- FindStat

Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000741: Graphs ⟶ ℤ

Values

[1,0] => [1] => [.,.] => ([],1) => 0
[1,0,1,0] => [1,2] => [.,[.,.]] => ([(0,1)],2) => 1
[1,1,0,0] => [2,1] => [[.,.],.] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,2,3] => [.,[.,[.,.]]] => ([(0,2),(1,2)],3) => 2
[1,0,1,1,0,0] => [1,3,2] => [.,[[.,.],.]] => ([(0,2),(1,2)],3) => 2
[1,1,0,0,1,0] => [2,1,3] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => 2
[1,1,0,1,0,0] => [2,3,1] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => 2
[1,1,1,0,0,0] => [3,2,1] => [[[.,.],.],.] => ([(0,2),(1,2)],3) => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [.,[.,[[.,.],.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(2,3)],4) => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(2,3)],4) => 2
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [.,[[[.,.],.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => 2
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [[[[.,.],.],.],.] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => [[.,.],[.,[[[.,.],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => [[.,.],[[[[.,.],.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,3,1,6,5,4] => [[.,.],[.,[[[.,.],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => [[.,.],[.,[.,[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => [[.,.],[.,[.,[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => [[.,.],[.,[.,[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [[.,.],[.,[.,[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,5,1] => [[.,.],[.,[.,[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => [[.,.],[.,[[.,[.,.]],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [[.,.],[.,[[[.,.],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => [[.,.],[[.,[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,6,3,5,4,1] => [[.,.],[[.,[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [[.,.],[[[[.,.],.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [[[.,.],.],[.,[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => [[[.,.],.],[.,[[.,.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => [[[.,.],.],[[[.,.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => [[[.,.],.],[.,[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => [[[.,.],.],[.,[[.,.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => [[[.,.],.],[.,[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [[[.,.],.],[.,[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,2,4,6,5,1] => [[[.,.],.],[.,[[.,.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,6,4,5,1] => [[[.,.],.],[[.,[.,.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [[[.,.],.],[[[.,.],.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => [[[[.,.],.],.],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => [[[[.,.],.],.],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => [[[[.,.],.],.],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [[[[.,.],.],.],[.,[.,.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [[[[.,.],.],.],[[.,.],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [[[[[.,.],.],.],.],[.,.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1

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$F_{1} = 1$

$F_{2} = 2\ q$

$F_{3} = 5\ q^{2}$

$F_{4} = 11\ q + 3\ q^{2}$

Description

The Colin de Verdière graph invariant.

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

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..