Identifier
Values
[1,2] => [[1,2]] => [1,2] => [1,2] => 0
[2,1] => [[1],[2]] => [2,1] => [2,1] => 0
[1,2,3] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [[1,2],[3]] => [3,1,2] => [2,3,1] => 0
[2,1,3] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [[1,2],[3]] => [3,1,2] => [2,3,1] => 0
[3,1,2] => [[1,3],[2]] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [[1,2,3],[4]] => [4,1,2,3] => [2,3,4,1] => 1
[1,3,2,4] => [[1,2,4],[3]] => [3,1,2,4] => [2,3,1,4] => 0
[1,3,4,2] => [[1,2,3],[4]] => [4,1,2,3] => [2,3,4,1] => 1
[1,4,2,3] => [[1,2,4],[3]] => [3,1,2,4] => [2,3,1,4] => 0
[1,4,3,2] => [[1,2],[3],[4]] => [4,3,1,2] => [3,4,2,1] => 1
[2,1,3,4] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => 1
[2,3,1,4] => [[1,2,4],[3]] => [3,1,2,4] => [2,3,1,4] => 0
[2,3,4,1] => [[1,2,3],[4]] => [4,1,2,3] => [2,3,4,1] => 1
[2,4,1,3] => [[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => 0
[2,4,3,1] => [[1,2],[3],[4]] => [4,3,1,2] => [3,4,2,1] => 1
[3,1,2,4] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 0
[3,1,4,2] => [[1,3],[2,4]] => [2,4,1,3] => [3,1,4,2] => 1
[3,2,1,4] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [[1,3],[2],[4]] => [4,2,1,3] => [3,2,4,1] => 1
[3,4,1,2] => [[1,2],[3,4]] => [3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [[1,2],[3],[4]] => [4,3,1,2] => [3,4,2,1] => 1
[4,1,2,3] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => 0
[4,1,3,2] => [[1,3],[2],[4]] => [4,2,1,3] => [3,2,4,1] => 1
[4,2,1,3] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 0
[4,2,3,1] => [[1,3],[2],[4]] => [4,2,1,3] => [3,2,4,1] => 1
[4,3,1,2] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => 0
[4,3,2,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,2,4,5,3] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,2,5,3,4] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,2,5,4,3] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[1,3,2,4,5] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [2,3,1,4,5] => 0
[1,3,2,5,4] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [3,4,1,5,2] => 0
[1,3,4,2,5] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,3,4,5,2] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,4,5,1,2] => 2
[1,3,5,4,2] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [2,3,1,4,5] => 0
[1,4,2,5,3] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [3,4,1,5,2] => 0
[1,4,3,2,5] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[1,4,3,5,2] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,4,2,5,1] => 1
[1,4,5,2,3] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,4,5,1,2] => 2
[1,4,5,3,2] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[1,5,2,3,4] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [2,3,1,4,5] => 0
[1,5,2,4,3] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,4,2,5,1] => 1
[1,5,3,2,4] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[1,5,3,4,2] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,4,2,5,1] => 1
[1,5,4,2,3] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[1,5,4,3,2] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [4,5,3,2,1] => 2
[2,1,3,4,5] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[2,1,3,5,4] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[2,1,4,3,5] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[2,1,4,5,3] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[2,1,5,3,4] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[2,1,5,4,3] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [4,2,5,3,1] => 2
[2,3,1,4,5] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [2,3,1,4,5] => 0
[2,3,1,5,4] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [3,4,1,5,2] => 0
[2,3,4,1,5] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[2,3,4,5,1] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[2,3,5,1,4] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,4,5,1,2] => 2
[2,3,5,4,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[2,4,1,3,5] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,1,2,5] => 0
[2,4,1,5,3] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [3,4,1,5,2] => 0
[2,4,3,1,5] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[2,4,3,5,1] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,4,2,5,1] => 1
[2,4,5,1,3] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,4,5,1,2] => 2
[2,4,5,3,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[2,5,1,3,4] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,1,2,5] => 0
[2,5,1,4,3] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [4,5,2,3,1] => 1
[2,5,3,1,4] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[2,5,3,4,1] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,4,2,5,1] => 1
[2,5,4,1,3] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [4,5,2,1,3] => 1
[2,5,4,3,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [4,5,3,2,1] => 2
[3,1,2,4,5] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[3,1,2,5,4] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[3,1,4,2,5] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[3,1,4,5,2] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[3,1,5,2,4] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[3,1,5,4,2] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [4,2,5,3,1] => 2
[3,2,1,4,5] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[3,2,1,5,4] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,2,1,5,3] => 0
[3,2,4,1,5] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[3,2,4,5,1] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[3,2,5,1,4] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [4,2,5,1,3] => 2
[3,2,5,4,1] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [4,2,5,3,1] => 2
[3,4,1,2,5] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,1,2,5] => 0
[3,4,1,5,2] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [3,4,1,5,2] => 0
[3,4,2,1,5] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[3,4,2,5,1] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [3,4,2,5,1] => 1
[3,4,5,1,2] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [3,4,5,1,2] => 2
[3,4,5,2,1] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[3,5,1,2,4] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,1,2,5] => 0
[3,5,1,4,2] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [4,5,2,3,1] => 1
[3,5,2,1,4] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [4,5,2,1,3] => 1
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [4,5,2,3,1] => 1
[3,5,4,1,2] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [4,5,2,1,3] => 1
[3,5,4,2,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [4,5,3,2,1] => 2
[4,1,2,3,5] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[4,1,2,5,3] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[4,1,3,2,5] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[4,1,3,5,2] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[4,1,5,2,3] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[4,1,5,3,2] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [4,2,5,3,1] => 2
[4,2,1,3,5] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[4,2,1,5,3] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,2,1,5,3] => 0
[4,2,3,1,5] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[4,2,3,5,1] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[4,2,5,1,3] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [4,2,5,1,3] => 2
[4,2,5,3,1] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [4,2,5,3,1] => 2
[4,3,1,2,5] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[4,3,1,5,2] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,2,1,5,3] => 0
[4,3,2,1,5] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[4,3,2,5,1] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,3,2,5,1] => 1
[4,3,5,1,2] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [4,2,5,1,3] => 2
[4,3,5,2,1] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[4,5,1,2,3] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,4,1,2,5] => 0
[4,5,1,3,2] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [4,5,2,3,1] => 1
[4,5,2,1,3] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [4,5,2,1,3] => 1
[4,5,2,3,1] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [4,5,2,3,1] => 1
[4,5,3,1,2] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [4,5,2,1,3] => 1
[4,5,3,2,1] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [4,5,3,2,1] => 2
[5,1,2,3,4] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[5,1,2,4,3] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[5,1,3,2,4] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[5,1,3,4,2] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[5,1,4,2,3] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[5,1,4,3,2] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[5,2,1,3,4] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[5,2,1,4,3] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,2,1,5,3] => 0
[5,2,3,1,4] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[5,2,3,4,1] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [3,2,4,5,1] => 1
[5,2,4,1,3] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [4,2,5,1,3] => 2
[5,2,4,3,1] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[5,3,1,2,4] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[5,3,1,4,2] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [4,2,1,5,3] => 0
[5,3,2,1,4] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[5,3,2,4,1] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,3,2,5,1] => 1
[5,3,4,1,2] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [4,2,5,1,3] => 2
[5,3,4,2,1] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [4,3,5,2,1] => 2
[5,4,1,2,3] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[5,4,1,3,2] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,3,2,5,1] => 1
[5,4,2,1,3] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[5,4,2,3,1] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [4,3,2,5,1] => 1
[5,4,3,1,2] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[5,4,3,2,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => 2
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Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the second entry is St001557The number of inversions of the second entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Map
inverse
Description
Sends a permutation to its inverse.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
Robinson-Schensted recording tableau
Description
Sends a permutation to its Robinson-Schensted recording tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding recording tableau.