Identifier
Values
[1,0,1,0] => [2,1] => [2,1] => 0
[1,1,0,0] => [1,2] => [1,2] => 0
[1,0,1,0,1,0] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0] => [2,3,1] => [2,3,1] => 1
[1,1,0,0,1,0] => [3,1,2] => [2,3,1] => 1
[1,1,0,1,0,0] => [2,1,3] => [2,1,3] => 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] => [4,3,2,1] => 2
[1,0,1,0,1,1,0,0] => [3,4,2,1] => [3,4,2,1] => 2
[1,0,1,1,0,0,1,0] => [4,2,3,1] => [3,4,2,1] => 2
[1,0,1,1,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0] => [4,3,1,2] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0] => [3,4,1,2] => [4,3,2,1] => 2
[1,1,0,1,0,0,1,0] => [4,2,1,3] => [3,2,4,1] => 1
[1,1,0,1,0,1,0,0] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 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] => [5,4,3,2,1] => 3
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => [4,5,3,2,1] => 3
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => [4,5,3,2,1] => 3
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => [4,3,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => [3,4,5,2,1] => 2
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => [4,5,3,2,1] => 3
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => [5,4,3,2,1] => 3
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => [4,3,5,2,1] => 2
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => [4,3,2,5,1] => 2
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => [3,4,2,5,1] => 2
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => [3,4,5,2,1] => 2
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => [3,4,2,5,1] => 2
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => [3,2,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => [4,5,3,2,1] => 3
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => [5,4,3,2,1] => 3
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => [5,4,3,2,1] => 3
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => [5,3,4,2,1] => 2
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [3,5,4,2,1] => 3
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => [3,4,5,2,1] => 2
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => [4,3,2,5,1] => 2
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => [4,3,2,1,5] => 2
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => [3,4,2,5,1] => 2
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => [3,2,4,1,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => [2,3,4,1,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [4,3,5,2,1] => 2
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => [3,4,2,5,1] => 2
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => [3,4,2,1,5] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => [4,3,2,1,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => [3,2,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => [2,3,1,4,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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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.
Map
catalanization
Description
The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].