Identifier
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Images
=>
Cc0002;cc-rep-2
[1]=>[1]=>[1] [1,2]=>[1,2]=>[1,1] [2,1]=>[2,1]=>[2] [1,2,3]=>[1,2,3]=>[1,1,1] [1,3,2]=>[1,3,2]=>[2,1] [2,1,3]=>[2,1,3]=>[2,1] [2,3,1]=>[2,3,1]=>[3] [3,1,2]=>[3,1,2]=>[3] [3,2,1]=>[3,2,1]=>[2,1] [1,2,3,4]=>[1,2,3,4]=>[1,1,1,1] [1,2,4,3]=>[1,2,4,3]=>[2,1,1] [1,3,2,4]=>[1,3,2,4]=>[2,1,1] [1,3,4,2]=>[1,3,4,2]=>[3,1] [1,4,2,3]=>[1,4,2,3]=>[3,1] [1,4,3,2]=>[1,4,3,2]=>[2,1,1] [2,1,3,4]=>[2,1,3,4]=>[2,1,1] [2,1,4,3]=>[2,1,4,3]=>[2,2] [2,3,1,4]=>[2,3,1,4]=>[3,1] [2,3,4,1]=>[2,3,4,1]=>[4] [2,4,1,3]=>[2,4,1,3]=>[4] [2,4,3,1]=>[2,4,3,1]=>[3,1] [3,1,2,4]=>[3,1,2,4]=>[3,1] [3,1,4,2]=>[3,1,4,2]=>[4] [3,2,1,4]=>[3,2,1,4]=>[2,1,1] [3,2,4,1]=>[3,2,4,1]=>[3,1] [3,4,1,2]=>[3,4,1,2]=>[2,2] [3,4,2,1]=>[3,4,2,1]=>[4] [4,1,2,3]=>[4,1,2,3]=>[4] [4,1,3,2]=>[4,1,3,2]=>[3,1] [4,2,1,3]=>[4,2,1,3]=>[3,1] [4,2,3,1]=>[4,2,3,1]=>[2,1,1] [4,3,1,2]=>[4,3,1,2]=>[4] [4,3,2,1]=>[4,3,2,1]=>[2,2] [1,2,3,4,5]=>[1,2,3,4,5]=>[1,1,1,1,1] [1,2,3,5,4]=>[1,2,3,5,4]=>[2,1,1,1] [1,2,4,3,5]=>[1,2,4,3,5]=>[2,1,1,1] [1,2,4,5,3]=>[1,2,4,5,3]=>[3,1,1] [1,2,5,3,4]=>[1,2,5,3,4]=>[3,1,1] [1,2,5,4,3]=>[1,2,5,4,3]=>[2,1,1,1] [1,3,2,4,5]=>[1,3,2,4,5]=>[2,1,1,1] [1,3,2,5,4]=>[1,3,2,5,4]=>[2,2,1] [1,3,4,2,5]=>[1,3,4,2,5]=>[3,1,1] [1,3,4,5,2]=>[1,3,4,5,2]=>[4,1] [1,3,5,2,4]=>[1,3,5,2,4]=>[4,1] [1,3,5,4,2]=>[1,3,5,4,2]=>[3,1,1] [1,4,2,3,5]=>[1,4,2,3,5]=>[3,1,1] [1,4,2,5,3]=>[1,4,2,5,3]=>[4,1] [1,4,3,2,5]=>[1,4,3,2,5]=>[2,1,1,1] [1,4,3,5,2]=>[1,4,3,5,2]=>[3,1,1] [1,4,5,2,3]=>[1,4,5,2,3]=>[2,2,1] [1,4,5,3,2]=>[1,4,5,3,2]=>[4,1] [1,5,2,3,4]=>[1,5,2,3,4]=>[4,1] [1,5,2,4,3]=>[1,5,2,4,3]=>[3,1,1] [1,5,3,2,4]=>[1,5,3,2,4]=>[3,1,1] [1,5,3,4,2]=>[1,5,3,4,2]=>[2,1,1,1] [1,5,4,2,3]=>[1,5,4,2,3]=>[4,1] [1,5,4,3,2]=>[1,5,4,3,2]=>[2,2,1] [2,1,3,4,5]=>[2,1,3,4,5]=>[2,1,1,1] [2,1,3,5,4]=>[2,1,3,5,4]=>[2,2,1] [2,1,4,3,5]=>[2,1,4,3,5]=>[2,2,1] [2,1,4,5,3]=>[2,1,4,5,3]=>[3,2] [2,1,5,3,4]=>[2,1,5,3,4]=>[3,2] [2,1,5,4,3]=>[2,1,5,4,3]=>[2,2,1] [2,3,1,4,5]=>[2,3,1,4,5]=>[3,1,1] [2,3,1,5,4]=>[2,3,1,5,4]=>[3,2] [2,3,4,1,5]=>[2,3,4,1,5]=>[4,1] [2,3,4,5,1]=>[2,3,4,5,1]=>[5] [2,3,5,1,4]=>[2,3,5,1,4]=>[5] [2,3,5,4,1]=>[2,3,5,4,1]=>[4,1] [2,4,1,3,5]=>[2,4,1,3,5]=>[4,1] [2,4,1,5,3]=>[2,4,1,5,3]=>[5] [2,4,3,1,5]=>[2,4,3,1,5]=>[3,1,1] [2,4,3,5,1]=>[2,4,3,5,1]=>[4,1] [2,4,5,1,3]=>[2,4,5,1,3]=>[3,2] [2,4,5,3,1]=>[2,4,5,3,1]=>[5] [2,5,1,3,4]=>[2,5,1,3,4]=>[5] [2,5,1,4,3]=>[2,5,1,4,3]=>[4,1] [2,5,3,1,4]=>[2,5,3,1,4]=>[4,1] [2,5,3,4,1]=>[2,5,3,4,1]=>[3,1,1] [2,5,4,1,3]=>[2,5,4,1,3]=>[5] [2,5,4,3,1]=>[2,5,4,3,1]=>[3,2] [3,1,2,4,5]=>[3,1,2,4,5]=>[3,1,1] [3,1,2,5,4]=>[3,1,2,5,4]=>[3,2] [3,1,4,2,5]=>[3,1,4,2,5]=>[4,1] [3,1,4,5,2]=>[3,1,4,5,2]=>[5] [3,1,5,2,4]=>[3,1,5,2,4]=>[5] [3,1,5,4,2]=>[3,1,5,4,2]=>[4,1] [3,2,1,4,5]=>[3,2,1,4,5]=>[2,1,1,1] [3,2,1,5,4]=>[3,2,1,5,4]=>[2,2,1] [3,2,4,1,5]=>[3,2,4,1,5]=>[3,1,1] [3,2,4,5,1]=>[3,2,4,5,1]=>[4,1] [3,2,5,1,4]=>[3,2,5,1,4]=>[4,1] [3,2,5,4,1]=>[3,2,5,4,1]=>[3,1,1] [3,4,1,2,5]=>[3,4,1,2,5]=>[2,2,1] [3,4,1,5,2]=>[3,4,1,5,2]=>[3,2] [3,4,2,1,5]=>[3,4,2,1,5]=>[4,1] [3,4,2,5,1]=>[3,4,2,5,1]=>[5] [3,4,5,1,2]=>[3,4,5,1,2]=>[5] [3,4,5,2,1]=>[3,4,5,2,1]=>[3,2] [3,5,1,2,4]=>[3,5,1,2,4]=>[3,2] [3,5,1,4,2]=>[3,5,1,4,2]=>[2,2,1] [3,5,2,1,4]=>[3,5,2,1,4]=>[5] [3,5,2,4,1]=>[3,5,2,4,1]=>[4,1] [3,5,4,1,2]=>[3,5,4,1,2]=>[3,2] [3,5,4,2,1]=>[3,5,4,2,1]=>[5] [4,1,2,3,5]=>[4,1,2,3,5]=>[4,1] [4,1,2,5,3]=>[4,1,2,5,3]=>[5] [4,1,3,2,5]=>[4,1,3,2,5]=>[3,1,1] [4,1,3,5,2]=>[4,1,3,5,2]=>[4,1] [4,1,5,2,3]=>[4,1,5,2,3]=>[3,2] [4,1,5,3,2]=>[4,1,5,3,2]=>[5] [4,2,1,3,5]=>[4,2,1,3,5]=>[3,1,1] [4,2,1,5,3]=>[4,2,1,5,3]=>[4,1] [4,2,3,1,5]=>[4,2,3,1,5]=>[2,1,1,1] [4,2,3,5,1]=>[4,2,3,5,1]=>[3,1,1] [4,2,5,1,3]=>[4,2,5,1,3]=>[2,2,1] [4,2,5,3,1]=>[4,2,5,3,1]=>[4,1] [4,3,1,2,5]=>[4,3,1,2,5]=>[4,1] [4,3,1,5,2]=>[4,3,1,5,2]=>[5] [4,3,2,1,5]=>[4,3,2,1,5]=>[2,2,1] [4,3,2,5,1]=>[4,3,2,5,1]=>[3,2] [4,3,5,1,2]=>[4,3,5,1,2]=>[3,2] [4,3,5,2,1]=>[4,3,5,2,1]=>[5] [4,5,1,2,3]=>[4,5,1,2,3]=>[5] [4,5,1,3,2]=>[4,5,1,3,2]=>[3,2] [4,5,2,1,3]=>[4,5,2,1,3]=>[3,2] [4,5,2,3,1]=>[4,5,2,3,1]=>[5] [4,5,3,1,2]=>[4,5,3,1,2]=>[2,2,1] [4,5,3,2,1]=>[4,5,3,2,1]=>[4,1] [5,1,2,3,4]=>[5,1,2,3,4]=>[5] [5,1,2,4,3]=>[5,1,2,4,3]=>[4,1] [5,1,3,2,4]=>[5,1,3,2,4]=>[4,1] [5,1,3,4,2]=>[5,1,3,4,2]=>[3,1,1] [5,1,4,2,3]=>[5,1,4,2,3]=>[5] [5,1,4,3,2]=>[5,1,4,3,2]=>[3,2] [5,2,1,3,4]=>[5,2,1,3,4]=>[4,1] [5,2,1,4,3]=>[5,2,1,4,3]=>[3,1,1] [5,2,3,1,4]=>[5,2,3,1,4]=>[3,1,1] [5,2,3,4,1]=>[5,2,3,4,1]=>[2,1,1,1] [5,2,4,1,3]=>[5,2,4,1,3]=>[4,1] [5,2,4,3,1]=>[5,2,4,3,1]=>[2,2,1] [5,3,1,2,4]=>[5,3,1,2,4]=>[5] [5,3,1,4,2]=>[5,3,1,4,2]=>[4,1] [5,3,2,1,4]=>[5,3,2,1,4]=>[3,2] [5,3,2,4,1]=>[5,3,2,4,1]=>[2,2,1] [5,3,4,1,2]=>[5,3,4,1,2]=>[5] [5,3,4,2,1]=>[5,3,4,2,1]=>[3,2] [5,4,1,2,3]=>[5,4,1,2,3]=>[3,2] [5,4,1,3,2]=>[5,4,1,3,2]=>[5] [5,4,2,1,3]=>[5,4,2,1,3]=>[5] [5,4,2,3,1]=>[5,4,2,3,1]=>[3,2] [5,4,3,1,2]=>[5,4,3,1,2]=>[4,1] [5,4,3,2,1]=>[5,4,3,2,1]=>[2,2,1] [1,4,5,6,2,3]=>[1,4,5,6,2,3]=>[5,1] [5,6,7,8,3,4,1,2]=>[5,6,7,8,3,4,1,2]=>[4,4] [5,6,7,8,4,1,2,3]=>[5,6,7,8,4,1,2,3]=>[8] [5,6,7,8,3,1,2,4]=>[5,6,7,8,3,1,2,4]=>[6,2] [3,4,7,8,1,2,5,6]=>[3,4,7,8,1,2,5,6]=>[4,4] [5,6,7,1,8,2,3,4]=>[5,6,7,1,8,2,3,4]=>[4,2,2] [1,2,5,6,7,8,3,4]=>[1,2,5,6,7,8,3,4]=>[3,3,1,1] [5,6,7,8,1,4,2,3]=>[5,6,7,8,1,4,2,3]=>[6,2]
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
even cycle type
Description
The partition corresponding to the even cycles.
A cycle of length $\ell$ of a signed permutation $\pi$ can be written in two line notation as
$$\begin{array}{cccc} a_1 & a_2 & \dots & a_\ell \\ \pi(a_1) & \pi(a_2) & \dots & \pi(a_\ell) \end{array}$$
where $a_i > 0$ for all $i$, $a_{i+1} = |\pi(a_i)|$ for $i < \ell$ and $a_1 = |\pi(a_\ell)|$.
The cycle is even, if the number of negative elements in the second row is even.
This map records the integer partition given by the lengths of the odd cycles.
The integer partition of even cycles together with the integer partition of the odd cycles determines the conjugacy class of the signed permutation.