Identifier
-
Mp00025:
Dyck paths
—to 132-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] => [3,2,1] => 2
[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] => [2,1,3] => 1
[1,1,1,0,0,0] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [4,3,2,1] => 16
[1,0,1,0,1,1,0,0] => [3,4,2,1] => 5
[1,0,1,1,0,0,1,0] => [4,2,3,1] => 6
[1,0,1,1,0,1,0,0] => [3,2,4,1] => 3
[1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0] => [4,3,1,2] => 5
[1,1,0,0,1,1,0,0] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0] => [4,2,1,3] => 3
[1,1,0,1,0,1,0,0] => [3,2,1,4] => 2
[1,1,0,1,1,0,0,0] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 768
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 168
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 216
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 70
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 14
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 216
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 56
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 90
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 35
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 9
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 20
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 10
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 4
[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] => [5,4,3,1,2] => 168
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 42
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 56
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 21
[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] => [5,4,2,1,3] => 70
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 21
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 35
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 16
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 5
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 10
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 6
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 14
[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] => [5,3,1,2,4] => 9
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 5
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 4
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 3
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 2
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 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] => [4,1,2,3,5] => 1
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 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] => [6,5,4,3,2,1] => 292864
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => 48048
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 64064
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => 15015
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 2112
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => 68640
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => 12870
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => 21450
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => 5775
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => 990
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 3520
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => 1155
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => 288
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 42
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 64064
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 11583
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 16016
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => 4158
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 660
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => 21450
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => 4455
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => 7700
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => 2310
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => 450
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => 1540
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => 567
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => 162
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => 28
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 3520
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 825
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => 1540
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => 525
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => 120
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => 448
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => 189
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => 64
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => 14
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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 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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