Identifier
-
Mp00127:
Permutations
—left-to-right-maxima to Dyck path⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001557: Permutations ⟶ ℤ
Values
[1,2] => [1,0,1,0] => [2,1] => 0
[2,1] => [1,1,0,0] => [1,2] => 0
[1,2,3] => [1,0,1,0,1,0] => [3,2,1] => 1
[1,3,2] => [1,0,1,1,0,0] => [2,3,1] => 1
[2,1,3] => [1,1,0,0,1,0] => [3,1,2] => 0
[2,3,1] => [1,1,0,1,0,0] => [2,1,3] => 0
[3,1,2] => [1,1,1,0,0,0] => [1,2,3] => 0
[3,2,1] => [1,1,1,0,0,0] => [1,2,3] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [3,2,1,4] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [2,3,1,4] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [3,1,2,4] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 0
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [2,1,3,4] => 0
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 3
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 3
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 3
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 3
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 2
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 2
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 2
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 2
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 3
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 3
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 2
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 3
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 3
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 2
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 2
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 1
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Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
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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