Seven ways to display the 24 permutations of 1, 2, 3, 4.
Type |
Chess-Board |
Tableaux |
Line & Cycle |
Tree |
Caterpillar |
Orbital |
A
[1] |
|
|
Line |
1-2-3-4 |
Cycle |
(1)(2)(3)(4) |
|
|
|
|
B
[2] |
|
|
Line |
1-2-4-3 |
Cycle |
(1)(2)(3,4) |
|
|
|
|
C
[3] |
|
|
Line |
1-3-2-4 |
Cycle |
(1)(2,3)(4) |
|
|
|
|
C
[4] |
|
|
Line |
1-3-4-2 |
Cycle |
(1)(2,3,4) |
|
|
|
|
C
[5] |
|
|
Line |
1-4-2-3 |
Cycle |
(1)(2,4,3) |
|
|
|
|
D
[6] |
|
|
Line |
2-1-3-4 |
Cycle |
(1,2)(3)(4) |
|
|
|
|
D
[7] |
|
|
Line |
2-3-4-1 |
Cycle |
(1,2,3,4) |
|
|
|
|
D
[8] |
|
|
Line |
3-1-2-4 |
Cycle |
(1,3,2)(4) |
|
|
|
|
D
[9] |
|
|
Line |
4-1-2-3 |
Cycle |
(1,4,3,2) |
|
|
|
|
E
[10] |
|
|
Line |
1-4-3-2 |
Cycle |
(1)(2,4)(3) |
|
|
|
|
F
[11] |
|
|
Line |
2-1-4-3 |
Cycle |
(1,2)(3,4) |
|
|
|
|
F
[12] |
|
|
Line |
2-4-3-1 |
Cycle |
(1,2,4)(3) |
|
|
|
|
F
[13] |
|
|
Line |
3-1-4-2 |
Cycle |
(1,3,4,2) |
|
|
|
|
F
[14] |
|
|
Line |
4-1-3-2 |
Cycle |
(1,4,2)(3) |
|
|
|
|
G
[15] |
|
|
Line |
2-3-1-4 |
Cycle |
(1,2,3)(4) |
|
|
|
|
G
[16] |
|
|
Line |
3-4-1-2 |
Cycle |
(1,3)(2,4) |
|
|
|
|
G
[17] |
|
|
Line |
2-4-1-3 |
Cycle |
(1,2,4,3) |
|
|
|
|
H
[18] |
|
|
Line |
3-2-1-4 |
Cycle |
(1,3)(2)(4) |
|
|
|
|
H
[19] |
|
|
Line |
3-2-4-1 |
Cycle |
(1,3,4)(2) |
|
|
|
|
H
[20] |
|
|
Line |
3-4-2-1 |
Cycle |
(1,3,2,4) |
|
|
|
|
H
[21] |
|
|
Line |
4-2-1-3 |
Cycle |
(1,4,3)(2) |
|
|
|
|
H
[22] |
|
|
Line |
4-2-3-1 |
Cycle |
(1,4)(2)(3) |
|
|
|
|
H
[23] |
|
|
Line |
4-3-1-2 |
Cycle |
(1,4,2,3) |
|
|
|
|
I
[24] |
|
|
Line |
4-3-2-1 |
Cycle |
(1,4)(2,3) |
|
|
|
|
The type of the permutation is the type of the unlabeled rooted tree it is
associated with. Labeled rooted trees are of the same type if they are equal under branch rotation.
In our example there are nine types (A,B,..,I), 1+1+3+4+1+4+3+6+1 = 24 = 4!. This is a refinement of the factorial numbers.
(See also Frank Ruskey's page).
This article is continued in Permutation Trees.