Identifier
-
Mp00143:
Dyck paths
—inverse promotion⟶
Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001683: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [1] => [1] => 0
[1,0,1,0] => [1,1,0,0] => [1,2] => [2,1] => 0
[1,1,0,0] => [1,0,1,0] => [2,1] => [1,2] => 0
[1,0,1,0,1,0] => [1,1,0,1,0,0] => [1,3,2] => [2,3,1] => 0
[1,0,1,1,0,0] => [1,1,1,0,0,0] => [1,2,3] => [3,2,1] => 0
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [2,3,1] => [1,3,2] => 1
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [2,1,3] => [3,1,2] => 0
[1,1,1,0,0,0] => [1,1,0,0,1,0] => [3,1,2] => [2,1,3] => 0
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [1,3,2,4] => [4,2,3,1] => 0
[1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,3,4,2] => [2,4,3,1] => 1
[1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [3,2,4,1] => 0
[1,0,1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,2,4,3] => [3,4,2,1] => 0
[1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => [4,1,3,2] => 1
[1,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,4,3,2] => 2
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [3,4,1,2] => 0
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [4,3,1,2] => 0
[1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [2,1,4,3] => 1
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => [2,4,1,3] => 1
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,0,1,0] => [3,1,2,4] => [4,2,1,3] => 0
[1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [3,2,1,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [2,5,4,3,1] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [3,5,2,4,1] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [5,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [4,3,2,5,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [4,3,5,2,1] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [4,5,1,3,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [4,1,5,3,2] => 2
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [5,4,1,3,2] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [5,1,4,3,2] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,5,4,3,2] => 3
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [3,5,1,4,2] => 2
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [3,1,5,4,2] => 2
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [3,5,4,1,2] => 1
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [4,3,1,5,2] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [4,3,5,1,2] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [2,5,1,4,3] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [2,1,5,4,3] => 2
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [5,2,1,4,3] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [5,2,4,1,3] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [2,5,4,1,3] => 2
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [4,2,1,5,3] => 1
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [4,2,5,1,3] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [4,5,2,1,3] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [5,4,2,1,3] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [3,2,1,5,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [3,2,5,1,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [3,5,2,1,4] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [5,3,2,1,4] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [4,3,2,1,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4,6] => [6,4,2,5,3,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,3,2,5,6,4] => [4,6,5,2,3,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,5,2,6,4] => [4,6,2,5,3,1] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,3,5,6,2,4] => [4,2,6,5,3,1] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [1,3,2,6,4,5] => [5,4,6,2,3,1] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,3,6,2,4,5] => [5,4,2,6,3,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,3,2,4,6,5] => [5,6,4,2,3,1] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,3,4,2,6,5] => [5,6,2,4,3,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,3,4,6,2,5] => [5,2,6,4,3,1] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,4,2,5,6] => [6,5,2,4,3,1] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,4,5,2,6] => [6,2,5,4,3,1] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,3,4,5,6,2] => [2,6,5,4,3,1] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => [6,3,2,5,4,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,5,6,3] => [3,6,5,2,4,1] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,5,2,6,3] => [3,6,2,5,4,1] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,5,6,2,3] => [3,2,6,5,4,1] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5] => [5,3,6,2,4,1] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [1,4,6,2,3,5] => [5,3,2,6,4,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,2,3,6,5] => [5,6,3,2,4,1] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,2,4,5,6,3] => [3,6,5,4,2,1] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,2,6,3,4] => [4,3,6,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,5,6,2,3,4] => [4,3,2,6,5,1] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,3,6,4] => [4,6,3,2,5,1] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,2,5,3,6,4] => [4,6,3,5,2,1] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,2,5,6,3,4] => [4,3,6,5,2,1] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,6] => [6,4,3,2,5,1] => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => 1
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Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
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
inverse promotion
Description
The inverse promotion of a Dyck path.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.
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