Identifier
-
Mp00068:
Permutations
—Simion-Schmidt map⟶
Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤ
Values
[1] => [1] => [.,.] => ([],1) => 0
[1,2] => [1,2] => [.,[.,.]] => ([(0,1)],2) => 1
[2,1] => [2,1] => [[.,.],.] => ([(0,1)],2) => 1
[6,4,2,1,3,5] => [6,4,2,1,5,3] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[6,4,2,1,5,3] => [6,4,2,1,5,3] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[6,4,2,3,1,5] => [6,4,2,5,1,3] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[6,4,2,3,5,1] => [6,4,2,5,3,1] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[6,4,2,5,1,3] => [6,4,2,5,1,3] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[6,4,2,5,3,1] => [6,4,2,5,3,1] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[6,4,5,2,1,3] => [6,4,5,2,1,3] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[6,4,5,2,3,1] => [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2
[6,3,1,2,4,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,2,4,7,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,2,5,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,2,5,7,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,2,7,4,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,2,7,5,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,4,2,5,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,4,2,7,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,4,5,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,4,5,7,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,4,7,2,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,4,7,5,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,5,2,4,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,5,2,7,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,5,4,2,7] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,5,4,7,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,5,7,2,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,5,7,4,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,7,2,4,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,7,2,5,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,7,4,2,5] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,7,4,5,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,7,5,2,4] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,1,7,5,4,2] => [6,3,1,7,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,1,4,5,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,1,4,7,5] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,1,5,4,7] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,1,5,7,4] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,1,7,4,5] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,1,7,5,4] => [6,3,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,4,1,5,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,4,1,7,5] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,4,5,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,4,5,7,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,4,7,1,5] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,4,7,5,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,5,1,4,7] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,5,1,7,4] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,5,4,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,5,4,7,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,5,7,1,4] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,5,7,4,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,7,1,4,5] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,7,1,5,4] => [6,3,2,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,7,4,1,5] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,7,4,5,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,7,5,1,4] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,2,7,5,4,1] => [6,3,2,7,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,1,2,5,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,1,2,7,5] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,1,5,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,1,5,7,2] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,1,7,2,5] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,1,7,5,2] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,2,1,5,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,2,1,7,5] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,2,5,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,2,5,7,1] => [6,3,7,2,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,2,7,1,5] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,2,7,5,1] => [6,3,7,2,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,5,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,5,1,7,2] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,5,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,5,2,7,1] => [6,3,7,5,2,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,5,7,1,2] => [6,3,7,5,4,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,5,7,2,1] => [6,3,7,5,4,2,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,7,1,2,5] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,7,1,5,2] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,7,2,1,5] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,7,2,5,1] => [6,3,7,5,2,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,7,5,1,2] => [6,3,7,5,4,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,4,7,5,2,1] => [6,3,7,5,4,2,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,1,2,4,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,1,2,7,4] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,1,4,2,7] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,1,4,7,2] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,1,7,2,4] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,1,7,4,2] => [6,3,7,1,5,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,2,1,4,7] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,2,1,7,4] => [6,3,7,2,1,5,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,2,4,1,7] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,2,4,7,1] => [6,3,7,2,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,2,7,1,4] => [6,3,7,2,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,2,7,4,1] => [6,3,7,2,5,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,4,1,2,7] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,4,1,7,2] => [6,3,7,5,1,4,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,4,2,1,7] => [6,3,7,5,2,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,4,2,7,1] => [6,3,7,5,2,4,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,4,7,1,2] => [6,3,7,5,4,1,2] => [[[.,[.,.]],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
[6,3,5,4,7,2,1] => [6,3,7,5,4,2,1] => [[[[.,.],.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 2
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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.
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.
Map
binary search tree: left to right
Description
Return the shape of the binary search tree of the permutation as a non labelled binary tree.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a 123-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between 132-avoiding permutations and 123-avoiding permutations, see [1, Proposition 19].
Details can be found in [1].
In particular, this is a bijection between 132-avoiding permutations and 123-avoiding permutations, see [1, Proposition 19].
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