Identifier
-
Mp00099:
Dyck paths
—bounce path⟶
Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001766: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [1] => [1] => 0
[1,0,1,0] => [1,0,1,0] => [2,1] => [1,2] => 0
[1,1,0,0] => [1,1,0,0] => [1,2] => [2,1] => 0
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [2,1,3] => [1,3,2] => 2
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 0
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [2,3,1] => [1,2,3] => 0
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,2,3] => [2,3,1] => 0
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [1,4,3,2] => 5
[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [1,4,2,3] => 4
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [1,3,4,2] => 4
[1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [1,4,2,3] => 4
[1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => [4,1,3,2] => 2
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,2,3] => 0
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [1,3,4,2] => 4
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,2,3] => 0
[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [3,4,1,2] => 0
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,2,3] => 0
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [1,4,3,5,2] => 7
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [1,4,2,5,3] => 7
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [1,5,3,4,2] => 9
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [1,4,2,5,3] => 7
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [1,5,2,3,4] => 6
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [1,4,5,3,2] => 8
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [1,4,5,2,3] => 7
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [1,5,3,4,2] => 9
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [1,4,5,2,3] => 7
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [1,5,2,3,4] => 6
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [1,3,4,5,2] => 6
[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [1,4,5,2,3] => 7
[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [1,5,2,3,4] => 6
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [4,1,3,5,2] => 4
[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [4,1,2,5,3] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [5,1,3,4,2] => 4
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [4,1,2,5,3] => 3
[1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => 0
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [1,4,5,3,2] => 8
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [1,4,5,2,3] => 7
[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [5,1,3,4,2] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [1,4,5,2,3] => 7
[1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => 0
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [1,3,4,5,2] => 6
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [1,4,5,2,3] => 7
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [4,5,1,3,2] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [5,1,3,4,2] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => 0
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => 0
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [1,3,4,5,2] => 6
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [3,4,5,1,2] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [1,4,3,6,5,2] => 12
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [1,4,2,6,5,3] => 12
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [1,5,3,6,4,2] => 13
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [1,4,2,6,5,3] => 12
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [1,5,2,6,3,4] => 11
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [1,4,3,5,6,2] => 9
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [1,4,2,5,6,3] => 10
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [1,5,3,6,4,2] => 13
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [1,4,2,5,6,3] => 10
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [1,5,2,6,3,4] => 11
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [1,6,3,4,5,2] => 13
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [1,4,2,5,6,3] => 10
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [1,5,2,6,3,4] => 11
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [1,6,2,3,4,5] => 8
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [1,4,6,3,5,2] => 13
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,4,6,2,5,3] => 12
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [1,5,6,3,4,2] => 13
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [1,4,6,2,5,3] => 12
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [1,5,6,2,3,4] => 10
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [1,4,3,5,6,2] => 9
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [1,4,2,5,6,3] => 10
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [1,5,6,3,4,2] => 13
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [1,4,2,5,6,3] => 10
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [1,5,6,2,3,4] => 10
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [1,6,3,4,5,2] => 13
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [1,4,2,5,6,3] => 10
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [1,5,6,2,3,4] => 10
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [1,6,2,3,4,5] => 8
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [1,4,5,6,3,2] => 11
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [1,4,5,6,2,3] => 10
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [1,5,6,3,4,2] => 13
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [1,4,5,6,2,3] => 10
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [1,5,6,2,3,4] => 10
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [1,6,3,4,5,2] => 13
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [1,4,5,6,2,3] => 10
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [1,5,6,2,3,4] => 10
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [1,6,2,3,4,5] => 8
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searching the database for statistics with the same generating function
Description
The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation.
Precisely the 132-avoiding permutations have only one associated reduced pipe dream.
Precisely the 132-avoiding permutations have only one associated reduced pipe dream.
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
Map
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
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