- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤ

Values

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

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

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

[1,2,3] => [.,[.,[.,.]]] => [3,2,1] => 6

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

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

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

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

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

[1,2,3,4] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => 24

[1,2,4,3] => [.,[.,[[.,.],.]]] => [3,4,2,1] => 12

[1,3,2,4] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 8

[1,3,4,2] => [.,[[.,[.,.]],.]] => [3,2,4,1] => 8

[1,4,2,3] => [.,[[.,.],[.,.]]] => [2,4,3,1] => 8

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

[2,1,3,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 6

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

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

[2,3,4,1] => [[.,[.,[.,.]]],.] => [3,2,1,4] => 6

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

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

[3,1,2,4] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 6

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

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

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

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

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

[4,1,2,3] => [[.,.],[.,[.,.]]] => [1,4,3,2] => 6

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

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

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

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

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

[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 120

[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 60

[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 40

[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 40

[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 40

[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 20

[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 30

[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 15

[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 20

[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 30

[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 20

[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 15

[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 30

[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 15

[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 10

[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 10

[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 20

[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 10

[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 30

[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 15

[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 10

[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 10

[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 10

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

[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 24

[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 12

[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 8

[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 8

[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 8

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

[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 12

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

[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 12

[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 24

[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 12

[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 12

[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 12

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

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

[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 8

[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 12

[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 8

[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 12

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

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

[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 8

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

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

[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 24

[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 12

[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 8

[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 8

[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 8

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

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

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

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

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

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

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

[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 12

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

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

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

[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 12

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

[3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 12

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

>>> Load all 2365 entries. <<<

search for individual values

searching the database for the individual values of this statistic

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searching the database for statistics with the same generating function

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$F_{1} = q$

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

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

$F_{4} = q + 6\ q^{2} + 4\ q^{3} + 4\ q^{4} + 4\ q^{6} + 3\ q^{8} + q^{12} + q^{24}$

$F_{5} = q + 10\ q^{2} + 10\ q^{3} + 20\ q^{4} + q^{5} + 20\ q^{6} + 15\ q^{8} + 6\ q^{10} + 15\ q^{12} + 4\ q^{15} + 4\ q^{20} + 5\ q^{24} + 4\ q^{30} + 3\ q^{40} + q^{60} + q^{120}$

$F_{6} = q + 15\ q^{2} + 20\ q^{3} + 60\ q^{4} + 6\ q^{5} + 81\ q^{6} + 75\ q^{8} + 10\ q^{9} + 36\ q^{10} + 85\ q^{12} + 24\ q^{15} + 45\ q^{16} + 30\ q^{18} + 24\ q^{20} + 50\ q^{24} + 25\ q^{30} + 30\ q^{36} + 18\ q^{40} + 30\ q^{48} + 12\ q^{60} + 15\ q^{72} + 4\ q^{90} + 10\ q^{120} + 5\ q^{144} + 4\ q^{180} + 3\ q^{240} + q^{360} + q^{720}$

Description

The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length

$0$ to the empty tree, and sending a permutation

$\sigma$ of length

$n \geq 1$ to a root node with two subtrees

$L$ and

$R$ by splitting

$\sigma$ at the index

$\sigma^{-1}(1)$ , normalizing both sides again to permutations and sending the permutations on the left and on the right of

$\sigma^{-1}(1)$ to the trees

$L$ and

$R$ , respectively.

Map

to 312-avoiding permutation

Description

Return a 312-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 minimal element of this Sylvester class.