Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
St000001: Permutations ⟶ ℤ
Values
[1,0] => [1] => 1
[1,0,1,0] => [2,1] => 1
[1,1,0,0] => [1,2] => 1
[1,0,1,0,1,0] => [2,1,3] => 1
[1,0,1,1,0,0] => [2,3,1] => 1
[1,1,0,0,1,0] => [3,1,2] => 1
[1,1,0,1,0,0] => [1,3,2] => 1
[1,1,1,0,0,0] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0] => [2,4,1,3] => 2
[1,0,1,1,0,0,1,0] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0] => [2,3,1,4] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0] => [3,1,4,2] => 2
[1,1,0,0,1,1,0,0] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0] => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => 2
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => 3
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => 5
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => 5
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => 3
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => 3
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => 3
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => 3
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => 2
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => 3
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => 5
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 5
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => 5
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => 5
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => 3
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => 2
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => 5
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 5
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => 8
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => 8
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => 16
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 16
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => 5
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => 5
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => 9
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => 14
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 14
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => 8
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 11
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => 10
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => 21
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => 21
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => 6
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => 9
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => 9
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => 4
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => 6
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => 6
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => 4
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => 4
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Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
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.
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