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Your data matches 63 different statistics following compositions of up to 3 maps.
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Matching statistic: St000939
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[1,3,2] => [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[2,1,3] => [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[2,3,1] => [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[3,1,2] => [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[3,2,1] => [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[1,1,3,3] => [2,2] => [2,2]
=> [2]
=> 1
[1,3,1,3] => [2,2] => [2,2]
=> [2]
=> 1
[1,3,3,1] => [2,2] => [2,2]
=> [2]
=> 1
[3,1,1,3] => [2,2] => [2,2]
=> [2]
=> 1
[3,1,3,1] => [2,2] => [2,2]
=> [2]
=> 1
[3,3,1,1] => [2,2] => [2,2]
=> [2]
=> 1
[1,1,3,4] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[1,1,4,3] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[1,3,1,4] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,1] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[1,4,1,3] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,1] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[3,1,1,4] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[3,1,4,1] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[3,4,1,1] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[4,1,1,3] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[4,1,3,1] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[4,3,1,1] => [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[1,2,2,4] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[1,2,4,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[2,1,2,4] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[2,1,4,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[2,2,1,4] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[2,2,4,1] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[2,4,1,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[2,4,2,1] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[4,1,2,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[4,2,1,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[4,2,2,1] => [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[1,2,3,3] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,3] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[1,3,3,2] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[2,1,3,3] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[2,3,1,3] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[2,3,3,1] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,1,2,3] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,1,3,2] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,2,1,3] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,2,3,1] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,3,1,2] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,3,2,1] => [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[1,2,3,4] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 3
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,3,2] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,1,3] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,3,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2
[3,1,2] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2
[3,2,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,2,2,4] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,2,4,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,4,2,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,1,2,4] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,1,4,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,2,1,4] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,2,4,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,4,1,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[2,4,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[4,1,2,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[4,2,1,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[4,2,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2
[1,2,3,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,3,2,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,3,3,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,1,3,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,3,1,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,3,3,1] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,1,2,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,1,3,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,2,1,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,2,3,1] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,3,1,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,3,2,1] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,2,2,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,2,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,2,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,3,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,3,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,3,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,3,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,3,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,3,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,6,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,6,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,2,6,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,3,2,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,3,2,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,3,6,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,6,2,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,6,2,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,6,3,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,2,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,2,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,2,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,6,3,3,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,2,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,2,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,2,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,6,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,6,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,6,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,1,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,1,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,1,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,3,1,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,3,1,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,3,3,1,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,3,3,6,1] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,2,3,6,1,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2 + 4
[1,3,2] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2 + 4
[2,1,3] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2 + 4
[2,3,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2 + 4
[3,1,2] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2 + 4
[3,2,1] => [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? = 2 + 4
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 4
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 4
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 4
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 4
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 4
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 4
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,1,4,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,4,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[4,1,1,3] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[4,1,3,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[4,3,1,1] => [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,2,2,4] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[1,2,4,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[1,4,2,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[2,1,2,4] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[2,1,4,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[2,2,1,4] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[2,2,4,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[2,4,1,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[2,4,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[4,1,2,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[4,2,1,2] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[4,2,2,1] => [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 4
[1,2,3,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,3,2,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,3,3,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[2,1,3,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[2,3,1,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[2,3,3,1] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,1,2,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,1,3,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,2,1,3] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,2,3,1] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,3,1,2] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[3,3,2,1] => [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 4
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 4
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 4
[1,2,2,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,2,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,2,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,3,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,3,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,3,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,3,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,3,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,3,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,6,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,6,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,2,6,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,2,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,2,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,2,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,2,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,2,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,2,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,3,2,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,3,2,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,3,6,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,6,2,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,6,2,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,3,6,3,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,2,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,2,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,2,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,3,2,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,3,2,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[1,6,3,3,2,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,2,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,2,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,2,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,3,2,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,3,2,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,3,3,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,3,3,6,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,3,6,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,3,6,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,6,2,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,6,3,2,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,1,6,3,3,2] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,2,1,3,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,2,1,3,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,2,1,6,3,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,2,3,1,3,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,2,3,1,6,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,2,3,3,1,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,2,3,3,6,1] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
[2,2,3,6,1,3] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 2 + 4
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001060
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 2 + 1
[1,3,2] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 2 + 1
[2,1,3] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 2 + 1
[2,3,1] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 2 + 1
[3,1,2] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 2 + 1
[3,2,1] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? = 2 + 1
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[3,1,4,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[3,4,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[4,1,1,3] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[4,1,3,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[4,3,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,2,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,2,4,2] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,4,2,2] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[2,1,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[2,1,4,2] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[2,2,1,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[2,2,4,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[2,4,1,2] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[2,4,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[4,1,2,2] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[4,2,1,2] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[4,2,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[1,2,3,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[1,3,2,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[1,3,3,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[2,1,3,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[2,3,1,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[2,3,3,1] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[3,1,2,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[3,1,3,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[3,2,1,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[3,2,3,1] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[3,3,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[3,3,2,1] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? = 3 + 1
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? = 3 + 1
[1,1,1,4,4,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,1,4,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,4,1,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,4,4,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,1,4,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,4,1,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,4,4,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,4,1,1,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,4,1,4,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,4,4,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,1,1,1,4,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,1,1,4,1,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,1,1,4,4,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,1,4,1,1,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,1,4,1,4,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,1,4,4,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,4,1,1,1,4] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,4,1,1,4,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,4,1,4,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[4,4,4,1,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,1,4,4,5] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,1,4,5,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,1,5,4,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,1,4,5] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,1,5,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,4,1,5] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,4,5,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,5,1,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,4,5,4,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,5,1,4,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,5,4,1,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,1,5,4,4,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,1,4,5] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,1,5,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,4,1,5] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,4,5,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,5,1,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,1,5,4,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,4,1,1,5] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,4,1,5,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,4,5,1,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,5,1,1,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,5,1,4,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,4,5,4,1,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,5,1,1,4,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,5,1,4,1,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,5,1,4,4,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,5,4,1,1,4] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,5,4,1,4,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
[1,5,4,4,1,1] => [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> 3 = 2 + 1
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001624
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,3,1] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,2] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,2,1] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,2,4,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,4,2,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,1,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,1,4,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,2,1,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,2,4,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,4,1,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,4,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,1,2,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,2,1,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,2,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,2,3,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,2,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,3,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,1,3,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,1,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,3,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,2,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,3,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,3,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3,2,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,3,2,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,3,4,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,4,2,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,4,3,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,1,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,1,4,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,3,1,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,3,4,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,4,1,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,4,3,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,1,2,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,1,4,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,2,1,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,2,4,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,4,1,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,4,2,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,1,2,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,1,3,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,2,1,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,2,3,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,3,1,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,3,2,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,4,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,4,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,4,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,1,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,1,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,4,1,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[4,1,1,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[4,1,1,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,2,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,2,5,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,4,2,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,4,5,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,5,2,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,5,4,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,1,5,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,1,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,5,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,1,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,4,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,1,2,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,1,5,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,2,1,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,2,5,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,5,1,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,5,2,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,1,2,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,1,4,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001630
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,3,1] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,2] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,2,1] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,2,4,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,4,2,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,1,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,1,4,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,2,1,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,2,4,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,4,1,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,4,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,1,2,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,2,1,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,2,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,2,3,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,2,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,3,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,1,3,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,1,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,3,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,2,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,3,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,3,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3,2,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,3,2,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,3,4,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,4,2,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,4,3,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,1,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,1,4,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,3,1,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,3,4,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,4,1,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,4,3,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,1,2,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,1,4,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,2,1,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,2,4,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,4,1,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,4,2,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,1,2,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,1,3,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,2,1,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,2,3,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,3,1,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,3,2,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,4,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,4,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,4,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,1,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,1,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,4,1,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[4,1,1,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[4,1,1,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,2,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,2,5,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,4,2,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,4,5,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,5,2,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,5,4,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,1,5,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,1,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,5,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,1,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,4,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,1,2,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,1,5,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,2,1,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,2,5,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,5,1,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,5,2,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,1,2,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,1,4,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,3,2] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,1,3] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[2,3,1] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,1,2] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[3,2,1] => [1,0,1,0,1,0]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,2,4,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,4,2,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,1,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,1,4,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,2,1,4] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,2,4,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,4,1,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[2,4,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,1,2,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,2,1,2] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[4,2,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
[1,2,3,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,2,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,3,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,1,3,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,1,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,3,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,2,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,1,3,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,1,3] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,2,3,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[3,3,2,1] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,3,2,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,3,4,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,4,2,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,4,3,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,1,3,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,1,4,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,3,1,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,3,4,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,4,1,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[2,4,3,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,1,2,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,1,4,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,2,1,4] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,2,4,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,4,1,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[3,4,2,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,1,2,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,1,3,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,2,1,3] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,2,3,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,3,1,2] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[4,3,2,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 3
[1,1,1,4,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,4,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,4,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,1,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,1,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,4,4,1,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[4,1,1,1,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[4,1,1,4,1] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1
[1,1,2,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,2,5,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,4,2,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,4,5,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,5,2,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,5,4,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,1,5,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,1,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,4,5,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,1,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,2,5,4,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,1,2,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,1,5,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,2,1,5] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,2,5,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,5,1,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,5,2,1] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,1,2,4] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,5,1,4,2] => [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000903
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00303: Parking functions —complement⟶ Parking functions
Mp00319: Parking functions —to composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Mp00319: Parking functions —to composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 40%
Values
[1,2,3] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,3,2] => [3,1,2] => [3,1,2] => 3 = 2 + 1
[2,1,3] => [2,3,1] => [2,3,1] => 3 = 2 + 1
[2,3,1] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[3,1,2] => [1,3,2] => [1,3,2] => 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[1,1,3,3] => [3,3,1,1] => [3,3,1,1] => 2 = 1 + 1
[1,3,1,3] => [3,1,3,1] => [3,1,3,1] => 2 = 1 + 1
[1,3,3,1] => [3,1,1,3] => [3,1,1,3] => 2 = 1 + 1
[3,1,1,3] => [1,3,3,1] => [1,3,3,1] => 2 = 1 + 1
[3,1,3,1] => [1,3,1,3] => [1,3,1,3] => 2 = 1 + 1
[3,3,1,1] => [1,1,3,3] => [1,1,3,3] => 2 = 1 + 1
[1,1,3,4] => [1,1,4,3] => [1,1,4,3] => 3 = 2 + 1
[1,1,4,3] => [1,1,3,4] => [1,1,3,4] => 3 = 2 + 1
[1,3,1,4] => [1,4,1,3] => [1,4,1,3] => 3 = 2 + 1
[1,3,4,1] => [1,4,3,1] => [1,4,3,1] => 3 = 2 + 1
[1,4,1,3] => [1,3,1,4] => [1,3,1,4] => 3 = 2 + 1
[1,4,3,1] => [1,3,4,1] => [1,3,4,1] => 3 = 2 + 1
[3,1,1,4] => [4,1,1,3] => [4,1,1,3] => 3 = 2 + 1
[3,1,4,1] => [4,1,3,1] => [4,1,3,1] => 3 = 2 + 1
[3,4,1,1] => [4,3,1,1] => [4,3,1,1] => 3 = 2 + 1
[4,1,1,3] => [3,1,1,4] => [3,1,1,4] => 3 = 2 + 1
[4,1,3,1] => [3,1,4,1] => [3,1,4,1] => 3 = 2 + 1
[4,3,1,1] => [3,4,1,1] => [3,4,1,1] => 3 = 2 + 1
[1,2,2,4] => [2,1,1,4] => [2,1,1,4] => 3 = 2 + 1
[1,2,4,2] => [2,1,4,1] => [2,1,4,1] => 3 = 2 + 1
[1,4,2,2] => [2,4,1,1] => [2,4,1,1] => 3 = 2 + 1
[2,1,2,4] => [1,2,1,4] => [1,2,1,4] => 3 = 2 + 1
[2,1,4,2] => [1,2,4,1] => [1,2,4,1] => 3 = 2 + 1
[2,2,1,4] => [1,1,2,4] => [1,1,2,4] => 3 = 2 + 1
[2,2,4,1] => [1,1,4,2] => [1,1,4,2] => 3 = 2 + 1
[2,4,1,2] => [1,4,2,1] => [1,4,2,1] => 3 = 2 + 1
[2,4,2,1] => [1,4,1,2] => [1,4,1,2] => 3 = 2 + 1
[4,1,2,2] => [4,2,1,1] => [4,2,1,1] => 3 = 2 + 1
[4,2,1,2] => [4,1,2,1] => [4,1,2,1] => 3 = 2 + 1
[4,2,2,1] => [4,1,1,2] => [4,1,1,2] => 3 = 2 + 1
[1,2,3,3] => [3,2,1,1] => [3,2,1,1] => 3 = 2 + 1
[1,3,2,3] => [3,1,2,1] => [3,1,2,1] => 3 = 2 + 1
[1,3,3,2] => [3,1,1,2] => [3,1,1,2] => 3 = 2 + 1
[2,1,3,3] => [2,3,1,1] => [2,3,1,1] => 3 = 2 + 1
[2,3,1,3] => [2,1,3,1] => [2,1,3,1] => 3 = 2 + 1
[2,3,3,1] => [2,1,1,3] => [2,1,1,3] => 3 = 2 + 1
[3,1,2,3] => [1,3,2,1] => [1,3,2,1] => 3 = 2 + 1
[3,1,3,2] => [1,3,1,2] => [1,3,1,2] => 3 = 2 + 1
[3,2,1,3] => [1,2,3,1] => [1,2,3,1] => 3 = 2 + 1
[3,2,3,1] => [1,2,1,3] => [1,2,1,3] => 3 = 2 + 1
[3,3,1,2] => [1,1,3,2] => [1,1,3,2] => 3 = 2 + 1
[3,3,2,1] => [1,1,2,3] => [1,1,2,3] => 3 = 2 + 1
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => ? = 3 + 1
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => ? = 3 + 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => ? = 3 + 1
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => ? = 3 + 1
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => ? = 3 + 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => ? = 3 + 1
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => ? = 3 + 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => ? = 3 + 1
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => ? = 3 + 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => ? = 3 + 1
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => ? = 3 + 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => ? = 3 + 1
[3,1,2,4] => [2,4,3,1] => [2,4,3,1] => ? = 3 + 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => ? = 3 + 1
[3,2,1,4] => [2,3,4,1] => [2,3,4,1] => ? = 3 + 1
[3,2,4,1] => [2,3,1,4] => [2,3,1,4] => ? = 3 + 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => ? = 3 + 1
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => ? = 3 + 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => ? = 3 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => ? = 3 + 1
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => ? = 3 + 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => ? = 3 + 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => ? = 3 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ? = 3 + 1
[1,1,1,4,4] => [1,1,1,4,4] => [1,1,1,4,4] => ? = 1 + 1
[1,1,4,1,4] => [1,1,4,1,4] => [1,1,4,1,4] => ? = 1 + 1
[1,1,4,4,1] => [1,1,4,4,1] => [1,1,4,4,1] => ? = 1 + 1
[1,4,1,1,4] => [1,4,1,1,4] => [1,4,1,1,4] => ? = 1 + 1
[1,4,1,4,1] => [1,4,1,4,1] => [1,4,1,4,1] => ? = 1 + 1
[1,4,4,1,1] => [1,4,4,1,1] => [1,4,4,1,1] => ? = 1 + 1
[4,1,1,1,4] => [4,1,1,1,4] => [4,1,1,1,4] => ? = 1 + 1
[4,1,1,4,1] => [4,1,1,4,1] => [4,1,1,4,1] => ? = 1 + 1
[4,1,4,1,1] => [4,1,4,1,1] => [4,1,4,1,1] => ? = 1 + 1
[4,4,1,1,1] => [4,4,1,1,1] => [4,4,1,1,1] => ? = 1 + 1
[1,1,1,4,5] => [1,1,1,4,3] => [1,1,1,4,3] => ? = 2 + 1
[1,1,1,5,4] => [1,1,1,3,4] => [1,1,1,3,4] => ? = 2 + 1
[1,1,4,1,5] => [1,1,4,1,3] => [1,1,4,1,3] => ? = 2 + 1
[1,1,4,5,1] => [1,1,4,3,1] => [1,1,4,3,1] => ? = 2 + 1
[1,1,5,1,4] => [1,1,3,1,4] => [1,1,3,1,4] => ? = 2 + 1
[1,1,5,4,1] => [1,1,3,4,1] => [1,1,3,4,1] => ? = 2 + 1
[1,4,1,1,5] => [1,4,1,1,3] => [1,4,1,1,3] => ? = 2 + 1
[1,4,1,5,1] => [1,4,1,3,1] => [1,4,1,3,1] => ? = 2 + 1
[1,4,5,1,1] => [1,4,3,1,1] => [1,4,3,1,1] => ? = 2 + 1
[1,5,1,1,4] => [1,3,1,1,4] => [1,3,1,1,4] => ? = 2 + 1
[1,5,1,4,1] => [1,3,1,4,1] => [1,3,1,4,1] => ? = 2 + 1
[1,5,4,1,1] => [1,3,4,1,1] => [1,3,4,1,1] => ? = 2 + 1
[4,1,1,1,5] => [4,1,1,1,3] => [4,1,1,1,3] => ? = 2 + 1
[4,1,1,5,1] => [4,1,1,3,1] => [4,1,1,3,1] => ? = 2 + 1
[4,1,5,1,1] => [4,1,3,1,1] => [4,1,3,1,1] => ? = 2 + 1
[4,5,1,1,1] => [4,3,1,1,1] => [4,3,1,1,1] => ? = 2 + 1
[1,1,3,3,3] => [3,3,1,1,1] => [3,3,1,1,1] => 2 = 1 + 1
[1,3,1,3,3] => [3,1,3,1,1] => [3,1,3,1,1] => 2 = 1 + 1
Description
The number of different parts of an integer composition.
Matching statistic: St001232
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 + 2
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 + 2
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 + 2
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 + 2
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 + 2
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 + 2
[1,1,3,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[1,3,1,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[1,3,3,1] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[3,1,1,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[3,1,3,1] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[3,3,1,1] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[1,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,1,4,3] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,3,1,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,3,4,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,4,1,3] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,4,3,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,1,1,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,1,4,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,4,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[4,1,1,3] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[4,1,3,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[4,3,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,2,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,2,4,2] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,4,2,2] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,1,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,1,4,2] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,2,1,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,2,4,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,4,1,2] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,4,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[4,1,2,2] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[4,2,1,2] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[4,2,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,2,3,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,3,2,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,3,3,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,1,3,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,3,1,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[2,3,3,1] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,1,2,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,1,3,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,2,1,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,2,3,1] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,3,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[3,3,2,1] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3 + 2
[1,2,4,3] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3 + 2
[1,1,3,3,5,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,3,5,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,3,5,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,5,3,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,5,3,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,1,5,5,3,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,1,3,5,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,1,5,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,1,5,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,3,1,5,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,3,5,1,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,3,5,5,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,5,1,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,5,1,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,5,3,1,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,5,3,5,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,5,5,1,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,3,5,5,3,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,1,3,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,1,3,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,1,5,3,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,3,1,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,3,1,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,3,3,1,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,3,3,5,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,3,5,1,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,3,5,3,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,5,1,3,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,5,3,1,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[1,5,5,3,3,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,1,3,5,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,1,5,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,1,5,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,3,1,5,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,3,5,1,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,3,5,5,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,5,1,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,5,1,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,5,3,1,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,5,3,5,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,5,5,1,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,1,5,5,3,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,3,1,1,5,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,3,1,5,1,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,3,1,5,5,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,3,5,1,1,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,3,5,1,5,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,3,5,5,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,5,1,1,3,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
[3,5,1,1,5,3] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 3 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000135
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
St000135: Parking functions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 60%
Values
[1,2,3] => 3 = 2 + 1
[1,3,2] => 3 = 2 + 1
[2,1,3] => 3 = 2 + 1
[2,3,1] => 3 = 2 + 1
[3,1,2] => 3 = 2 + 1
[3,2,1] => 3 = 2 + 1
[1,1,3,3] => 2 = 1 + 1
[1,3,1,3] => 2 = 1 + 1
[1,3,3,1] => 2 = 1 + 1
[3,1,1,3] => 2 = 1 + 1
[3,1,3,1] => 2 = 1 + 1
[3,3,1,1] => 2 = 1 + 1
[1,1,3,4] => 3 = 2 + 1
[1,1,4,3] => 3 = 2 + 1
[1,3,1,4] => 3 = 2 + 1
[1,3,4,1] => 3 = 2 + 1
[1,4,1,3] => 3 = 2 + 1
[1,4,3,1] => 3 = 2 + 1
[3,1,1,4] => 3 = 2 + 1
[3,1,4,1] => 3 = 2 + 1
[3,4,1,1] => 3 = 2 + 1
[4,1,1,3] => 3 = 2 + 1
[4,1,3,1] => 3 = 2 + 1
[4,3,1,1] => 3 = 2 + 1
[1,2,2,4] => 3 = 2 + 1
[1,2,4,2] => 3 = 2 + 1
[1,4,2,2] => 3 = 2 + 1
[2,1,2,4] => 3 = 2 + 1
[2,1,4,2] => 3 = 2 + 1
[2,2,1,4] => 3 = 2 + 1
[2,2,4,1] => 3 = 2 + 1
[2,4,1,2] => 3 = 2 + 1
[2,4,2,1] => 3 = 2 + 1
[4,1,2,2] => 3 = 2 + 1
[4,2,1,2] => 3 = 2 + 1
[4,2,2,1] => 3 = 2 + 1
[1,2,3,3] => 3 = 2 + 1
[1,3,2,3] => 3 = 2 + 1
[1,3,3,2] => 3 = 2 + 1
[2,1,3,3] => 3 = 2 + 1
[2,3,1,3] => 3 = 2 + 1
[2,3,3,1] => 3 = 2 + 1
[3,1,2,3] => 3 = 2 + 1
[3,1,3,2] => 3 = 2 + 1
[3,2,1,3] => 3 = 2 + 1
[3,2,3,1] => 3 = 2 + 1
[3,3,1,2] => 3 = 2 + 1
[3,3,2,1] => 3 = 2 + 1
[1,2,3,4] => 4 = 3 + 1
[1,2,4,3] => 4 = 3 + 1
[1,1,1,4,4] => ? = 1 + 1
[1,1,4,1,4] => ? = 1 + 1
[1,1,4,4,1] => ? = 1 + 1
[1,4,1,1,4] => ? = 1 + 1
[1,4,1,4,1] => ? = 1 + 1
[1,4,4,1,1] => ? = 1 + 1
[4,1,1,1,4] => ? = 1 + 1
[4,1,1,4,1] => ? = 1 + 1
[4,1,4,1,1] => ? = 1 + 1
[4,4,1,1,1] => ? = 1 + 1
[1,1,1,4,5] => ? = 2 + 1
[1,1,1,5,4] => ? = 2 + 1
[1,1,4,1,5] => ? = 2 + 1
[1,1,4,5,1] => ? = 2 + 1
[1,1,5,1,4] => ? = 2 + 1
[1,1,5,4,1] => ? = 2 + 1
[1,4,1,1,5] => ? = 2 + 1
[1,4,1,5,1] => ? = 2 + 1
[1,4,5,1,1] => ? = 2 + 1
[1,5,1,1,4] => ? = 2 + 1
[1,5,1,4,1] => ? = 2 + 1
[1,5,4,1,1] => ? = 2 + 1
[4,1,1,1,5] => ? = 2 + 1
[4,1,1,5,1] => ? = 2 + 1
[4,1,5,1,1] => ? = 2 + 1
[4,5,1,1,1] => ? = 2 + 1
[5,1,1,1,4] => ? = 2 + 1
[5,1,1,4,1] => ? = 2 + 1
[5,1,4,1,1] => ? = 2 + 1
[5,4,1,1,1] => ? = 2 + 1
[1,1,2,4,4] => ? = 1 + 1
[1,1,4,2,4] => ? = 1 + 1
[1,1,4,4,2] => ? = 1 + 1
[1,2,1,4,4] => ? = 1 + 1
[1,2,4,1,4] => ? = 1 + 1
[1,2,4,4,1] => ? = 1 + 1
[1,4,1,2,4] => ? = 1 + 1
[1,4,1,4,2] => ? = 1 + 1
[1,4,2,1,4] => ? = 1 + 1
[1,4,2,4,1] => ? = 1 + 1
[1,4,4,1,2] => ? = 1 + 1
[1,4,4,2,1] => ? = 1 + 1
[2,1,1,4,4] => ? = 1 + 1
[2,1,4,1,4] => ? = 1 + 1
[2,1,4,4,1] => ? = 1 + 1
[2,4,1,1,4] => ? = 1 + 1
[2,4,1,4,1] => ? = 1 + 1
[2,4,4,1,1] => ? = 1 + 1
[4,1,1,2,4] => ? = 1 + 1
[4,1,1,4,2] => ? = 1 + 1
Description
The number of lucky cars of the parking function.
A lucky car is a car that was able to park in its prefered spot.
The generating function,
$$
q\prod_{i=1}^{n-1} (i + (n-i+1)q)
$$
was established in [1].
The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001712The number of natural descents of a standard Young tableau. St001946The number of descents in a parking function. 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 $eA$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000291The number of descents of a binary word. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001821The sorting index of a signed permutation. St001896The number of right descents of a signed permutations. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000015The number of peaks of a Dyck path. St000390The number of runs of ones in a binary word. St000538The number of even inversions of a permutation. St000670The reversal length of a permutation. St000702The number of weak deficiencies of a permutation. St000710The number of big deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St000991The number of right-to-left minima of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001415The length of the longest palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001424The number of distinct squares in a binary word. St001520The number of strict 3-descents. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001948The number of augmented double ascents of a permutation. St000820The number of compositions obtained by rotating the composition. St001267The length of the Lyndon factorization of the binary word. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000761The number of ascents in an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St001820The size of the image of the pop stack sorting operator. St000630The length of the shortest palindromic decomposition of a binary word. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001488The number of corners of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000335The difference of lower and upper interactions. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001645The pebbling number of a connected graph.
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