Identifier
-
Mp00122:
Dyck paths
—Elizalde-Deutsch bijection⟶
Dyck paths
St000014: Dyck paths ⟶ ℤ
Values
[1,0] => [1,0] => 1
[1,0,1,0] => [1,1,0,0] => 1
[1,1,0,0] => [1,0,1,0] => 2
[1,0,1,0,1,0] => [1,1,0,0,1,0] => 3
[1,0,1,1,0,0] => [1,1,0,1,0,0] => 3
[1,1,0,0,1,0] => [1,1,1,0,0,0] => 1
[1,1,0,1,0,0] => [1,0,1,1,0,0] => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 6
[1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 6
[1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 6
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => 12
[1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 12
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 12
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => 4
[1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0] => 4
[1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => 12
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 12
[1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0] => 4
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => 4
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 12
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 24
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 30
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 30
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 30
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 30
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 10
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 10
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 30
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 30
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 30
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 30
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 60
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 60
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 60
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 60
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 20
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 5
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 20
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 5
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 5
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 10
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 5
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 10
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 30
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 30
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 20
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 60
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 60
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 60
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 10
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 20
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 20
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 5
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => 20
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 20
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 60
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 60
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => 20
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 20
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 20
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 60
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 120
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 90
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 90
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 90
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 90
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 60
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 180
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 180
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 180
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 180
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 60
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 180
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 180
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 60
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 60
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 60
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 180
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 15
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 90
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 15
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 60
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 180
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 180
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 180
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 180
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 180
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 180
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 180
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 60
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 60
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 180
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 60
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 60
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 180
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 180
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 180
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => 180
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 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
Elizalde-Deutsch bijection
Description
The Elizalde-Deutsch bijection on Dyck paths.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.
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