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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> []
=> [1,0]
=> [1,0]
=> 0
[[]]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 5
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 6
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 3
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5
[[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> 7
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> 8
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> 7
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> 7
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> 7
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> 5
[[[],[],[],[]],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> 6
[[[],[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 4
[[[],[],[],[[]]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[[[[],[],[],[]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001823
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001823: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 44%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001823: Signed permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 44%
Values
[]
=> []
=> [1] => [1] => 0
[[]]
=> [1,0]
=> [2,1] => [2,1] => 1
[[],[]]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => ? = 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [5,4,1,2,3] => ? = 3
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 3
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,2,3,6,4] => ? = 5
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,6,3,5] => ? = 5
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,2,4,5] => ? = 5
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,4,2,3,5] => ? = 6
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,2,3,4] => ? = 3
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 4
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [5,4,1,2,6,3] => ? = 5
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 3
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [6,1,2,3,4,7,5] => ? = 6
[[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,2,3,7,4,6] => ? = 5
[[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [4,1,2,7,3,5,6] => ? = 7
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,2,6,3,7,5] => ? = 8
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [7,1,2,5,3,4,6] => ? = 7
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [7,1,2,6,3,4,5] => ? = 4
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [3,1,6,2,4,7,5] => ? = 7
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [6,1,7,2,3,4,5] => ? = 5
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [6,1,5,2,3,7,4] => ? = 6
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,4,6,2,3,5] => ? = 7
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => ? = 5
[[[],[],[],[]],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [5,7,1,2,3,4,6] => ? = 6
[[[],[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ? = 4
[[[],[],[],[[]]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [5,6,1,2,3,7,4] => ? = 6
[[[[],[],[],[]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => [6,7,5,1,2,3,4] => ? = 5
Description
The Stasinski-Voll length of a signed permutation.
The Stasinski-Voll length of a signed permutation $\sigma$ is
$$
L(\sigma) = \frac{1}{2} \#\{(i,j) ~\mid -n \leq i < j \leq n,~ i \not\equiv j \operatorname{mod} 2,~ \sigma(i) > \sigma(j)\},
$$
where $n$ is the size of $\sigma$.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0
[[]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 3
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5
[[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[[],[],[],[[]],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[],[],[[]],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7
[[],[],[[]],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 8
[[],[],[[],[]],[]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7
[[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[],[[]],[],[[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 7
[[],[[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[],[[],[],[[]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[[],[[[],[]],[]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7
[[[],[],[]],[],[]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[[],[],[],[]],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[[[],[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(6,5)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[],[],[],[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,6),(6,5)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 6
[[[[],[],[],[]]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,4)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
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: St001582
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[]
=> []
=> [1,0]
=> [1] => ? = 0
[[]]
=> [1,0]
=> [1,1,0,0]
=> [1,2] => 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ? = 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => ? = 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => ? = 3
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => ? = 3
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => ? = 5
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => ? = 5
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => ? = 5
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5] => ? = 6
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5] => ? = 3
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => ? = 4
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => ? = 5
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6] => ? = 3
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,7,6] => ? = 6
[[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,6] => ? = 5
[[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,7] => ? = 7
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,4,6,7] => ? = 8
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,5,6,4,7] => ? = 7
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,2,6,4,7] => ? = 4
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5] => ? = 7
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5,7] => ? = 5
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5,7] => ? = 6
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,4,2,5,7,6] => ? = 7
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5,7] => ? = 5
[[[],[],[],[]],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5,7] => ? = 6
[[[],[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,7] => ? = 4
[[[],[],[],[[]]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5,7] => ? = 6
[[[[],[],[],[]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,2,3,5,4,7,6] => ? = 5
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001941
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001941: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001941: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[]
=> []
=> ?
=> ? => ? = 0
[[]]
=> [1,0]
=> [[1],[2]]
=> [2,1] => 1
[[],[]]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ? = 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => ? = 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => ? = 3
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => ? = 3
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [2,4,6,9,10,1,3,5,7,8] => ? = 5
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [2,4,7,8,10,1,3,5,6,9] => ? = 5
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [2,5,6,8,10,1,3,4,7,9] => ? = 5
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [2,5,7,8,10,1,3,4,6,9] => ? = 6
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,5,7,9,10,1,3,4,6,8] => ? = 3
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [3,5,7,9,10,1,2,4,6,8] => ? = 4
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [3,5,8,9,10,1,2,4,6,7] => ? = 5
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => ? = 3
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> [2,4,6,8,11,12,1,3,5,7,9,10] => ? = 6
[[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> [2,4,6,9,10,12,1,3,5,7,8,11] => ? = 5
[[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> [2,4,7,8,10,12,1,3,5,6,9,11] => ? = 7
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,6,9,10],[2,4,7,8,11,12]]
=> [2,4,7,8,11,12,1,3,5,6,9,10] => ? = 8
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,3,5,6,8,11],[2,4,7,9,10,12]]
=> [2,4,7,9,10,12,1,3,5,6,8,11] => ? = 7
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> [2,4,7,9,11,12,1,3,5,6,8,10] => ? = 4
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,4,7,9,10],[2,5,6,8,11,12]]
=> [2,5,6,8,11,12,1,3,4,7,9,10] => ? = 7
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> [2,5,7,9,11,12,1,3,4,6,8,10] => ? = 5
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[1,3,4,6,8,9],[2,5,7,10,11,12]]
=> [2,5,7,10,11,12,1,3,4,6,8,9] => ? = 6
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,3,4,5,7,10],[2,6,8,9,11,12]]
=> [2,6,8,9,11,12,1,3,4,5,7,10] => ? = 7
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,6,9,11],[3,5,7,8,10,12]]
=> [3,5,7,8,10,12,1,2,4,6,9,11] => ? = 5
[[[],[],[],[]],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,8,11],[3,5,7,9,10,12]]
=> [3,5,7,9,10,12,1,2,4,6,8,11] => ? = 6
[[[],[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> [3,5,7,9,11,12,1,2,4,6,8,10] => ? = 4
[[[],[],[],[[]]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,8,9],[3,5,7,10,11,12]]
=> [3,5,7,10,11,12,1,2,4,6,8,9] => ? = 6
[[[[],[],[],[]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,6,8,10,11,12,1,2,3,5,7,9] => ? = 5
Description
The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5.3]) with parameters given by the identity and the permutation.
Also the number of paths in the Bruhat order from the identity to the permutation that are increasing with respect to a given reflection ordering as defined in Björner and Brenti [1, Section 5.2].
Matching statistic: St000608
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000608: Set partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000608: Set partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[]
=> []
=> []
=> ?
=> ? = 0 - 1
[[]]
=> [1,0]
=> [(1,2)]
=> {{1,2}}
=> 0 = 1 - 1
[[],[]]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> 2 = 3 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> ? = 3 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> ? = 3 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9}}
=> ? = 5 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10}}
=> ? = 5 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> {{1,2},{3,6},{4,5},{7,8},{9,10}}
=> ? = 5 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> {{1,2},{3,8},{4,5},{6,7},{9,10}}
=> ? = 6 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> ? = 3 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> ? = 4 - 1
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> {{1,10},{2,3},{4,5},{6,9},{7,8}}
=> ? = 5 - 1
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12}}
=> ? = 3 - 1
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> {{1,2},{3,4},{5,6},{7,8},{9,12},{10,11}}
=> ? = 6 - 1
[[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9},{11,12}}
=> ? = 5 - 1
[[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10},{11,12}}
=> ? = 7 - 1
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> {{1,2},{3,4},{5,8},{6,7},{9,12},{10,11}}
=> ? = 8 - 1
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9},{11,12}}
=> ? = 7 - 1
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> {{1,2},{3,4},{5,12},{6,7},{8,9},{10,11}}
=> ? = 4 - 1
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> {{1,2},{3,6},{4,5},{7,8},{9,12},{10,11}}
=> ? = 7 - 1
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> {{1,2},{3,12},{4,5},{6,7},{8,9},{10,11}}
=> ? = 5 - 1
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> {{1,2},{3,12},{4,5},{6,7},{8,11},{9,10}}
=> ? = 6 - 1
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> {{1,2},{3,12},{4,9},{5,6},{7,8},{10,11}}
=> ? = 7 - 1
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> {{1,8},{2,3},{4,5},{6,7},{9,10},{11,12}}
=> ? = 5 - 1
[[[],[],[],[]],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9},{11,12}}
=> ? = 6 - 1
[[[],[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> {{1,12},{2,3},{4,5},{6,7},{8,9},{10,11}}
=> ? = 4 - 1
[[[],[],[],[[]]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]
=> {{1,12},{2,3},{4,5},{6,7},{8,11},{9,10}}
=> ? = 6 - 1
[[[[],[],[],[]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> {{1,12},{2,11},{3,4},{5,6},{7,8},{9,10}}
=> ? = 5 - 1
Description
The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal.
Matching statistic: St000649
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[]
=> []
=> ?
=> ? => ? = 0 - 1
[[]]
=> [1,0]
=> [[1],[2]]
=> [2,1] => 0 = 1 - 1
[[],[]]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 0 = 1 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => 2 = 3 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ? = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [2,4,7,8,1,3,5,6] => ? = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [3,5,7,8,1,2,4,6] => ? = 3 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,4,6,8,10,1,3,5,7,9] => ? = 3 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [2,4,6,9,10,1,3,5,7,8] => ? = 5 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [2,4,7,8,10,1,3,5,6,9] => ? = 5 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [2,5,6,8,10,1,3,4,7,9] => ? = 5 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [2,5,7,8,10,1,3,4,6,9] => ? = 6 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [2,5,7,9,10,1,3,4,6,8] => ? = 3 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [3,5,7,9,10,1,2,4,6,8] => ? = 4 - 1
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,7],[3,5,8,9,10]]
=> [3,5,8,9,10,1,2,4,6,7] => ? = 5 - 1
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => ? = 3 - 1
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> [2,4,6,8,11,12,1,3,5,7,9,10] => ? = 6 - 1
[[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> [2,4,6,9,10,12,1,3,5,7,8,11] => ? = 5 - 1
[[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> [2,4,7,8,10,12,1,3,5,6,9,11] => ? = 7 - 1
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,3,5,6,9,10],[2,4,7,8,11,12]]
=> [2,4,7,8,11,12,1,3,5,6,9,10] => ? = 8 - 1
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,3,5,6,8,11],[2,4,7,9,10,12]]
=> [2,4,7,9,10,12,1,3,5,6,8,11] => ? = 7 - 1
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> [2,4,7,9,11,12,1,3,5,6,8,10] => ? = 4 - 1
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,3,4,7,9,10],[2,5,6,8,11,12]]
=> [2,5,6,8,11,12,1,3,4,7,9,10] => ? = 7 - 1
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,3,4,6,8,10],[2,5,7,9,11,12]]
=> [2,5,7,9,11,12,1,3,4,6,8,10] => ? = 5 - 1
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [[1,3,4,6,8,9],[2,5,7,10,11,12]]
=> [2,5,7,10,11,12,1,3,4,6,8,9] => ? = 6 - 1
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,3,4,5,7,10],[2,6,8,9,11,12]]
=> [2,6,8,9,11,12,1,3,4,5,7,10] => ? = 7 - 1
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,6,9,11],[3,5,7,8,10,12]]
=> [3,5,7,8,10,12,1,2,4,6,9,11] => ? = 5 - 1
[[[],[],[],[]],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,8,11],[3,5,7,9,10,12]]
=> [3,5,7,9,10,12,1,2,4,6,8,11] => ? = 6 - 1
[[[],[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8,10],[3,5,7,9,11,12]]
=> [3,5,7,9,11,12,1,2,4,6,8,10] => ? = 4 - 1
[[[],[],[],[[]]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[1,2,4,6,8,9],[3,5,7,10,11,12]]
=> [3,5,7,10,11,12,1,2,4,6,8,9] => ? = 6 - 1
[[[[],[],[],[]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7,9],[4,6,8,10,11,12]]
=> [4,6,8,10,11,12,1,2,3,5,7,9] => ? = 5 - 1
Description
The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Matching statistic: St001327
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001327: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001327: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[]
=> .
=> ?
=> ?
=> ? = 0 - 1
[[]]
=> [.,.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[],[]]
=> [[.,.],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0 = 1 - 1
[[],[],[]]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 2 - 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 3 - 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 2 - 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ? = 4 - 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 3 - 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [[[[[[],[]],[]],[]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 3 - 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 5 - 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 5 - 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 5 - 1
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 6 - 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 3 - 1
[[[],[],[],[]]]
=> [.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 4 - 1
[[[],[],[[]]]]
=> [.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 5 - 1
[[],[],[],[],[],[]]
=> [[[[[[.,.],.],.],.],.],.]
=> [[[[[[[],[]],[]],[]],[]],[]],[]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 3 - 1
[[],[],[],[],[[]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> [[[[[[],[]],[]],[]],[]],[[],[]]]
=> ([(0,12),(1,11),(2,11),(3,8),(4,8),(5,9),(6,10),(7,11),(7,12),(8,10),(9,10),(9,12)],13)
=> ? = 6 - 1
[[],[],[],[[]],[]]
=> [[[[[.,.],.],.],[.,.]],.]
=> [[[[[[],[]],[]],[]],[[],[]]],[]]
=> ([(0,9),(1,9),(2,10),(3,11),(4,8),(5,8),(6,7),(7,12),(8,12),(9,11),(10,11),(10,12)],13)
=> ? = 5 - 1
[[],[],[[]],[],[]]
=> [[[[[.,.],.],[.,.]],.],.]
=> [[[[[[],[]],[]],[[],[]]],[]],[]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,9),(5,10),(6,7),(7,12),(8,11),(9,10),(10,11),(11,12)],13)
=> ? = 7 - 1
[[],[],[[]],[[]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> [[[[[],[]],[]],[[],[]]],[[],[]]]
=> ([(0,11),(1,11),(2,9),(3,9),(4,10),(5,8),(6,8),(7,11),(7,12),(8,12),(9,10),(10,12)],13)
=> ? = 8 - 1
[[],[],[[],[]],[]]
=> [[[[.,.],.],[[.,.],.]],.]
=> [[[[[],[]],[]],[[[],[]],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,9),(4,11),(5,10),(6,7),(7,12),(8,10),(9,11),(10,12),(11,12)],13)
=> ? = 7 - 1
[[],[],[[],[],[]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> [[[[],[]],[]],[[[[],[]],[]],[]]]
=> ([(0,12),(1,11),(2,9),(3,9),(4,10),(5,8),(6,8),(7,11),(7,12),(8,11),(9,10),(10,12)],13)
=> ? = 4 - 1
[[],[[]],[],[[]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> [[[[[],[]],[[],[]]],[]],[[],[]]]
=> ([(0,11),(1,12),(2,12),(3,9),(4,9),(5,8),(6,8),(7,11),(7,12),(8,10),(9,10),(10,11)],13)
=> ? = 7 - 1
[[],[[],[],[],[]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> [[[],[]],[[[[[],[]],[]],[]],[]]]
=> ([(0,12),(1,11),(2,11),(3,8),(4,8),(5,9),(6,10),(7,11),(7,12),(8,10),(9,10),(9,12)],13)
=> ? = 5 - 1
[[],[[],[],[[]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [[[],[]],[[[[],[]],[]],[[],[]]]]
=> ([(0,11),(1,11),(2,9),(3,9),(4,10),(5,8),(6,8),(7,11),(7,12),(8,12),(9,10),(10,12)],13)
=> ? = 6 - 1
[[],[[[],[]],[]]]
=> [[.,.],[[.,[[.,.],.]],.]]
=> [[[],[]],[[[],[[[],[]],[]]],[]]]
=> ([(0,12),(1,11),(2,11),(3,8),(4,8),(5,9),(6,10),(7,11),(7,12),(8,10),(9,10),(9,12)],13)
=> ? = 7 - 1
[[[],[],[]],[],[]]
=> [[[.,[[[.,.],.],.]],.],.]
=> [[[[],[[[[],[]],[]],[]]],[]],[]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 5 - 1
[[[],[],[],[]],[]]
=> [[.,[[[[.,.],.],.],.]],.]
=> [[[],[[[[[],[]],[]],[]],[]]],[]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 6 - 1
[[[],[],[],[],[]]]
=> [.,[[[[[.,.],.],.],.],.]]
=> [[],[[[[[[],[]],[]],[]],[]],[]]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 4 - 1
[[[],[],[],[[]]]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [[],[[[[[],[]],[]],[]],[[],[]]]]
=> ([(0,9),(1,9),(2,10),(3,11),(4,8),(5,8),(6,7),(7,12),(8,12),(9,11),(10,11),(10,12)],13)
=> ? = 6 - 1
[[[[],[],[],[]]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> [[],[[],[[[[[],[]],[]],[]],[]]]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 5 - 1
Description
The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph.
A graph is a split graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge and $(b,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001566
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[]
=> []
=> []
=> ? => ? = 0 + 1
[[]]
=> [1,0]
=> [(1,2)]
=> [2,1] => 2 = 1 + 1
[[],[]]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2 = 1 + 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 3 = 2 + 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => 4 = 3 + 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? = 2 + 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => ? = 4 + 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => ? = 3 + 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => ? = 3 + 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => ? = 5 + 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => ? = 5 + 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => ? = 5 + 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => ? = 6 + 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => ? = 3 + 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => ? = 4 + 1
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => ? = 5 + 1
[[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 3 + 1
[[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,12,11,10,9] => ? = 6 + 1
[[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,10,9,8,7,12,11] => ? = 5 + 1
[[],[],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,8,7,6,5,10,9,12,11] => ? = 7 + 1
[[],[],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,8,7,6,5,12,11,10,9] => ? = 8 + 1
[[],[],[[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,10,7,6,9,8,5,12,11] => ? = 7 + 1
[[],[],[[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,12,7,6,9,8,11,10,5] => ? = 4 + 1
[[],[[]],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,6,5,4,3,8,7,12,11,10,9] => ? = 7 + 1
[[],[[],[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,12,5,4,7,6,9,8,11,10,3] => ? = 5 + 1
[[],[[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,12,5,4,7,6,11,10,9,8,3] => ? = 6 + 1
[[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)]
=> [2,1,12,9,6,5,8,7,4,11,10,3] => ? = 7 + 1
[[[],[],[]],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> [8,3,2,5,4,7,6,1,10,9,12,11] => ? = 5 + 1
[[[],[],[],[]],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [10,3,2,5,4,7,6,9,8,1,12,11] => ? = 6 + 1
[[[],[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> [12,3,2,5,4,7,6,9,8,11,10,1] => ? = 4 + 1
[[[],[],[],[[]]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]
=> [12,3,2,5,4,7,6,11,10,9,8,1] => ? = 6 + 1
[[[[],[],[],[]]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> [12,11,4,3,6,5,8,7,10,9,2,1] => ? = 5 + 1
Description
The length of the longest arithmetic progression in a permutation.
For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with
$$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
Matching statistic: St001672
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001672: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001672: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[]
=> .
=> ?
=> ?
=> ? = 0 + 2
[[]]
=> [.,.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 3 = 1 + 2
[[],[]]
=> [[.,.],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 1 + 2
[[],[],[]]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 4 = 2 + 2
[[],[[]]]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 5 = 3 + 2
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 2 + 2
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ? = 4 + 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ? = 3 + 2
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [[[[[[],[]],[]],[]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 3 + 2
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 5 + 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 5 + 2
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 5 + 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 6 + 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ? = 3 + 2
[[[],[],[],[]]]
=> [.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ? = 4 + 2
[[[],[],[[]]]]
=> [.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ? = 5 + 2
[[],[],[],[],[],[]]
=> [[[[[[.,.],.],.],.],.],.]
=> [[[[[[[],[]],[]],[]],[]],[]],[]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 3 + 2
[[],[],[],[],[[]]]
=> [[[[[.,.],.],.],.],[.,.]]
=> [[[[[[],[]],[]],[]],[]],[[],[]]]
=> ([(0,12),(1,11),(2,11),(3,8),(4,8),(5,9),(6,10),(7,11),(7,12),(8,10),(9,10),(9,12)],13)
=> ? = 6 + 2
[[],[],[],[[]],[]]
=> [[[[[.,.],.],.],[.,.]],.]
=> [[[[[[],[]],[]],[]],[[],[]]],[]]
=> ([(0,9),(1,9),(2,10),(3,11),(4,8),(5,8),(6,7),(7,12),(8,12),(9,11),(10,11),(10,12)],13)
=> ? = 5 + 2
[[],[],[[]],[],[]]
=> [[[[[.,.],.],[.,.]],.],.]
=> [[[[[[],[]],[]],[[],[]]],[]],[]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,9),(5,10),(6,7),(7,12),(8,11),(9,10),(10,11),(11,12)],13)
=> ? = 7 + 2
[[],[],[[]],[[]]]
=> [[[[.,.],.],[.,.]],[.,.]]
=> [[[[[],[]],[]],[[],[]]],[[],[]]]
=> ([(0,11),(1,11),(2,9),(3,9),(4,10),(5,8),(6,8),(7,11),(7,12),(8,12),(9,10),(10,12)],13)
=> ? = 8 + 2
[[],[],[[],[]],[]]
=> [[[[.,.],.],[[.,.],.]],.]
=> [[[[[],[]],[]],[[[],[]],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,9),(4,11),(5,10),(6,7),(7,12),(8,10),(9,11),(10,12),(11,12)],13)
=> ? = 7 + 2
[[],[],[[],[],[]]]
=> [[[.,.],.],[[[.,.],.],.]]
=> [[[[],[]],[]],[[[[],[]],[]],[]]]
=> ([(0,12),(1,11),(2,9),(3,9),(4,10),(5,8),(6,8),(7,11),(7,12),(8,11),(9,10),(10,12)],13)
=> ? = 4 + 2
[[],[[]],[],[[]]]
=> [[[[.,.],[.,.]],.],[.,.]]
=> [[[[[],[]],[[],[]]],[]],[[],[]]]
=> ([(0,11),(1,12),(2,12),(3,9),(4,9),(5,8),(6,8),(7,11),(7,12),(8,10),(9,10),(10,11)],13)
=> ? = 7 + 2
[[],[[],[],[],[]]]
=> [[.,.],[[[[.,.],.],.],.]]
=> [[[],[]],[[[[[],[]],[]],[]],[]]]
=> ([(0,12),(1,11),(2,11),(3,8),(4,8),(5,9),(6,10),(7,11),(7,12),(8,10),(9,10),(9,12)],13)
=> ? = 5 + 2
[[],[[],[],[[]]]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [[[],[]],[[[[],[]],[]],[[],[]]]]
=> ([(0,11),(1,11),(2,9),(3,9),(4,10),(5,8),(6,8),(7,11),(7,12),(8,12),(9,10),(10,12)],13)
=> ? = 6 + 2
[[],[[[],[]],[]]]
=> [[.,.],[[.,[[.,.],.]],.]]
=> [[[],[]],[[[],[[[],[]],[]]],[]]]
=> ([(0,12),(1,11),(2,11),(3,8),(4,8),(5,9),(6,10),(7,11),(7,12),(8,10),(9,10),(9,12)],13)
=> ? = 7 + 2
[[[],[],[]],[],[]]
=> [[[.,[[[.,.],.],.]],.],.]
=> [[[[],[[[[],[]],[]],[]]],[]],[]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 5 + 2
[[[],[],[],[]],[]]
=> [[.,[[[[.,.],.],.],.]],.]
=> [[[],[[[[[],[]],[]],[]],[]]],[]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 6 + 2
[[[],[],[],[],[]]]
=> [.,[[[[[.,.],.],.],.],.]]
=> [[],[[[[[[],[]],[]],[]],[]],[]]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 4 + 2
[[[],[],[],[[]]]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [[],[[[[[],[]],[]],[]],[[],[]]]]
=> ([(0,9),(1,9),(2,10),(3,11),(4,8),(5,8),(6,7),(7,12),(8,12),(9,11),(10,11),(10,12)],13)
=> ? = 6 + 2
[[[[],[],[],[]]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> [[],[[],[[[[[],[]],[]],[]],[]]]]
=> ([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)
=> ? = 5 + 2
Description
The restrained domination number of a graph.
This is the minimal size of a set of vertices $D$ such that every vertex not in $D$ is adjacent to a vertex in $D$ and to a vertex not in $D$.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001644The dimension of a graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000907The number of maximal antichains of minimal length in a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
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