- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001461: Permutations ⟶ ℤ

Values

[[]] => [1,0] => [1] => 1

[[],[]] => [1,0,1,0] => [1,2] => 2

[[[]]] => [1,1,0,0] => [2,1] => 1

[[],[],[]] => [1,0,1,0,1,0] => [1,2,3] => 3

[[],[[]]] => [1,0,1,1,0,0] => [1,3,2] => 2

[[[]],[]] => [1,1,0,0,1,0] => [2,1,3] => 2

[[[],[]]] => [1,1,0,1,0,0] => [2,3,1] => 1

[[[[]]]] => [1,1,1,0,0,0] => [3,1,2] => 1

[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 4

[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 3

[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 3

[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 2

[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => 2

[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 3

[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 2

[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 2

[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [3,1,2,4] => 2

[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 1

[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [2,4,1,3] => 1

[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => 1

[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => 1

[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => 1

[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 5

[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 4

[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 4

[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 3

[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 3

[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 4

[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 3

[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 3

[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => 3

[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 2

[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => 2

[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 2

[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => 2

[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => 2

[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 4

[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 3

[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 3

[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 2

[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => 2

[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 3

[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => 3

[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 2

[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => 2

[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 2

[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => 2

[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => 2

[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => 2

[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => 2

[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 1

[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => 1

[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => 1

[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => 1

[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => 1

[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 1

[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => 1

[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => 1

[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => 1

[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => 1

[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => 1

[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => 1

[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => 1

[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = q$

$F_{3} = q + q^{2}$

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

$F_{5} = 5\ q + 5\ q^{2} + 3\ q^{3} + q^{4}$

$F_{6} = 14\ q + 14\ q^{2} + 9\ q^{3} + 4\ q^{4} + q^{5}$

$F_{7} = 42\ q + 42\ q^{2} + 28\ q^{3} + 14\ q^{4} + 5\ q^{5} + q^{6}$

Description

The number of topologically connected components of the chord diagram of a permutation.
The chord diagram of a permutation

$\pi\in\mathfrak S_n$ is obtained by placing labels

$1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from

$i$ to

$\pi(i)$ for every label

$i$ .
This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component.
The permutation

$\pi\in\mathfrak S_n$ stabilizes an interval

$I=\{a,a+1,\dots,b\}$ if

$\pi(I)=I$ . It is stabilized-interval-free, if the only interval

$\pi$ stablizes is

$\{1,\dots,n\}$ . Thus, this statistic is

$1$ if

$\pi$ is stabilized-interval-free.

Map

to 321-avoiding permutation (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength

$n$ in an

$n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.

Map

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.