Identifier
Mp00170: Permutations to signed permutationSigned 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] [1,2,3,4,5,6]=>[1,2,3,4,5,6]=>[1,1,1,1,1,1] [1,2,3,6,4,5]=>[1,2,3,6,4,5]=>[3,1,1,1] [1,2,5,3,4,6]=>[1,2,5,3,4,6]=>[3,1,1,1] [1,2,5,6,3,4]=>[1,2,5,6,3,4]=>[2,2,1,1] [1,4,2,3,5,6]=>[1,4,2,3,5,6]=>[3,1,1,1] [1,4,2,6,3,5]=>[1,4,2,6,3,5]=>[5,1] [1,4,5,2,3,6]=>[1,4,5,2,3,6]=>[2,2,1,1] [1,4,5,6,2,3]=>[1,4,5,6,2,3]=>[5,1] [1,5,2,3,4,6]=>[1,5,2,3,4,6]=>[4,1,1] [1,5,6,2,3,4]=>[1,5,6,2,3,4]=>[5,1] [1,6,2,3,4,5]=>[1,6,2,3,4,5]=>[5,1] [1,6,2,5,3,4]=>[1,6,2,5,3,4]=>[5,1] [1,6,4,2,3,5]=>[1,6,4,2,3,5]=>[5,1] [1,6,4,5,2,3]=>[1,6,4,5,2,3]=>[5,1] [3,1,2,4,5,6]=>[3,1,2,4,5,6]=>[3,1,1,1] [3,1,2,6,4,5]=>[3,1,2,6,4,5]=>[3,3] [3,1,5,2,4,6]=>[3,1,5,2,4,6]=>[5,1] [3,1,5,6,2,4]=>[3,1,5,6,2,4]=>[4,2] [3,4,1,2,5,6]=>[3,4,1,2,5,6]=>[2,2,1,1] [3,4,1,6,2,5]=>[3,4,1,6,2,5]=>[4,2] [3,4,5,1,2,6]=>[3,4,5,1,2,6]=>[5,1] [3,4,5,6,1,2]=>[3,4,5,6,1,2]=>[3,3] [3,4,6,1,2,5]=>[3,4,6,1,2,5]=>[6] [3,5,1,2,4,6]=>[3,5,1,2,4,6]=>[3,2,1] [3,5,6,1,2,4]=>[3,5,6,1,2,4]=>[4,2] [3,6,1,2,4,5]=>[3,6,1,2,4,5]=>[4,2] [3,6,1,5,2,4]=>[3,6,1,5,2,4]=>[4,2] [3,6,4,1,2,5]=>[3,6,4,1,2,5]=>[3,3] [3,6,4,5,1,2]=>[3,6,4,5,1,2]=>[4,2] [4,1,2,3,5,6]=>[4,1,2,3,5,6]=>[4,1,1] [4,5,1,2,3,6]=>[4,5,1,2,3,6]=>[5,1] [4,5,6,1,2,3]=>[4,5,6,1,2,3]=>[2,2,2] [4,6,1,2,3,5]=>[4,6,1,2,3,5]=>[6] [4,6,5,1,2,3]=>[4,6,5,1,2,3]=>[4,2] [5,1,2,3,4,6]=>[5,1,2,3,4,6]=>[5,1] [5,1,2,6,3,4]=>[5,1,2,6,3,4]=>[4,2] [5,1,4,2,3,6]=>[5,1,4,2,3,6]=>[5,1] [5,1,4,6,2,3]=>[5,1,4,6,2,3]=>[3,3] [5,1,6,2,3,4]=>[5,1,6,2,3,4]=>[6] [5,3,1,2,4,6]=>[5,3,1,2,4,6]=>[5,1] [5,3,1,6,2,4]=>[5,3,1,6,2,4]=>[4,2] [5,3,4,1,2,6]=>[5,3,4,1,2,6]=>[5,1] [5,3,4,6,1,2]=>[5,3,4,6,1,2]=>[4,2] [5,6,1,2,3,4]=>[5,6,1,2,3,4]=>[3,3] [5,6,1,4,2,3]=>[5,6,1,4,2,3]=>[5,1] [5,6,3,1,2,4]=>[5,6,3,1,2,4]=>[5,1] [5,6,3,4,1,2]=>[5,6,3,4,1,2]=>[2,2,1,1] [6,1,2,3,4,5]=>[6,1,2,3,4,5]=>[6] [6,2,1,5,3,4]=>[6,2,1,5,3,4]=>[5,1] [6,2,4,1,3,5]=>[6,2,4,1,3,5]=>[5,1] [6,3,1,2,4,5]=>[6,3,1,2,4,5]=>[6] [1,2,3,4,7,5,6]=>[1,2,3,4,7,5,6]=>[3,1,1,1,1] [1,2,5,3,7,4,6]=>[1,2,5,3,7,4,6]=>[5,1,1] [1,2,7,3,4,5,6]=>[1,2,7,3,4,5,6]=>[5,1,1] [1,2,7,3,6,4,5]=>[1,2,7,3,6,4,5]=>[5,1,1] [1,4,7,2,3,5,6]=>[1,4,7,2,3,5,6]=>[4,2,1] [1,5,7,2,3,4,6]=>[1,5,7,2,3,4,6]=>[6,1] [1,6,7,2,3,4,5]=>[1,6,7,2,3,4,5]=>[3,3,1] [1,6,7,2,5,3,4]=>[1,6,7,2,5,3,4]=>[5,1,1] [1,6,7,4,2,3,5]=>[1,6,7,4,2,3,5]=>[5,1,1] [3,1,7,2,4,5,6]=>[3,1,7,2,4,5,6]=>[7] [3,1,7,5,2,4,6]=>[3,1,7,5,2,4,6]=>[7] [3,4,7,1,2,5,6]=>[3,4,7,1,2,5,6]=>[7] [3,4,7,1,6,2,5]=>[3,4,7,1,6,2,5]=>[7] [3,4,7,5,1,2,6]=>[3,4,7,5,1,2,6]=>[7] [3,4,7,5,6,1,2]=>[3,4,7,5,6,1,2]=>[7] [3,5,7,1,2,4,6]=>[3,5,7,1,2,4,6]=>[5,2] [3,6,1,2,7,4,5]=>[3,6,1,2,7,4,5]=>[3,2,2] [3,7,4,1,2,5,6]=>[3,7,4,1,2,5,6]=>[4,3] [3,7,4,5,1,2,6]=>[3,7,4,5,1,2,6]=>[4,3] [3,7,4,5,6,1,2]=>[3,7,4,5,6,1,2]=>[5,2] [3,7,4,6,1,2,5]=>[3,7,4,6,1,2,5]=>[7] [3,7,5,1,2,4,6]=>[3,7,5,1,2,4,6]=>[7] [3,7,5,6,1,2,4]=>[3,7,5,6,1,2,4]=>[4,3] [3,7,6,1,2,4,5]=>[3,7,6,1,2,4,5]=>[4,3] [4,7,5,6,1,2,3]=>[4,7,5,6,1,2,3]=>[7] [5,3,4,7,1,2,6]=>[5,3,4,7,1,2,6]=>[5,2] [5,3,7,1,2,4,6]=>[5,3,7,1,2,4,6]=>[7] [5,4,7,1,2,3,6]=>[5,4,7,1,2,3,6]=>[4,3] [7,1,2,5,3,4,6]=>[7,1,2,5,3,4,6]=>[7] [7,1,3,2,4,5,6]=>[7,1,3,2,4,5,6]=>[6,1] [7,1,4,2,3,5,6]=>[7,1,4,2,3,5,6]=>[7] [7,1,4,2,6,3,5]=>[7,1,4,2,6,3,5]=>[7] [7,1,4,5,2,3,6]=>[7,1,4,5,2,3,6]=>[7] [7,1,4,6,2,3,5]=>[7,1,4,6,2,3,5]=>[4,3] [7,1,5,2,3,4,6]=>[7,1,5,2,3,4,6]=>[5,2] [7,1,5,6,2,3,4]=>[7,1,5,6,2,3,4]=>[7] [7,1,6,2,3,4,5]=>[7,1,6,2,3,4,5]=>[7] [7,1,6,4,5,2,3]=>[7,1,6,4,5,2,3]=>[5,1,1] [7,2,4,1,3,5,6]=>[7,2,4,1,3,5,6]=>[6,1] [7,2,5,1,3,4,6]=>[7,2,5,1,3,4,6]=>[4,2,1] [7,2,6,1,3,4,5]=>[7,2,6,1,3,4,5]=>[6,1] [7,3,1,2,4,5,6]=>[7,3,1,2,4,5,6]=>[7] [7,3,5,1,2,4,6]=>[7,3,5,1,2,4,6]=>[4,3] [7,3,5,6,1,2,4]=>[7,3,5,6,1,2,4]=>[7] [7,3,6,1,2,4,5]=>[7,3,6,1,2,4,5]=>[7] [7,5,1,4,2,3,6]=>[7,5,1,4,2,3,6]=>[4,2,1] [7,8,5,6,3,4,1,2]=>[7,8,5,6,3,4,1,2]=>[2,2,2,2] [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.