Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001640: Permutations ⟶ ℤ
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [1,2] => 1
[1,1,0,0] => [1,2] => [2,1] => 0
[1,0,1,0,1,0] => [2,1,3] => [3,2,1] => 0
[1,0,1,1,0,0] => [2,3,1] => [3,1,2] => 1
[1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 0
[1,1,0,1,0,0] => [1,3,2] => [2,1,3] => 1
[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] => 1
[1,0,1,0,1,1,0,0] => [2,4,1,3] => [3,1,4,2] => 0
[1,0,1,1,0,0,1,0] => [2,1,3,4] => [3,2,4,1] => 0
[1,0,1,1,0,1,0,0] => [2,3,1,4] => [3,4,2,1] => 0
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [3,4,1,2] => 1
[1,1,0,0,1,0,1,0] => [3,1,4,2] => [4,2,1,3] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,1,3,2] => 0
[1,1,0,1,0,0,1,0] => [3,1,2,4] => [4,2,3,1] => 0
[1,1,0,1,0,1,0,0] => [1,3,2,4] => [2,4,3,1] => 0
[1,1,0,1,1,0,0,0] => [1,3,4,2] => [2,4,1,3] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,3,4,2] => 0
[1,1,1,0,0,1,0,0] => [1,4,2,3] => [2,1,4,3] => 0
[1,1,1,0,1,0,0,0] => [1,2,4,3] => [2,3,1,4] => 1
[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] => 0
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [3,5,2,4,1] => 0
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [3,2,5,1,4] => 1
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [3,5,2,1,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [3,5,1,4,2] => 0
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [3,2,1,5,4] => 0
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [3,1,4,5,2] => 0
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [3,2,4,1,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [3,4,2,1,5] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [3,4,1,5,2] => 0
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [3,2,4,5,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [3,4,5,1,2] => 1
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [4,2,5,3,1] => 0
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [4,2,5,1,3] => 1
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [4,5,2,1,3] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [4,5,1,3,2] => 0
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [4,2,1,5,3] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [4,1,3,5,2] => 0
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [4,2,3,1,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [2,4,3,1,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [2,4,1,5,3] => 0
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [4,2,3,5,1] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [2,4,3,5,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [2,4,5,3,1] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [2,4,5,1,3] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [5,2,1,4,3] => 0
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5,1,3,4,2] => 0
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [5,2,3,1,4] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [2,5,3,1,4] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [2,5,1,4,3] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [5,2,3,4,1] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [2,5,3,4,1] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [2,3,5,4,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [2,3,5,1,4] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,3,4,5,2] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [2,1,4,5,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [2,3,1,5,4] => 0
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [2,3,4,1,5] => 1
[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] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [3,5,2,4,1,6] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [3,2,5,1,6,4] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [3,5,2,1,6,4] => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [3,5,1,4,6,2] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [3,2,5,4,6,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [3,5,2,4,6,1] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [3,2,5,6,4,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [3,5,2,6,4,1] => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [3,5,6,2,4,1] => 0
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [3,2,5,6,1,4] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [3,5,2,6,1,4] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [3,5,6,2,1,4] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [3,5,6,1,4,2] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [3,2,6,4,1,5] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,2,4,1,5] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [3,2,6,1,5,4] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [3,6,2,1,5,4] => 0
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [3,6,1,4,5,2] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [3,2,6,4,5,1] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => [3,6,2,4,5,1] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [3,2,4,6,5,1] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [3,4,2,6,5,1] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [3,4,6,2,5,1] => 0
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => [3,2,4,6,1,5] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [3,4,2,6,1,5] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [3,4,6,2,1,5] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => [3,4,6,1,5,2] => 0
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [3,2,1,5,6,4] => 0
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => 0
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [3,2,4,1,6,5] => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [3,4,2,1,6,5] => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => [3,4,1,5,6,2] => 0
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [3,2,4,5,1,6] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [3,4,2,5,1,6] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [3,4,5,2,1,6] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [3,4,5,1,6,2] => 0
>>> Load all 197 entries. <<<
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Description
The number of ascent tops in the permutation such that all smaller elements appear before.
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
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.
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