- FindStat

Identifier

Mp00297: Parking functions —ordered tree⟶ Ordered trees
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000991: Permutations ⟶ ℤ

Values

[1] => [[]] => [.,.] => [1] => 1

[1,1] => [[],[]] => [[.,.],.] => [1,2] => 2

[1,2] => [[[]]] => [.,[.,.]] => [2,1] => 1

[2,1] => [[[]]] => [.,[.,.]] => [2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

$F_{4} = 64\ q + 48\ q^{2} + 12\ q^{3} + q^{4}$

$F_{5} = 625\ q + 500\ q^{2} + 150\ q^{3} + 20\ q^{4} + q^{5}$

$F_{6} = 7776\ q + 6480\ q^{2} + 2160\ q^{3} + 360\ q^{4} + 30\ q^{5} + q^{6}$

Description

The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see St000314The number of left-to-right-maxima of a permutation..

Map

ordered tree

Description

The ordered tree associated with the parking function by Pollak.
Let

$(p_1,\dots,p_n)$ be a parking function. Set

$c_i = p_{i+1}-p_i \mod (n+1)$ to obtain a Prüfer code, and return the underlying ordered tree rooted at

$0$ .

Map

to binary tree: left brother = left child

Description

Return a binary tree of size

$n-1$ (where

$n$ is the size of

$t$ , and where

$t$ is an ordered tree) by the following recursive rule:
- if

$x$ is the left brother of

$y$ in

$t$ , then

$x$ becomes the left child of

$y$ ;
- if

$x$ is the last child of

$y$ in

$t$ , then

$x$ becomes the right child of

$y$ ,
and removing the root of

$t$ .

Map

to 132-avoiding permutation

Description

Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.