Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000235: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [1,2] => 2
[1,1,0,0] => [1,2] => [2,1] => 0
[1,0,1,0,1,0] => [2,1,3] => [3,2,1] => 2
[1,0,1,1,0,0] => [2,3,1] => [1,2,3] => 3
[1,1,0,0,1,0] => [3,1,2] => [3,1,2] => 3
[1,1,0,1,0,0] => [1,3,2] => [2,1,3] => 2
[1,1,1,0,0,0] => [1,2,3] => [2,3,1] => 0
[1,0,1,0,1,0,1,0] => [2,1,4,3] => [3,2,1,4] => 4
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [4,2,1,3] => 4
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [3,2,4,1] => 2
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [4,2,3,1] => 3
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 4
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [3,1,2,4] => 4
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,2,3] => 4
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [3,4,2,1] => 3
[1,1,0,1,0,1,0,0] => [1,3,2,4] => [2,4,3,1] => 2
[1,1,0,1,1,0,0,0] => [1,3,4,2] => [2,1,3,4] => 3
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [3,4,1,2] => 4
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [2,4,1,3] => 3
[1,1,1,0,1,0,0,0] => [1,2,4,3] => [2,3,1,4] => 2
[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] => [2,1,4,3,5] => [3,2,5,4,1] => 4
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [4,2,5,3,1] => 4
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [3,2,1,4,5] => 5
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [4,2,1,3,5] => 5
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [5,2,1,3,4] => 5
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [3,2,5,1,4] => 5
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [4,2,5,1,3] => 5
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [3,2,4,1,5] => 4
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [4,2,3,1,5] => 5
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5,2,3,1,4] => 5
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [3,2,4,5,1] => 2
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [4,2,3,5,1] => 3
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [5,2,3,4,1] => 4
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 5
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [3,5,2,4,1] => 4
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [4,5,2,3,1] => 4
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [3,1,2,4,5] => 5
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [4,1,2,3,5] => 5
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => 5
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [3,5,2,1,4] => 5
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [4,5,2,1,3] => 5
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [3,4,2,1,5] => 5
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [2,4,3,1,5] => 4
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [2,5,3,1,4] => 4
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [3,4,2,5,1] => 3
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [2,4,3,5,1] => 2
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [2,5,3,4,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [2,1,3,4,5] => 4
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [3,5,1,2,4] => 5
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [4,5,1,2,3] => 5
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [3,4,1,2,5] => 5
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [2,4,1,3,5] => 4
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [2,5,1,3,4] => 4
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [3,4,5,2,1] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [2,4,5,3,1] => 3
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [2,3,5,4,1] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [2,3,1,4,5] => 3
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [3,4,5,1,2] => 5
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [2,4,5,1,3] => 4
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [2,3,5,1,4] => 3
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [2,3,4,1,5] => 2
[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] => [2,1,4,3,6,5] => [3,2,5,4,1,6] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [4,2,5,3,1,6] => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [3,2,6,4,1,5] => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [4,2,6,3,1,5] => 6
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [5,2,6,3,1,4] => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [3,2,5,4,6,1] => 4
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [4,2,5,3,6,1] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [3,2,6,4,5,1] => 5
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [4,2,6,3,5,1] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [5,2,6,3,4,1] => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [3,2,1,4,5,6] => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [4,2,1,3,5,6] => 6
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [5,2,1,3,4,6] => 6
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [6,2,1,3,4,5] => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [3,2,5,1,4,6] => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [4,2,5,1,3,6] => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [3,2,6,1,4,5] => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [4,2,6,1,3,5] => 6
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [5,2,6,1,3,4] => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [3,2,5,6,4,1] => 5
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => [4,2,5,6,3,1] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [3,2,4,6,5,1] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [4,2,3,6,5,1] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [5,2,3,6,4,1] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => [3,2,4,1,5,6] => 5
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [4,2,3,1,5,6] => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [5,2,3,1,4,6] => 6
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => [6,2,3,1,4,5] => 6
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [3,2,5,6,1,4] => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [4,2,5,6,1,3] => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [3,2,4,6,1,5] => 5
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [4,2,3,6,1,5] => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => [5,2,3,6,1,4] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [3,2,4,5,1,6] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [4,2,3,5,1,6] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [5,2,3,4,1,6] => 6
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 6
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Description
The number of indices that are not cyclical small weak excedances.
A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
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
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
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