Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,1] => [1,2] => [1,2] => [1,2] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,1,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,1] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[1,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,1] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,2,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,2,1] => [3,1,2] => [3,2,1] => [3,2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 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
The number of edges in the reduced word graph of a signed permutation.
The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Map
to car permutation
Description
Maps a parking function to the permutation of cars that correspond to that parking function.
For example, the image $[2, 4, 5, 6, 3, 1, 7]$ means that car 2 takes spots 1, car 4 takes spot 2, and so on.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.