Identifier
Values
[1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 0
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => 0
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => 0
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,5,1,3,4] => 0
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,4,5,2] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,6,1,3,4,5] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,4,2] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,1,4,3,5] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,4,5,6,2] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,4,6,1,5,2] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,3,5,1,4] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,2,5,4] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,2,5,3] => 0
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [4,6,1,2,5,3] => 1
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,4,6,5,1,3] => 1
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,4,3,6,5,2] => 0
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,1,5,3,4,6] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 0
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,6,3,2,5,4] => 0
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,1,4,5,3,6] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,6,2,4,3] => 1
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,6,1,4,5] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,4,6,5,3] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,4,1,2,6,5] => 1
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,1,3,6,4,2] => 0
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,3,5,1,4] => 0
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,2,6,3,5,4] => 0
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,2,5,6,3] => 0
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => 0
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,3,4,6,5,2] => 0
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,4,1,6,3,5] => 0
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,2,6,4,5,3] => 0
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,1,5,6,3,4] => 1
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,5,2,6,4] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,2,3,5,4] => 0
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 0
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => [1,3,5,7,2,6,4] => 1
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [1,4,5,2,7,6,3] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,3,4,6,1,5] => 0
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,4,2,6,5] => 0
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,2,3,6,5] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,3,6,4] => 0
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => [1,3,5,2,7,6,4] => 0
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [1,3,7,4,2,6,5] => 0
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,4,2,7,3,6,5] => 0
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => [1,4,5,3,7,6,2] => 0
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [1,2,3,5,7,4,6] => 0
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [1,3,7,5,2,6,4] => 0
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => [1,3,2,4,7,6,5] => 0
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [1,2,3,6,4,7,5] => 0
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [1,2,4,5,7,6,3] => 0
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,2,3,7,5,6,4] => 0
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,2,7,3,4,6,5] => 0
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [1,3,4,5,7,6,2] => 0
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,2,4,7,5,6,3] => 0
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [1,3,2,5,4,7,6] => 0
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,2,7,3,5,6,4] => 0
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,3,4,5,2,7,6] => 0
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [1,2,4,3,5,7,6] => 0
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0
[] => [] => [1] => [1] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement St000830The total displacement of a permutation. is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length St000216The absolute length of a permutation. is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions St000018The number of inversions of a permutation. of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.