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Your data matches 49 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [1,1]
=> 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1]
=> 1
[3,2,1] => [1,2,3] => [1,2,3] => [3]
=> 0
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [2,1,1]
=> 2
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [2,2]
=> 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,1]
=> 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [3,1]
=> 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [3,1]
=> 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4]
=> 0
[3,5,1,4,2] => [2,4,1,5,3] => [3,5,1,4,2] => [2,2,1]
=> 3
[3,5,2,4,1] => [1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> 2
[4,2,5,1,3] => [3,1,5,2,4] => [4,2,5,1,3] => [2,2,1]
=> 3
[4,2,5,3,1] => [1,3,5,2,4] => [1,4,5,2,3] => [3,2]
=> 2
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> 2
[4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => [3,2]
=> 2
[4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> 1
[5,2,4,1,3] => [3,1,4,2,5] => [4,2,3,1,5] => [3,1,1]
=> 2
[5,3,1,4,2] => [2,4,1,3,5] => [3,4,1,2,5] => [3,2]
=> 2
[5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => [3,2]
=> 2
[5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> 1
[5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> 1
[5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => [4,1]
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => [6,2,4,3,5,1] => [3,1,1,1]
=> 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [2,2,2]
=> 4
[4,6,2,5,3,1] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [3,2,1]
=> 3
[4,6,3,1,5,2] => [2,5,1,3,6,4] => [3,6,1,4,5,2] => [3,2,1]
=> 3
[4,6,3,5,1,2] => [2,1,5,3,6,4] => [2,1,6,4,5,3] => [3,2,1]
=> 3
[4,6,3,5,2,1] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [4,1,1]
=> 2
[5,2,6,4,1,3] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [3,2,1]
=> 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => [4,5,6,1,2,3] => [3,3]
=> 3
[5,3,6,2,4,1] => [1,4,2,6,3,5] => [1,5,3,6,2,4] => [3,2,1]
=> 3
[5,3,6,4,1,2] => [2,1,4,6,3,5] => [2,1,5,6,3,4] => [3,3]
=> 3
[5,3,6,4,2,1] => [1,2,4,6,3,5] => [1,2,5,6,3,4] => [4,2]
=> 2
[5,6,2,4,1,3] => [3,1,4,2,6,5] => [4,2,3,1,6,5] => [3,2,1]
=> 3
[5,6,3,1,4,2] => [2,4,1,3,6,5] => [3,4,1,2,6,5] => [3,3]
=> 3
[5,6,3,4,2,1] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> 2
[5,6,4,2,3,1] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [4,2]
=> 2
[5,6,4,3,2,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,1]
=> 1
[6,3,5,1,4,2] => [2,4,1,5,3,6] => [3,5,1,4,2,6] => [3,2,1]
=> 3
[6,3,5,2,4,1] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [4,1,1]
=> 2
[6,4,2,5,1,3] => [3,1,5,2,4,6] => [4,2,5,1,3,6] => [3,2,1]
=> 3
[6,4,2,5,3,1] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [4,2]
=> 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> 2
[6,4,5,3,1,2] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [4,2]
=> 2
[6,4,5,3,2,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [5,1]
=> 1
[6,5,2,4,1,3] => [3,1,4,2,5,6] => [4,2,3,1,5,6] => [4,1,1]
=> 2
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00282: Posets —Dedekind-MacNeille completion⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[4,5,2,3,1] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[4,5,3,1,2] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[4,5,3,2,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[5,3,1,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[5,3,4,1,2] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[5,3,4,2,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[5,4,2,3,1] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[5,4,3,1,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
=> ([(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)
=> 4
[4,6,2,5,3,1] => [1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[4,6,3,1,5,2] => [2,5,1,3,6,4] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[4,6,3,5,1,2] => [2,1,5,3,6,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 3
[4,6,3,5,2,1] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[5,2,6,4,1,3] => [3,1,4,6,2,5] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 3
[5,3,6,2,4,1] => [1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[5,3,6,4,1,2] => [2,1,4,6,3,5] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 3
[5,3,6,4,2,1] => [1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 2
[5,6,2,4,1,3] => [3,1,4,2,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 3
[5,6,3,1,4,2] => [2,4,1,3,6,5] => ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 3
[5,6,3,4,2,1] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 2
[5,6,4,2,3,1] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 2
[5,6,4,3,2,1] => [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 1
[6,3,5,1,4,2] => [2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[6,4,2,5,1,3] => [3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> 3
[6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[6,4,5,3,1,2] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
[6,4,5,3,2,1] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[6,5,2,4,1,3] => [3,1,4,2,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St000362
Mp00064: Permutations —reverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[2,3,1] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[3,1,2] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[3,5,1,4,2] => [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[3,5,2,4,1] => [1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,2,5,1,3] => [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[4,2,5,3,1] => [1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[5,2,4,1,3] => [3,1,4,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,3,1,4,2] => [2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 1
[5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[4,6,2,5,3,1] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[4,6,3,1,5,2] => [2,5,1,3,6,4] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,6,3,5,1,2] => [2,1,5,3,6,4] => [2,1,6,4,5,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,6,3,5,2,1] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[5,2,6,4,1,3] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3
[5,3,6,2,4,1] => [1,4,2,6,3,5] => [1,5,3,6,2,4] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[5,3,6,4,1,2] => [2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[5,3,6,4,2,1] => [1,2,4,6,3,5] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[5,6,2,4,1,3] => [3,1,4,2,6,5] => [4,2,3,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[5,6,3,1,4,2] => [2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[5,6,3,4,2,1] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 2
[5,6,4,2,3,1] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 2
[5,6,4,3,2,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> 1
[6,3,5,1,4,2] => [2,4,1,5,3,6] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[6,3,5,2,4,1] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[6,4,2,5,1,3] => [3,1,5,2,4,6] => [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[6,4,2,5,3,1] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 2
[6,4,5,3,1,2] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 2
[6,4,5,3,2,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> 1
[6,5,2,4,1,3] => [3,1,4,2,5,6] => [4,2,3,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[8,6,4,7,5,3,2,1] => [1,2,3,5,7,4,6,8] => [1,2,3,6,7,4,5,8] => ([(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
[8,5,7,4,6,3,2,1] => [1,2,3,6,4,7,5,8] => [1,2,3,7,5,6,4,8] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St001725
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => ([],3)
=> ([],3)
=> 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 3 = 2 + 1
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => ([],4)
=> ([],4)
=> 1 = 0 + 1
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[4,5,3,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[5,3,4,2,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[5,4,2,3,1] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[5,4,3,1,2] => ([(3,4)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
[5,4,3,2,1] => ([],5)
=> ([],5)
=> 1 = 0 + 1
[3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[4,2,6,1,5,3] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 4 + 1
[4,6,2,5,3,1] => ([(1,4),(1,5),(2,3),(2,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4 = 3 + 1
[4,6,3,1,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4 = 3 + 1
[4,6,3,5,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[4,6,3,5,2,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4 = 3 + 1
[5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[5,3,6,2,4,1] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4 = 3 + 1
[5,3,6,4,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[5,6,2,4,1,3] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[5,6,3,1,4,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[5,6,3,4,2,1] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[5,6,4,2,3,1] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[5,6,4,3,1,2] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[5,6,4,3,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> 2 = 1 + 1
[6,3,5,1,4,2] => ([(1,4),(1,5),(2,3),(2,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4 = 3 + 1
[6,3,5,2,4,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[6,4,2,5,1,3] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4 = 3 + 1
[6,4,2,5,3,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[6,4,5,3,1,2] => ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> 3 = 2 + 1
[6,4,5,3,2,1] => ([(4,5)],6)
=> ([(4,5)],6)
=> 2 = 1 + 1
[6,5,2,4,1,3] => ([(2,5),(3,4),(3,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 2 + 1
[5,2,7,4,1,6,3] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(3,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ? = 4 + 1
[5,3,7,2,6,4,1] => ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 4 + 1
[7,4,2,6,1,5,3] => ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 4 + 1
[8,6,4,7,5,3,2,1] => ([(4,7),(5,6),(5,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
[8,5,7,4,6,3,2,1] => ([(4,7),(5,6),(5,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 2 + 1
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St000918
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St000918: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 93%●distinct values known / distinct values provided: 80%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St000918: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 93%●distinct values known / distinct values provided: 80%
Values
[1,2] => [2,1] => ([(0,1)],2)
=> ([],1)
=> 1
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> ([],1)
=> 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> ([],1)
=> 1
[3,2,1] => [1,2,3] => ([],3)
=> ([],0)
=> ? = 0
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> 2
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> ([],1)
=> 1
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ([],1)
=> 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> ([],1)
=> 1
[4,3,2,1] => [1,2,3,4] => ([],4)
=> ([],0)
=> ? = 0
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[3,5,2,4,1] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,2,5,3,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[4,5,2,3,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> 2
[4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],2)
=> 2
[4,5,3,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([],1)
=> 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[5,3,1,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
[5,3,4,1,2] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],2)
=> 2
[5,3,4,2,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([],1)
=> 1
[5,4,2,3,1] => [1,3,2,4,5] => ([(3,4)],5)
=> ([],1)
=> 1
[5,4,3,1,2] => [2,1,3,4,5] => ([(3,4)],5)
=> ([],1)
=> 1
[5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ([],0)
=> ? = 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 4
[4,6,2,5,3,1] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[4,6,3,1,5,2] => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[4,6,3,5,1,2] => [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 3
[4,6,3,5,2,1] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[5,2,6,4,1,3] => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[5,3,6,2,4,1] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[5,3,6,4,1,2] => [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 3
[5,3,6,4,2,1] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[5,6,2,4,1,3] => [3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 3
[5,6,3,1,4,2] => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 3
[5,6,3,4,2,1] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[5,6,4,2,3,1] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[5,6,4,3,2,1] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ([],1)
=> 1
[6,3,5,1,4,2] => [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[6,3,5,2,4,1] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[6,4,2,5,1,3] => [3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[6,4,2,5,3,1] => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[6,4,5,3,1,2] => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[6,4,5,3,2,1] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ([],1)
=> 1
[6,5,2,4,1,3] => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[6,5,3,1,4,2] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
[6,5,3,4,1,2] => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ([],2)
=> 2
[6,5,3,4,2,1] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ([],1)
=> 1
[6,5,4,3,2,1] => [1,2,3,4,5,6] => ([],6)
=> ([],0)
=> ? = 0
[5,2,7,4,1,6,3] => [3,6,1,4,7,2,5] => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,6),(4,8),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4
[7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([],7)
=> ([],0)
=> ? = 0
[8,6,4,7,5,3,2,1] => [1,2,3,5,7,4,6,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 2
[8,5,7,4,6,3,2,1] => [1,2,3,6,4,7,5,8] => ([(4,7),(5,6),(6,7)],8)
=> ?
=> ? = 2
Description
The 2-limited packing number of a graph.
A subset $B$ of the set of vertices of a graph is a $k$-limited packing set if its intersection with the (closed) neighbourhood of any vertex is at most $k$. The $k$-limited packing number is the largest number of vertices in a $k$-limited packing set.
Matching statistic: St000670
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => [3,2,4,1] => 2
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => [3,4,2,1] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,5,1,4,2] => [3,1,5,2,4] => [5,4,2,1,3] => [4,3,5,2,1] => 3
[3,5,2,4,1] => [3,1,4,2,5] => [4,2,1,3,5] => [3,2,4,1,5] => 2
[4,2,5,1,3] => [2,4,1,5,3] => [5,3,1,2,4] => [3,4,2,5,1] => 3
[4,2,5,3,1] => [2,4,1,3,5] => [4,3,1,2,5] => [3,4,2,1,5] => 2
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,3,2,4] => [1,4,3,5,2] => 2
[5,3,1,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => [1,4,5,3,2] => 2
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => [4,2,1,6,3,5] => [3,2,6,5,1,4] => 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [2,5,4,1,6,3] => 4
[4,6,2,5,3,1] => [3,1,5,2,4,6] => [5,4,2,1,3,6] => [4,3,5,2,1,6] => 3
[4,6,3,1,5,2] => [3,1,4,6,2,5] => [6,5,2,1,3,4] => [4,3,5,6,2,1] => 3
[4,6,3,5,1,2] => [3,1,4,2,6,5] => [4,2,1,3,6,5] => [3,2,4,1,6,5] => 3
[4,6,3,5,2,1] => [3,1,4,2,5,6] => [4,2,1,3,5,6] => [3,2,4,1,5,6] => 2
[5,2,6,4,1,3] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => [4,5,3,2,6,1] => 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => [4,1,2,6,5,3] => [2,3,4,6,5,1] => 3
[5,3,6,2,4,1] => [2,4,1,5,3,6] => [5,3,1,2,4,6] => [3,4,2,5,1,6] => 3
[5,3,6,4,1,2] => [2,4,1,3,6,5] => [4,3,1,2,6,5] => [3,4,2,1,6,5] => 3
[5,3,6,4,2,1] => [2,4,1,3,5,6] => [4,3,1,2,5,6] => [3,4,2,1,5,6] => 2
[5,6,2,4,1,3] => [2,1,5,3,6,4] => [2,1,6,4,3,5] => [2,1,5,4,6,3] => 3
[5,6,3,1,4,2] => [2,1,4,6,3,5] => [2,1,6,5,3,4] => [2,1,5,6,4,3] => 3
[5,6,3,4,2,1] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[5,6,4,2,3,1] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2
[5,6,4,3,2,1] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[6,3,5,1,4,2] => [1,4,2,6,3,5] => [1,6,5,3,2,4] => [1,5,4,6,3,2] => 3
[6,3,5,2,4,1] => [1,4,2,5,3,6] => [1,5,3,2,4,6] => [1,4,3,5,2,6] => 2
[6,4,2,5,1,3] => [1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,4,5,3,6,2] => 3
[6,4,2,5,3,1] => [1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,4,5,3,2,6] => 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[6,4,5,3,1,2] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[6,4,5,3,2,1] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[6,5,2,4,1,3] => [1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,2,5,4,6,3] => 2
[4,7,3,6,2,5,1] => [4,1,5,2,6,3,7] => [4,2,1,6,3,5,7] => [3,2,6,5,1,4,7] => ? = 3
[5,2,7,4,1,6,3] => [3,6,1,4,7,2,5] => [3,1,4,6,2,7,5] => [2,3,6,5,1,7,4] => ? = 4
[5,3,7,2,6,4,1] => [3,5,1,6,2,4,7] => [3,1,5,2,6,4,7] => [2,5,4,1,6,3,7] => ? = 4
[5,7,3,6,4,2,1] => [3,1,5,2,4,6,7] => [5,4,2,1,3,6,7] => [4,3,5,2,1,6,7] => ? = 3
[5,7,4,2,6,3,1] => [3,1,4,6,2,5,7] => [6,5,2,1,3,4,7] => [4,3,5,6,2,1,7] => ? = 3
[5,7,4,6,2,3,1] => [3,1,4,2,6,5,7] => [4,2,1,3,6,5,7] => [3,2,4,1,6,5,7] => ? = 3
[5,7,4,6,3,1,2] => [3,1,4,2,5,7,6] => [4,2,1,3,5,7,6] => [3,2,4,1,5,7,6] => ? = 3
[5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => [4,2,1,3,5,6,7] => [3,2,4,1,5,6,7] => ? = 2
[6,3,7,5,2,4,1] => [2,5,1,3,6,4,7] => [6,4,3,1,2,5,7] => [4,5,3,2,6,1,7] => ? = 3
[6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => [4,1,2,6,5,3,7] => [2,3,4,6,5,1,7] => ? = 3
[6,4,7,3,5,2,1] => [2,4,1,5,3,6,7] => [5,3,1,2,4,6,7] => [3,4,2,5,1,6,7] => ? = 3
[6,4,7,5,2,3,1] => [2,4,1,3,6,5,7] => [4,3,1,2,6,5,7] => [3,4,2,1,6,5,7] => ? = 3
[6,4,7,5,3,1,2] => [2,4,1,3,5,7,6] => [4,3,1,2,5,7,6] => [3,4,2,1,5,7,6] => ? = 3
[6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => [4,3,1,2,5,6,7] => [3,4,2,1,5,6,7] => ? = 2
[6,7,3,5,2,4,1] => [2,1,5,3,6,4,7] => [2,1,6,4,3,5,7] => [2,1,5,4,6,3,7] => ? = 3
[6,7,4,2,5,3,1] => [2,1,4,6,3,5,7] => [2,1,6,5,3,4,7] => [2,1,5,6,4,3,7] => ? = 3
[6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[6,7,5,2,4,1,3] => [2,1,3,6,4,7,5] => [2,1,3,7,5,4,6] => [2,1,3,6,5,7,4] => ? = 3
[6,7,5,3,1,4,2] => [2,1,3,5,7,4,6] => [2,1,3,7,6,4,5] => [2,1,3,6,7,5,4] => ? = 3
[6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[7,4,6,2,5,3,1] => [1,4,2,6,3,5,7] => [1,6,5,3,2,4,7] => [1,5,4,6,3,2,7] => ? = 3
[7,4,6,3,1,5,2] => [1,4,2,5,7,3,6] => [1,7,6,3,2,4,5] => [1,5,4,6,7,3,2] => ? = 3
[7,5,2,6,4,1,3] => [1,3,6,2,4,7,5] => [1,7,5,4,2,3,6] => [1,5,6,4,3,7,2] => ? = 3
[8,6,4,7,5,3,2,1] => [1,3,5,2,4,6,7,8] => [1,5,4,2,3,6,7,8] => ? => ? = 2
[8,5,7,4,6,3,2,1] => [1,4,2,5,3,6,7,8] => [1,5,3,2,4,6,7,8] => ? => ? = 2
Description
The reversal length of a permutation.
A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$.
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.
Matching statistic: St000354
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [4,1,3,2] => 2
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [2,4,1,3] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,5,1,4,2] => [3,1,5,2,4] => [4,2,5,1,3] => [3,2,5,1,4] => 3
[3,5,2,4,1] => [3,1,4,2,5] => [4,2,3,1,5] => [4,1,3,2,5] => 2
[4,2,5,1,3] => [2,4,1,5,3] => [3,5,1,4,2] => [2,5,1,4,3] => 3
[4,2,5,3,1] => [2,4,1,3,5] => [3,4,1,2,5] => [2,4,1,3,5] => 2
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,2,4,3] => 2
[5,3,1,4,2] => [1,3,5,2,4] => [1,4,5,2,3] => [1,3,5,2,4] => 2
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => [6,2,4,3,5,1] => [6,1,3,2,5,4] => 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [3,6,2,5,1,4] => 4
[4,6,2,5,3,1] => [3,1,5,2,4,6] => [4,2,5,1,3,6] => [3,2,5,1,4,6] => 3
[4,6,3,1,5,2] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [4,1,3,6,2,5] => 3
[4,6,3,5,1,2] => [3,1,4,2,6,5] => [4,2,3,1,6,5] => [4,1,3,2,6,5] => 3
[4,6,3,5,2,1] => [3,1,4,2,5,6] => [4,2,3,1,5,6] => [4,1,3,2,5,6] => 2
[5,2,6,4,1,3] => [2,5,1,3,6,4] => [3,6,1,4,5,2] => [2,6,1,3,5,4] => 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => [4,5,6,1,2,3] => [2,4,6,1,3,5] => 3
[5,3,6,2,4,1] => [2,4,1,5,3,6] => [3,5,1,4,2,6] => [2,5,1,4,3,6] => 3
[5,3,6,4,1,2] => [2,4,1,3,6,5] => [3,4,1,2,6,5] => [2,4,1,3,6,5] => 3
[5,3,6,4,2,1] => [2,4,1,3,5,6] => [3,4,1,2,5,6] => [2,4,1,3,5,6] => 2
[5,6,2,4,1,3] => [2,1,5,3,6,4] => [2,1,6,4,5,3] => [2,1,6,3,5,4] => 3
[5,6,3,1,4,2] => [2,1,4,6,3,5] => [2,1,5,6,3,4] => [2,1,4,6,3,5] => 3
[5,6,3,4,2,1] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[5,6,4,2,3,1] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2
[5,6,4,3,2,1] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[6,3,5,1,4,2] => [1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,4,3,6,2,5] => 3
[6,3,5,2,4,1] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,5,2,4,3,6] => 2
[6,4,2,5,1,3] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,3,6,2,5,4] => 3
[6,4,2,5,3,1] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,3,5,2,4,6] => 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[6,4,5,3,1,2] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[6,4,5,3,2,1] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[6,5,2,4,1,3] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,6,3,5,4] => 2
[4,7,3,6,2,5,1] => [4,1,5,2,6,3,7] => [6,2,4,3,5,1,7] => [6,1,3,2,5,4,7] => ? = 3
[5,2,7,4,1,6,3] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [4,7,1,3,6,2,5] => ? = 4
[5,3,7,2,6,4,1] => [3,5,1,6,2,4,7] => [5,6,3,4,1,2,7] => [3,6,2,5,1,4,7] => ? = 4
[5,7,3,6,4,2,1] => [3,1,5,2,4,6,7] => [4,2,5,1,3,6,7] => [3,2,5,1,4,6,7] => ? = 3
[5,7,4,2,6,3,1] => [3,1,4,6,2,5,7] => [5,2,3,6,1,4,7] => [4,1,3,6,2,5,7] => ? = 3
[5,7,4,6,2,3,1] => [3,1,4,2,6,5,7] => [4,2,3,1,6,5,7] => [4,1,3,2,6,5,7] => ? = 3
[5,7,4,6,3,1,2] => [3,1,4,2,5,7,6] => [4,2,3,1,5,7,6] => [4,1,3,2,5,7,6] => ? = 3
[5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => [4,2,3,1,5,6,7] => [4,1,3,2,5,6,7] => ? = 2
[6,3,7,5,2,4,1] => [2,5,1,3,6,4,7] => [3,6,1,4,5,2,7] => [2,6,1,3,5,4,7] => ? = 3
[6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => [4,5,6,1,2,3,7] => [2,4,6,1,3,5,7] => ? = 3
[6,4,7,3,5,2,1] => [2,4,1,5,3,6,7] => [3,5,1,4,2,6,7] => [2,5,1,4,3,6,7] => ? = 3
[6,4,7,5,2,3,1] => [2,4,1,3,6,5,7] => [3,4,1,2,6,5,7] => [2,4,1,3,6,5,7] => ? = 3
[6,4,7,5,3,1,2] => [2,4,1,3,5,7,6] => [3,4,1,2,5,7,6] => [2,4,1,3,5,7,6] => ? = 3
[6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => [3,4,1,2,5,6,7] => [2,4,1,3,5,6,7] => ? = 2
[6,7,3,5,2,4,1] => [2,1,5,3,6,4,7] => [2,1,6,4,5,3,7] => [2,1,6,3,5,4,7] => ? = 3
[6,7,4,2,5,3,1] => [2,1,4,6,3,5,7] => [2,1,5,6,3,4,7] => [2,1,4,6,3,5,7] => ? = 3
[6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[6,7,5,2,4,1,3] => [2,1,3,6,4,7,5] => [2,1,3,7,5,6,4] => [2,1,3,7,4,6,5] => ? = 3
[6,7,5,3,1,4,2] => [2,1,3,5,7,4,6] => [2,1,3,6,7,4,5] => [2,1,3,5,7,4,6] => ? = 3
[6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[7,3,6,2,5,1,4] => [1,5,2,6,3,7,4] => [1,7,3,5,4,6,2] => [1,7,2,4,3,6,5] => ? = 3
[7,4,2,6,1,5,3] => [1,4,6,2,7,3,5] => [1,6,7,4,5,2,3] => [1,4,7,3,6,2,5] => ? = 4
[7,4,6,3,1,5,2] => [1,4,2,5,7,3,6] => [1,6,3,4,7,2,5] => [1,5,2,4,7,3,6] => ? = 3
[7,4,6,3,5,1,2] => [1,4,2,5,3,7,6] => [1,5,3,4,2,7,6] => [1,5,2,4,3,7,6] => ? = 3
[7,4,6,3,5,2,1] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 2
[8,6,4,7,5,3,2,1] => [1,3,5,2,4,6,7,8] => [1,4,5,2,3,6,7,8] => ? => ? = 2
[8,5,7,4,6,3,2,1] => [1,4,2,5,3,6,7,8] => [1,5,3,4,2,6,7,8] => ? => ? = 2
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000829
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000829: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [4,1,3,2] => 2
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [2,4,1,3] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,5,1,4,2] => [3,1,5,2,4] => [4,2,5,1,3] => [3,2,5,1,4] => 3
[3,5,2,4,1] => [3,1,4,2,5] => [4,2,3,1,5] => [4,1,3,2,5] => 2
[4,2,5,1,3] => [2,4,1,5,3] => [3,5,1,4,2] => [2,5,1,4,3] => 3
[4,2,5,3,1] => [2,4,1,3,5] => [3,4,1,2,5] => [2,4,1,3,5] => 2
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,2,4,3] => 2
[5,3,1,4,2] => [1,3,5,2,4] => [1,4,5,2,3] => [1,3,5,2,4] => 2
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => [6,2,4,3,5,1] => [6,1,3,2,5,4] => 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [3,6,2,5,1,4] => 4
[4,6,2,5,3,1] => [3,1,5,2,4,6] => [4,2,5,1,3,6] => [3,2,5,1,4,6] => 3
[4,6,3,1,5,2] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [4,1,3,6,2,5] => 3
[4,6,3,5,1,2] => [3,1,4,2,6,5] => [4,2,3,1,6,5] => [4,1,3,2,6,5] => 3
[4,6,3,5,2,1] => [3,1,4,2,5,6] => [4,2,3,1,5,6] => [4,1,3,2,5,6] => 2
[5,2,6,4,1,3] => [2,5,1,3,6,4] => [3,6,1,4,5,2] => [2,6,1,3,5,4] => 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => [4,5,6,1,2,3] => [2,4,6,1,3,5] => 3
[5,3,6,2,4,1] => [2,4,1,5,3,6] => [3,5,1,4,2,6] => [2,5,1,4,3,6] => 3
[5,3,6,4,1,2] => [2,4,1,3,6,5] => [3,4,1,2,6,5] => [2,4,1,3,6,5] => 3
[5,3,6,4,2,1] => [2,4,1,3,5,6] => [3,4,1,2,5,6] => [2,4,1,3,5,6] => 2
[5,6,2,4,1,3] => [2,1,5,3,6,4] => [2,1,6,4,5,3] => [2,1,6,3,5,4] => 3
[5,6,3,1,4,2] => [2,1,4,6,3,5] => [2,1,5,6,3,4] => [2,1,4,6,3,5] => 3
[5,6,3,4,2,1] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[5,6,4,2,3,1] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2
[5,6,4,3,2,1] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[6,3,5,1,4,2] => [1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,4,3,6,2,5] => 3
[6,3,5,2,4,1] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,5,2,4,3,6] => 2
[6,4,2,5,1,3] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,3,6,2,5,4] => 3
[6,4,2,5,3,1] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,3,5,2,4,6] => 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[6,4,5,3,1,2] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[6,4,5,3,2,1] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[6,5,2,4,1,3] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,6,3,5,4] => 2
[4,7,3,6,2,5,1] => [4,1,5,2,6,3,7] => [6,2,4,3,5,1,7] => [6,1,3,2,5,4,7] => ? = 3
[5,2,7,4,1,6,3] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [4,7,1,3,6,2,5] => ? = 4
[5,3,7,2,6,4,1] => [3,5,1,6,2,4,7] => [5,6,3,4,1,2,7] => [3,6,2,5,1,4,7] => ? = 4
[5,7,3,6,4,2,1] => [3,1,5,2,4,6,7] => [4,2,5,1,3,6,7] => [3,2,5,1,4,6,7] => ? = 3
[5,7,4,2,6,3,1] => [3,1,4,6,2,5,7] => [5,2,3,6,1,4,7] => [4,1,3,6,2,5,7] => ? = 3
[5,7,4,6,2,3,1] => [3,1,4,2,6,5,7] => [4,2,3,1,6,5,7] => [4,1,3,2,6,5,7] => ? = 3
[5,7,4,6,3,1,2] => [3,1,4,2,5,7,6] => [4,2,3,1,5,7,6] => [4,1,3,2,5,7,6] => ? = 3
[5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => [4,2,3,1,5,6,7] => [4,1,3,2,5,6,7] => ? = 2
[6,3,7,5,2,4,1] => [2,5,1,3,6,4,7] => [3,6,1,4,5,2,7] => [2,6,1,3,5,4,7] => ? = 3
[6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => [4,5,6,1,2,3,7] => [2,4,6,1,3,5,7] => ? = 3
[6,4,7,3,5,2,1] => [2,4,1,5,3,6,7] => [3,5,1,4,2,6,7] => [2,5,1,4,3,6,7] => ? = 3
[6,4,7,5,2,3,1] => [2,4,1,3,6,5,7] => [3,4,1,2,6,5,7] => [2,4,1,3,6,5,7] => ? = 3
[6,4,7,5,3,1,2] => [2,4,1,3,5,7,6] => [3,4,1,2,5,7,6] => [2,4,1,3,5,7,6] => ? = 3
[6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => [3,4,1,2,5,6,7] => [2,4,1,3,5,6,7] => ? = 2
[6,7,3,5,2,4,1] => [2,1,5,3,6,4,7] => [2,1,6,4,5,3,7] => [2,1,6,3,5,4,7] => ? = 3
[6,7,4,2,5,3,1] => [2,1,4,6,3,5,7] => [2,1,5,6,3,4,7] => [2,1,4,6,3,5,7] => ? = 3
[6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[6,7,5,2,4,1,3] => [2,1,3,6,4,7,5] => [2,1,3,7,5,6,4] => [2,1,3,7,4,6,5] => ? = 3
[6,7,5,3,1,4,2] => [2,1,3,5,7,4,6] => [2,1,3,6,7,4,5] => [2,1,3,5,7,4,6] => ? = 3
[6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[7,3,6,2,5,1,4] => [1,5,2,6,3,7,4] => [1,7,3,5,4,6,2] => [1,7,2,4,3,6,5] => ? = 3
[7,4,2,6,1,5,3] => [1,4,6,2,7,3,5] => [1,6,7,4,5,2,3] => [1,4,7,3,6,2,5] => ? = 4
[7,4,6,3,1,5,2] => [1,4,2,5,7,3,6] => [1,6,3,4,7,2,5] => [1,5,2,4,7,3,6] => ? = 3
[7,4,6,3,5,1,2] => [1,4,2,5,3,7,6] => [1,5,3,4,2,7,6] => [1,5,2,4,3,7,6] => ? = 3
[7,4,6,3,5,2,1] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 2
[8,6,4,7,5,3,2,1] => [1,3,5,2,4,6,7,8] => [1,4,5,2,3,6,7,8] => ? => ? = 2
[8,5,7,4,6,3,2,1] => [1,4,2,5,3,6,7,8] => [1,5,3,4,2,6,7,8] => ? => ? = 2
Description
The Ulam distance of a permutation to the identity permutation.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus [[St000062]] equals $n$.
Matching statistic: St001489
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [4,1,3,2] => 2
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [2,4,1,3] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,5,1,4,2] => [3,1,5,2,4] => [4,2,5,1,3] => [3,2,5,1,4] => 3
[3,5,2,4,1] => [3,1,4,2,5] => [4,2,3,1,5] => [4,1,3,2,5] => 2
[4,2,5,1,3] => [2,4,1,5,3] => [3,5,1,4,2] => [2,5,1,4,3] => 3
[4,2,5,3,1] => [2,4,1,3,5] => [3,4,1,2,5] => [2,4,1,3,5] => 2
[4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[5,2,4,1,3] => [1,4,2,5,3] => [1,5,3,4,2] => [1,5,2,4,3] => 2
[5,3,1,4,2] => [1,3,5,2,4] => [1,4,5,2,3] => [1,3,5,2,4] => 2
[5,3,4,1,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[5,4,3,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,6,2,5,1,4] => [4,1,5,2,6,3] => [6,2,4,3,5,1] => [6,1,3,2,5,4] => 3
[4,2,6,1,5,3] => [3,5,1,6,2,4] => [5,6,3,4,1,2] => [3,6,2,5,1,4] => 4
[4,6,2,5,3,1] => [3,1,5,2,4,6] => [4,2,5,1,3,6] => [3,2,5,1,4,6] => 3
[4,6,3,1,5,2] => [3,1,4,6,2,5] => [5,2,3,6,1,4] => [4,1,3,6,2,5] => 3
[4,6,3,5,1,2] => [3,1,4,2,6,5] => [4,2,3,1,6,5] => [4,1,3,2,6,5] => 3
[4,6,3,5,2,1] => [3,1,4,2,5,6] => [4,2,3,1,5,6] => [4,1,3,2,5,6] => 2
[5,2,6,4,1,3] => [2,5,1,3,6,4] => [3,6,1,4,5,2] => [2,6,1,3,5,4] => 3
[5,3,1,6,4,2] => [2,4,6,1,3,5] => [4,5,6,1,2,3] => [2,4,6,1,3,5] => 3
[5,3,6,2,4,1] => [2,4,1,5,3,6] => [3,5,1,4,2,6] => [2,5,1,4,3,6] => 3
[5,3,6,4,1,2] => [2,4,1,3,6,5] => [3,4,1,2,6,5] => [2,4,1,3,6,5] => 3
[5,3,6,4,2,1] => [2,4,1,3,5,6] => [3,4,1,2,5,6] => [2,4,1,3,5,6] => 2
[5,6,2,4,1,3] => [2,1,5,3,6,4] => [2,1,6,4,5,3] => [2,1,6,3,5,4] => 3
[5,6,3,1,4,2] => [2,1,4,6,3,5] => [2,1,5,6,3,4] => [2,1,4,6,3,5] => 3
[5,6,3,4,2,1] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[5,6,4,2,3,1] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => [2,1,3,5,4,6] => 2
[5,6,4,3,1,2] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2
[5,6,4,3,2,1] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[6,3,5,1,4,2] => [1,4,2,6,3,5] => [1,5,3,6,2,4] => [1,4,3,6,2,5] => 3
[6,3,5,2,4,1] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,5,2,4,3,6] => 2
[6,4,2,5,1,3] => [1,3,5,2,6,4] => [1,4,6,2,5,3] => [1,3,6,2,5,4] => 3
[6,4,2,5,3,1] => [1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,3,5,2,4,6] => 2
[6,4,5,2,3,1] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2
[6,4,5,3,1,2] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[6,4,5,3,2,1] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[6,5,2,4,1,3] => [1,2,5,3,6,4] => [1,2,6,4,5,3] => [1,2,6,3,5,4] => 2
[4,7,3,6,2,5,1] => [4,1,5,2,6,3,7] => [6,2,4,3,5,1,7] => [6,1,3,2,5,4,7] => ? = 3
[5,2,7,4,1,6,3] => [3,6,1,4,7,2,5] => [6,7,3,4,5,1,2] => [4,7,1,3,6,2,5] => ? = 4
[5,3,7,2,6,4,1] => [3,5,1,6,2,4,7] => [5,6,3,4,1,2,7] => [3,6,2,5,1,4,7] => ? = 4
[5,7,3,6,4,2,1] => [3,1,5,2,4,6,7] => [4,2,5,1,3,6,7] => [3,2,5,1,4,6,7] => ? = 3
[5,7,4,2,6,3,1] => [3,1,4,6,2,5,7] => [5,2,3,6,1,4,7] => [4,1,3,6,2,5,7] => ? = 3
[5,7,4,6,2,3,1] => [3,1,4,2,6,5,7] => [4,2,3,1,6,5,7] => [4,1,3,2,6,5,7] => ? = 3
[5,7,4,6,3,1,2] => [3,1,4,2,5,7,6] => [4,2,3,1,5,7,6] => [4,1,3,2,5,7,6] => ? = 3
[5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => [4,2,3,1,5,6,7] => [4,1,3,2,5,6,7] => ? = 2
[6,3,7,5,2,4,1] => [2,5,1,3,6,4,7] => [3,6,1,4,5,2,7] => [2,6,1,3,5,4,7] => ? = 3
[6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => [4,5,6,1,2,3,7] => [2,4,6,1,3,5,7] => ? = 3
[6,4,7,3,5,2,1] => [2,4,1,5,3,6,7] => [3,5,1,4,2,6,7] => [2,5,1,4,3,6,7] => ? = 3
[6,4,7,5,2,3,1] => [2,4,1,3,6,5,7] => [3,4,1,2,6,5,7] => [2,4,1,3,6,5,7] => ? = 3
[6,4,7,5,3,1,2] => [2,4,1,3,5,7,6] => [3,4,1,2,5,7,6] => [2,4,1,3,5,7,6] => ? = 3
[6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => [3,4,1,2,5,6,7] => [2,4,1,3,5,6,7] => ? = 2
[6,7,3,5,2,4,1] => [2,1,5,3,6,4,7] => [2,1,6,4,5,3,7] => [2,1,6,3,5,4,7] => ? = 3
[6,7,4,2,5,3,1] => [2,1,4,6,3,5,7] => [2,1,5,6,3,4,7] => [2,1,4,6,3,5,7] => ? = 3
[6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[6,7,5,2,4,1,3] => [2,1,3,6,4,7,5] => [2,1,3,7,5,6,4] => [2,1,3,7,4,6,5] => ? = 3
[6,7,5,3,1,4,2] => [2,1,3,5,7,4,6] => [2,1,3,6,7,4,5] => [2,1,3,5,7,4,6] => ? = 3
[6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 2
[6,7,5,4,2,3,1] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 2
[6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[7,3,6,2,5,1,4] => [1,5,2,6,3,7,4] => [1,7,3,5,4,6,2] => [1,7,2,4,3,6,5] => ? = 3
[7,4,2,6,1,5,3] => [1,4,6,2,7,3,5] => [1,6,7,4,5,2,3] => [1,4,7,3,6,2,5] => ? = 4
[7,4,6,3,1,5,2] => [1,4,2,5,7,3,6] => [1,6,3,4,7,2,5] => [1,5,2,4,7,3,6] => ? = 3
[7,4,6,3,5,1,2] => [1,4,2,5,3,7,6] => [1,5,3,4,2,7,6] => [1,5,2,4,3,7,6] => ? = 3
[7,4,6,3,5,2,1] => [1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 2
[8,6,4,7,5,3,2,1] => [1,3,5,2,4,6,7,8] => [1,4,5,2,3,6,7,8] => ? => ? = 2
[8,5,7,4,6,3,2,1] => [1,4,2,5,3,6,7,8] => [1,5,3,4,2,6,7,8] => ? => ? = 2
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St001875
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 1
[2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 1
[3,2,1] => ([],3)
=> ([],1)
=> ? = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,2,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[4,3,1,2] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 1
[4,3,2,1] => ([],4)
=> ([],1)
=> ? = 0 + 1
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([],1)
=> ? = 3 + 1
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ? = 3 + 1
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,5,3,2,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3,4,2,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[5,4,2,3,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[5,4,3,1,2] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 1
[5,4,3,2,1] => ([],5)
=> ([],1)
=> ? = 0 + 1
[3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[4,2,6,1,5,3] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ? = 4 + 1
[4,6,2,5,3,1] => ([(1,4),(1,5),(2,3),(2,5)],6)
=> ([],1)
=> ? = 3 + 1
[4,6,3,1,5,2] => ([(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 3 + 1
[4,6,3,5,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[4,6,3,5,2,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[5,2,6,4,1,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
[5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[5,3,6,2,4,1] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],1)
=> ? = 3 + 1
[5,3,6,4,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[5,6,2,4,1,3] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[5,6,3,1,4,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[5,6,3,4,2,1] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,6,4,2,3,1] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,6,4,3,1,2] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,6,4,3,2,1] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[6,3,5,1,4,2] => ([(1,4),(1,5),(2,3),(2,5)],6)
=> ([],1)
=> ? = 3 + 1
[6,3,5,2,4,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[6,4,2,5,1,3] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([],1)
=> ? = 3 + 1
[6,4,2,5,3,1] => ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[6,4,5,3,1,2] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[6,4,5,3,2,1] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[6,5,2,4,1,3] => ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[6,5,3,1,4,2] => ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[6,5,3,4,2,1] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[6,5,4,2,3,1] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[6,5,4,3,1,2] => ([(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 1
[6,5,4,3,2,1] => ([],6)
=> ([],1)
=> ? = 0 + 1
[4,7,3,6,2,5,1] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[5,2,7,4,1,6,3] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(3,6)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
[5,3,7,2,6,4,1] => ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1)],2)
=> ? = 4 + 1
[5,7,3,6,4,2,1] => ([(2,5),(2,6),(3,4),(3,6)],7)
=> ([],1)
=> ? = 3 + 1
[5,7,4,2,6,3,1] => ([(1,6),(2,5),(2,6),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 3 + 1
[5,7,4,6,2,3,1] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[5,7,4,6,3,1,2] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[5,7,4,6,3,2,1] => ([(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[6,3,7,5,2,4,1] => ([(1,6),(2,5),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
[6,4,2,7,5,3,1] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[6,4,7,3,5,2,1] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],1)
=> ? = 3 + 1
[6,4,7,5,2,3,1] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[6,4,7,5,3,1,2] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[6,4,7,5,3,2,1] => ([(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[6,7,3,5,2,4,1] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[6,7,4,2,5,3,1] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[6,7,4,5,3,2,1] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[6,7,5,2,4,1,3] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[6,7,5,3,1,4,2] => ([(1,6),(2,5),(3,4),(3,6)],7)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
[6,7,5,3,4,2,1] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[6,7,5,4,2,3,1] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[6,7,5,4,3,1,2] => ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[6,7,5,4,3,2,1] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 1 + 1
[7,3,6,2,5,1,4] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[7,4,2,6,1,5,3] => ([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1)],2)
=> ? = 4 + 1
[7,4,6,2,5,3,1] => ([(2,5),(2,6),(3,4),(3,6)],7)
=> ([],1)
=> ? = 3 + 1
[7,5,3,6,2,4,1] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],1)
=> ? = 3 + 1
[7,5,6,4,3,2,1] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 1 + 1
[7,6,3,5,1,4,2] => ([(2,5),(2,6),(3,4),(3,6)],7)
=> ([],1)
=> ? = 3 + 1
[7,6,4,2,5,1,3] => ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([],1)
=> ? = 3 + 1
[7,6,4,5,3,2,1] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 1 + 1
[7,6,5,3,4,2,1] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 1 + 1
[7,6,5,4,2,3,1] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 1 + 1
[7,6,5,4,3,1,2] => ([(5,6)],7)
=> ([(0,1)],2)
=> ? = 1 + 1
[7,6,5,4,3,2,1] => ([],7)
=> ([],1)
=> ? = 0 + 1
Description
The number of simple modules with projective dimension at most 1.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000672The number of minimal elements in Bruhat order not less than the permutation. St001812The biclique partition number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St001668The number of points of the poset minus the width of the poset. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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 $eA$. St000264The girth of a graph, which is not a tree. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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