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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000658
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 0
Description
The number of rises of length 2 of a Dyck path.
This is also the number of (1,1) steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St000454
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 50%
Values
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,5,6,7,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,5,3,4,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,3,7,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,1,5,7,3,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,5,1,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,1,6,7,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,5,6,1,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,5,1,3,4,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,5,1,3,7,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,5,1,3,4,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,6,2,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,6,7,2,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,4,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,2,5,3,6,7,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,2,5,3,6,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,3,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,2,5,3,7,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 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.
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