Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
St000014: Dyck paths ⟶ ℤ
Values
[.,.] => [1,0] => 1
[.,[.,.]] => [1,1,0,0] => 1
[[.,.],.] => [1,0,1,0] => 2
[.,[.,[.,.]]] => [1,1,1,0,0,0] => 1
[.,[[.,.],.]] => [1,1,0,1,0,0] => 3
[[.,.],[.,.]] => [1,0,1,1,0,0] => 3
[[.,[.,.]],.] => [1,1,0,0,1,0] => 3
[[[.,.],.],.] => [1,0,1,0,1,0] => 6
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 1
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 4
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 6
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => 4
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => 12
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 4
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 12
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 6
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => 12
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => 4
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => 12
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => 12
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => 12
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => 24
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 1
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 5
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 10
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 5
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => 20
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 10
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 30
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 10
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 30
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => 5
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => 20
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => 30
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 20
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => 60
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 5
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 20
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 30
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 20
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => 60
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 10
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 30
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => 20
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => 60
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 10
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => 30
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => 30
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => 30
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => 60
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 5
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => 20
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => 30
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => 20
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => 60
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => 20
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => 60
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => 30
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => 60
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => 20
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => 60
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => 60
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => 60
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => 120
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => 6
[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 15
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => 6
[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => 30
[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => 20
[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => 60
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 15
[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => 60
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => 6
[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => 30
[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => 60
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => 30
[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => 120
[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => 15
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 60
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 90
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 60
[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => 180
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => 20
[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => 60
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 60
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => 180
[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 15
[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => 60
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 90
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => 60
[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => 180
[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => 6
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 30
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 60
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 30
[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => 120
[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => 60
[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => 180
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 60
[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => 180
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Description
The number of parking functions supported by a Dyck path.
One representation of a parking function is as a pair consisting of a Dyck path and a permutation $\pi$ such that if $[a_0, a_1, \dots, a_{n-1}]$ is the area sequence of the Dyck path then the permutation $\pi$ satisfies $pi_i < pi_{i+1}$ whenever $a_{i} < a_{i+1}$. This statistic counts the number of permutations $\pi$ which satisfy this condition.
One representation of a parking function is as a pair consisting of a Dyck path and a permutation $\pi$ such that if $[a_0, a_1, \dots, a_{n-1}]$ is the area sequence of the Dyck path then the permutation $\pi$ satisfies $pi_i < pi_{i+1}$ whenever $a_{i} < a_{i+1}$. This statistic counts the number of permutations $\pi$ which satisfy this condition.
Map
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
The bijection is defined recursively as follows:
- a leaf is associated with an empty Dyck path,
- a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.
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