Identifier
-
Mp00025:
Dyck paths
—to 132-avoiding permutation⟶
Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000021: Permutations ⟶ ℤ (values match St000325The width of the tree associated to a permutation., St000470The number of runs in a permutation.)
Values
[1,0] => [1] => [1] => 0
[1,0,1,0] => [2,1] => [1,2] => 0
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [3,2,1] => [1,3,2] => 1
[1,0,1,1,0,0] => [2,3,1] => [1,2,3] => 0
[1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 1
[1,1,0,1,0,0] => [2,1,3] => [1,2,3] => 0
[1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,4,2,3] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [1,4,2,3] => 1
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [1,4,2,3] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [1,4,3,2] => 2
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [1,4,2,5,3] => 2
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [1,5,2,3,4] => 1
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [1,4,2,3,5] => 1
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [1,3,5,2,4] => 1
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [1,5,2,4,3] => 2
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [1,4,3,2,5] => 2
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [1,5,2,3,4] => 1
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [1,4,5,2,3] => 1
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [1,3,2,4,5] => 1
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [1,5,2,3,4] => 1
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [1,4,5,2,3] => 1
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [1,3,4,5,2] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [1,4,2,5,3] => 2
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [1,4,2,3,5] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [1,3,5,2,4] => 1
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [1,5,3,2,4] => 2
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [1,4,2,5,3] => 2
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [1,4,2,3,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [1,3,4,2,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [1,5,3,2,4] => 2
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,4,2,5,3] => 2
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [1,5,4,2,3] => 2
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [1,4,2,3,5] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [1,3,2,4,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [1,4,3,2,5] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 3
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [1,4,3,2,5] => 2
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => [1,5,2,6,3,4] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => [1,6,2,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => [1,5,2,4,3,6] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => [1,4,3,6,2,5] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => [1,6,2,5,3,4] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => [1,5,2,6,3,4] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => [1,6,2,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => [1,5,2,4,6,3] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => [1,4,6,2,5,3] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => [1,6,2,3,4,5] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => [1,5,2,3,4,6] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => [1,4,6,2,3,5] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => [1,3,5,2,4,6] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => [1,6,2,5,3,4] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => [1,5,3,4,2,6] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => [1,6,2,4,3,5] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => [1,5,3,6,2,4] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => [1,4,2,5,3,6] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => [1,6,2,5,4,3] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => [1,5,4,2,6,3] => 3
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => [1,6,2,4,3,5] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => [1,5,6,2,4,3] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => [1,4,2,5,6,3] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => [1,6,2,3,4,5] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => [1,5,6,2,3,4] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => [1,4,2,3,5,6] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => [1,3,5,6,2,4] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => [1,6,2,5,4,3] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => [1,5,4,3,2,6] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => [1,6,2,4,3,5] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => [1,5,6,2,4,3] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => [1,4,3,2,5,6] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => [1,6,2,3,4,5] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => [1,5,6,2,3,4] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => [1,4,5,6,2,3] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => [1,3,2,4,5,6] => 1
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Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
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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