 1
[1,4,3,6,5,2] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => [3,5,1,2,4,6] => ? = 1 - 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [3,5,6,1,2,4] => ? = 1 - 1
[1,4,5,2,6,3] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [3,5,6,1,2,4] => ? = 1 - 1
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [3,5,6,1,2,4] => ? = 1 - 1
[1,4,5,3,6,2] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => [3,5,6,1,2,4] => ? = 1 - 1
[1,4,5,6,2,3] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => [3,5,1,2,4,6] => ? = 1 - 1
[1,4,5,6,3,2] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => [3,5,1,2,4,6] => ? = 1 - 1
[1,4,6,2,3,5] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => [5,3,6,1,2,4] => ? = 2 - 1
[1,4,6,2,5,3] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => [5,3,6,1,2,4] => ? = 2 - 1
[1,4,6,3,2,5] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => [5,3,6,1,2,4] => ? = 2 - 1
[1,4,6,3,5,2] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => [5,3,6,1,2,4] => ? = 2 - 1
[1,4,6,5,2,3] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => [3,6,1,2,4,5] => ? = 1 - 1
[1,4,6,5,3,2] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => [3,6,1,2,4,5] => ? = 1 - 1
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => [3,6,4,1,2,5] => ? = 2 - 1
[1,5,2,3,6,4] => [1,2,5,6,4,3] => [3,4,6,5,2,1] => [4,3,1,2,5,6] => ? = 2 - 1
[1,5,2,4,3,6] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => [3,4,6,1,2,5] => ? = 1 - 1
[1,5,2,4,6,3] => [1,2,5,6,3,4] => [4,3,6,5,2,1] => [3,4,1,2,5,6] => ? = 1 - 1
[1,5,2,6,3,4] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => [3,4,6,1,2,5] => ? = 1 - 1
[1,5,2,6,4,3] => [1,2,5,4,6,3] => [3,6,4,5,2,1] => [4,6,3,1,2,5] => ? = 2 - 1
[1,5,3,2,4,6] => [1,2,5,4,3,6] => [6,3,4,5,2,1] => [3,6,4,1,2,5] => ? = 2 - 1
[1,5,3,2,6,4] => [1,2,5,6,4,3] => [3,4,6,5,2,1] => [4,3,1,2,5,6] => ? = 2 - 1
[1,5,3,4,2,6] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => [3,4,6,1,2,5] => ? = 1 - 1
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001625
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 40%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2 - 1
[4,2,1,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,5,2,4,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
[1,5,3,4,2] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0 = 1 - 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
[2,5,1,4,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2 - 1
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1 - 1
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 2 - 1
[3,2,5,1,4] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 2 - 1
[3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2 - 1
[3,5,1,4,2] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2 - 1
[3,5,2,1,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2 - 1
[3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2 - 1
[4,1,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[4,1,2,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 - 1
[4,1,5,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2 - 1
[4,2,1,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[4,2,1,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 - 1
[4,2,5,3,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2 - 1
[4,5,1,2,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2 - 1
[4,5,1,3,2] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[4,5,2,1,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2 - 1
[4,5,2,3,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2 - 1
[4,5,3,1,2] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2 - 1
[4,5,3,2,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2 - 1
[5,1,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 3 - 1
[5,1,2,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2 - 1
[5,1,3,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 - 1
[5,1,4,2,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 - 1
[5,2,1,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 3 - 1
[5,2,1,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2 - 1
[5,2,3,1,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 - 1
[5,2,4,1,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 - 1
[5,3,1,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 - 1
[5,3,2,1,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2 - 1
[5,4,1,2,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2 - 1
[5,4,1,3,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 2 - 1
[5,4,2,1,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2 - 1
[5,4,2,3,1] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 2 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
Description
The Möbius invariant of a lattice.
The '''Möbius invariant''' of a lattice $L$ is the value of the Möbius function applied to least and greatest element, that is $\mu(L)=\mu_L(\hat{0},\hat{1})$, where $\hat{0}$ is the least element of $L$ and $\hat{1}$ is the greatest element of $L$.
For the definition of the Möbius function, see [[St000914]].
Matching statistic: St001878
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> ? = 0
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[2,1] => [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,4,3,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2
[4,2,1,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[3,1,5,2,4] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 2
[3,2,5,1,4] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 2
[3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2
[3,5,1,4,2] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2
[3,5,2,1,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2
[3,5,2,4,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2
[4,1,2,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2
[4,1,5,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2
[4,2,1,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2
[4,2,5,3,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2
[4,5,1,2,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2
[4,5,2,1,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2
[4,5,2,3,1] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? = 2
[4,5,3,1,2] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2
[4,5,3,2,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 2
[5,1,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 3
[5,1,2,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2
[5,1,3,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2
[5,1,4,2,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2
[5,2,1,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
=> ? = 3
[5,2,1,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2
[5,2,3,1,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2
[5,2,4,1,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2
[5,3,1,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2
[5,3,2,1,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
=> ? = 2
[5,4,1,2,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2
[5,4,1,3,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 2
[5,4,2,1,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? = 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000633
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[2,5,3,1,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[3,2,5,1,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2
[3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[3,5,2,1,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[3,5,2,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2
[4,1,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,1,2,5,3] => [1,4,5,3,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[4,1,5,3,2] => [1,4,3,5,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[4,2,1,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,2,1,5,3] => [1,4,5,3,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[4,2,5,3,1] => [1,4,3,5,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[4,5,1,2,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[4,5,1,3,2] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,5,2,1,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[4,5,2,3,1] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2
[4,5,3,1,2] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[4,5,3,2,1] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[5,1,2,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3
[5,1,2,4,3] => [1,5,3,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[5,1,3,2,4] => [1,5,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[5,2,1,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3
[5,2,1,4,3] => [1,5,3,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2
[5,2,3,1,4] => [1,5,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The size of the automorphism group of a poset.
A poset automorphism is a permutation of the elements of the poset preserving the order relation.
Matching statistic: St000850
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000850: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000850: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[2,5,3,1,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 - 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 - 1
[3,2,5,1,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 2 - 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 - 1
[3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 - 1
[3,5,2,1,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 - 1
[3,5,2,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 2 - 1
[4,1,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 - 1
[4,1,2,5,3] => [1,4,5,3,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2 - 1
[4,1,5,3,2] => [1,4,3,5,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[4,2,1,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 - 1
[4,2,1,5,3] => [1,4,5,3,2] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2 - 1
[4,2,5,3,1] => [1,4,3,5,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[4,5,1,2,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[4,5,1,3,2] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 - 1
[4,5,2,1,3] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[4,5,2,3,1] => [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 2 - 1
[4,5,3,1,2] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[4,5,3,2,1] => [1,4,2,5,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[5,1,2,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
[5,1,2,4,3] => [1,5,3,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2 - 1
[5,1,3,2,4] => [1,5,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[5,2,1,3,4] => [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 3 - 1
[5,2,1,4,3] => [1,5,3,2,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2 - 1
[5,2,3,1,4] => [1,5,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The number of 1/2-balanced pairs in a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$.
Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains.
Olson and Sagan [3] exhibit various posets that have a $1/2$-balanced pair.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000635The number of strictly order preserving maps of a poset into itself. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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 $K[x]/(x^n)$. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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