Identifier
Values
[1] => [1] => [1] => [1] => 0
[-1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[1,-2] => [1,2] => [1,2] => [1,2] => 0
[-1,2] => [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
[2,-1] => [2,1] => [2,1] => [2,1] => 0
[-2,1] => [2,1] => [2,1] => [2,1] => 0
[-2,-1] => [2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,-2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[-1,-2,-3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,3,-2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,-3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[1,-3,-2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[-1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[-1,3,-2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[-1,-3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[-1,-3,-2] => [1,3,2] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,1,-3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,-1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,-1,-3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[-2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[-2,1,-3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[-2,-1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[-2,-1,-3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[2,3,-1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[2,-3,1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[2,-3,-1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[-2,3,1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[-2,3,-1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[-2,-3,1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[-2,-3,-1] => [2,3,1] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[3,1,-2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[3,-1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[3,-1,-2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[-3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[-3,1,-2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[-3,-1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[-3,-1,-2] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[3,2,-1] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[3,-2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[3,-2,-1] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[-3,2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[-3,2,-1] => [3,2,1] => [2,3,1] => [2,3,1] => 0
[-3,-2,1] => [3,2,1] => [2,3,1] => [2,3,1] => 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
Alexandersson Kebede
Description
Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation $\pi$ of length $n$. The mapping looks for a smallest odd integer $i\in[n-1]$ such that swapping the entries $\pi(i)$ and $\pi(i+1)$ preserves the set of right-to-left minima. Otherwise, $\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly $\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in $S_n$ fixed under this map, with exactly $k$ right-to-left minima, see Lemma 35 in [1].
Map
permutation
Description
The permutation obtained by forgetting the colours.
Map
to signed permutation
Description
The signed permutation with all signs positive.