Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000061: Binary trees ⟶ ℤ
Values
[1,0,1,0] => [2,1] => [2,1] => [[.,.],.] => 2
[1,1,0,0] => [1,2] => [1,2] => [.,[.,.]] => 1
[1,0,1,0,1,0] => [3,2,1] => [3,2,1] => [[[.,.],.],.] => 3
[1,0,1,1,0,0] => [2,3,1] => [2,3,1] => [[.,.],[.,.]] => 2
[1,1,0,0,1,0] => [3,1,2] => [1,3,2] => [.,[[.,.],.]] => 1
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => [[.,.],[.,.]] => 2
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => [.,[.,[.,.]]] => 1
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.] => 4
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,4,2,1] => [[[.,.],.],[.,.]] => 3
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [2,4,3,1] => [[.,.],[[.,.],.]] => 2
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => [[[.,.],.],[.,.]] => 3
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]] => 2
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,4,3,2] => [.,[[[.,.],.],.]] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [3,1,4,2] => [[.,[.,.]],[.,.]] => 2
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [2,1,4,3] => [[.,.],[[.,.],.]] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]] => 3
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]] => 2
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]] => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 1
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => 5
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]] => 4
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]] => 3
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]] => 4
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]] => 3
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]] => 2
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]] => 3
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]] => 3
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]] => 4
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]] => 3
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]] => 2
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]] => 2
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]] => 3
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]] => 2
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => 1
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [4,1,5,3,2] => [[.,[[.,.],.]],[.,.]] => 2
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [3,1,5,4,2] => [[.,[.,.]],[[.,.],.]] => 2
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [4,3,1,5,2] => [[[.,[.,.]],.],[.,.]] => 3
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => 2
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [4,2,1,5,3] => [[[.,.],[.,.]],[.,.]] => 3
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => 3
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => 4
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]] => 3
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]] => 2
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]] => 2
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]] => 2
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]] => 1
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => 1
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]] => 2
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => 2
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => 3
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]] => 2
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 1
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [5,6,4,3,2,1] => [[[[[.,.],.],.],.],[.,.]] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [4,6,5,3,2,1] => [[[[.,.],.],.],[[.,.],.]] => 4
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [5,4,6,3,2,1] => [[[[[.,.],.],.],.],[.,.]] => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [4,5,6,3,2,1] => [[[[.,.],.],.],[.,[.,.]]] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => [[[.,.],.],[[[.,.],.],.]] => 3
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [5,3,6,4,2,1] => [[[[.,.],.],[.,.]],[.,.]] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [4,3,6,5,2,1] => [[[[.,.],.],.],[[.,.],.]] => 4
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [5,4,3,6,2,1] => [[[[[.,.],.],.],.],[.,.]] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [4,5,3,6,2,1] => [[[[.,.],.],.],[.,[.,.]]] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [3,4,6,5,2,1] => [[[.,.],.],[.,[[.,.],.]]] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [3,5,4,6,2,1] => [[[.,.],.],[[.,.],[.,.]]] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [4,3,5,6,2,1] => [[[[.,.],.],.],[.,[.,.]]] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [3,4,5,6,2,1] => [[[.,.],.],[.,[.,[.,.]]]] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [2,6,5,4,3,1] => [[.,.],[[[[.,.],.],.],.]] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [5,2,6,4,3,1] => [[[.,.],[[.,.],.]],[.,.]] => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [4,2,6,5,3,1] => [[[.,.],[.,.]],[[.,.],.]] => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [5,4,2,6,3,1] => [[[[.,.],[.,.]],.],[.,.]] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [4,5,2,6,3,1] => [[[.,.],[.,.]],[.,[.,.]]] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [3,2,6,5,4,1] => [[[.,.],.],[[[.,.],.],.]] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [5,3,2,6,4,1] => [[[[.,.],.],[.,.]],[.,.]] => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [4,3,2,6,5,1] => [[[[.,.],.],.],[[.,.],.]] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [5,4,3,2,6,1] => [[[[[.,.],.],.],.],[.,.]] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [4,5,3,2,6,1] => [[[[.,.],.],.],[.,[.,.]]] => 4
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [3,4,2,6,5,1] => [[[.,.],.],[.,[[.,.],.]]] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [3,5,4,2,6,1] => [[[.,.],.],[[.,.],[.,.]]] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [4,3,5,2,6,1] => [[[[.,.],.],.],[.,[.,.]]] => 4
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [3,4,5,2,6,1] => [[[.,.],.],[.,[.,[.,.]]]] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [2,3,6,5,4,1] => [[.,.],[.,[[[.,.],.],.]]] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [2,5,3,6,4,1] => [[.,.],[[.,[.,.]],[.,.]]] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [2,4,3,6,5,1] => [[.,.],[[.,.],[[.,.],.]]] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [2,5,4,3,6,1] => [[.,.],[[[.,.],.],[.,.]]] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [4,2,5,3,6,1] => [[[.,.],[.,.]],[.,[.,.]]] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [3,2,4,6,5,1] => [[[.,.],.],[.,[[.,.],.]]] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [3,2,5,4,6,1] => [[[.,.],.],[[.,.],[.,.]]] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [4,3,2,5,6,1] => [[[[.,.],.],.],[.,[.,.]]] => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [3,4,2,5,6,1] => [[[.,.],.],[.,[.,[.,.]]]] => 3
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => [2,3,4,6,5,1] => [[.,.],[.,[.,[[.,.],.]]]] => 2
>>> Load all 195 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 nodes on the left branch of a binary tree.
Also corresponds to ST000011 after applying the Tamari bijection between binary trees and Dyck path.
Also corresponds to ST000011 after applying the Tamari bijection between binary trees and Dyck path.
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
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
See Mp00067Foata bijection.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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!