Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
St000407: Permutations ⟶ ℤ
Values
[.,.] => [1] => 0
[.,[.,.]] => [2,1] => 0
[[.,.],.] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => 0
[[.,.],[.,.]] => [1,3,2] => 0
[[.,[.,.]],.] => [2,1,3] => 0
[[[.,.],.],.] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => 0
[.,[[.,.],[.,.]]] => [2,4,3,1] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => 0
[[.,[.,.]],[.,.]] => [2,1,4,3] => 1
[[[.,.],.],[.,.]] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => 0
[[[.,.],[.,.]],.] => [1,3,2,4] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 0
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 1
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 0
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 0
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 0
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 0
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 3
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 0
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 3
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 1
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 1
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 0
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 0
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 0
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 1
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 0
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 0
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 0
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 0
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 0
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 0
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 0
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 0
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 0
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 3
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 2
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 0
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 3
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 2
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 1
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 0
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 0
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 1
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 0
>>> Load all 305 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The number of occurrences of the pattern 2143 in a permutation.
A permutation $\pi$ avoids this pattern if and only if it is vexillary as introduced in [1].
A permutation $\pi$ avoids this pattern if and only if it is vexillary as introduced in [1].
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.
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!