Identifier
-
Mp00108:
Permutations
—cycle type⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001557: Permutations ⟶ ℤ
Values
[1] => [1] => [1,0,1,0] => [3,1,2] => 0
[1,2] => [1,1] => [1,0,1,1,0,0] => [3,1,4,2] => 0
[2,1] => [2] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[1,2,3] => [1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[1,3,2] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
[2,1,3] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
[2,3,1] => [3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[3,1,2] => [3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 1
[3,2,1] => [2,1] => [1,0,1,0,1,0] => [4,1,2,3] => 0
[1,2,4,3] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[1,3,2,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[1,3,4,2] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[1,4,2,3] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[1,4,3,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[2,1,3,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[2,3,1,4] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[2,4,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[3,1,2,4] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[3,2,1,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[3,2,4,1] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[4,1,3,2] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[4,2,1,3] => [3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 2
[4,2,3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[4,3,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 2
[1,2,4,5,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,2,5,3,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,3,2,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,3,4,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,3,5,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,4,2,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,4,3,5,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,4,5,2,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,5,2,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,5,3,2,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[1,5,4,3,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,1,3,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,1,4,3,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,1,4,5,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[2,1,5,3,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[2,1,5,4,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,3,1,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[2,3,1,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[2,4,3,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[2,4,5,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[2,5,3,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[2,5,4,3,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[3,1,2,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[3,1,2,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[3,2,1,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[3,2,4,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[3,2,5,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[3,4,1,2,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[3,4,1,5,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[3,4,5,2,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[3,5,1,2,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[3,5,1,4,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[3,5,4,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[4,1,3,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[4,1,5,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[4,2,1,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[4,2,3,5,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[4,2,5,1,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[4,3,2,1,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[4,3,2,5,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[4,3,5,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[4,5,1,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[4,5,2,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[4,5,3,1,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[5,1,3,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[5,1,4,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[5,2,1,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[5,2,3,1,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[5,2,4,3,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[5,3,2,1,4] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[5,3,2,4,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[5,3,4,2,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[5,4,1,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[5,4,2,3,1] => [3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 3
[5,4,3,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[1,3,2,5,6,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,3,2,6,4,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,3,4,2,6,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,3,5,6,2,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,3,6,5,4,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,4,2,3,6,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,4,5,2,6,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,4,5,6,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,4,6,2,3,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,4,6,5,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,5,2,6,3,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,5,4,3,6,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,5,4,6,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,5,6,2,4,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,5,6,3,2,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,6,2,5,4,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,6,4,3,2,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,6,4,5,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,6,5,2,3,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[1,6,5,3,4,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 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.
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
cycle type
Description
The cycle type of a permutation as a partition.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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