5.4 Permutation Functions

Bijections on a finite set A to itself is called a permutation of A A permutation is one to one and onto. -A function is one to one if a unique element of A is a function with a unique element of the second set of A -A function is onto if each element of A is a function with each element of the second set of A A = {1,2,3}, then all permutations of A with corresponding functions values are: 1A P1 P2 P3 P4 P5

The number of permutations of a set with 3 elements is 3! (3 factorial). 3! = 6 1A is the identity function P1 The finite set is on top row and the corresponding function values are on the lower row.

P4 = {(1,3), (2,1), (3,2)} The inverse of P4: P4-1 = {(3,1), (1,2), (2,3)} We put this in increasing order: P4-1 = {(1,2), (2,3), (3,1)} We see that P4-1 = P3

To find the composition: P3 P2 composition = Starting at the first column of the second permutation, 1 points to 2, go to the column labeled 2 of the first permutation. Under 2 is the number 3. Put 3 under the first column of the result permutation. Next, the second column of the second permutation 2 points to 1. Go to the column labeled 1 of the first permutation. Under 1 is the number 2. Put 2 under the second column of the result permutation. Next, the third column of the second permutation 3 points to 3. Go to the column labeled 3 of the first permutation. Under 3 is the number 1. Put 1 under the third column of the result permutation. 1 2 3

Cyclic permutations are cycles (not like the digraph cycles) Cyclic permutations are cycles (not like the digraph cycles). Cycles are not required to include all the numbers of a set. Cycle (3,2,1,4) could be a permutation of the set {1,2,3,4} or of {1,2,3,4,5,6,7,8} Cycle (3,5,8,2) = (5,8,2,3) = (8,2,3,5) = (2,3,5,8) 3 2 5 8

The composition of cycles (4,1,3,5) (5,6,3) Where A = {1,2,3,4,5,6} 4 R 1, 1R3, 3R5, 5R4 Cycle (5,6,3) 5R6, 6R3, 3R5 = 1 2 3 4 5 6

If we perform (5,6,3) (4,1,3,5) = We see that: (4,1,3,5) (5,6,3) = (5,6,3) (4,1,35)

Two cycles are said to be disjoint if there is no like element of A that appears in both cycles. Cycles (1,2,5) and (3,4,6) are disjoint. There are no like elements.

Permutations are used to produce transposition codes Permutations are used to produce transposition codes. A common transposition code is the keyword columnar transposition. In this code, a keyword is needed: Message: The fifth goblet contains the gold Keyword: JONES JONES THEFI FTHGO BLETC ONTAI NSTHE GOLDX The message is padded with an X to fill the row.

1. Assign a number to each letter of the keyword. 1 2 3 4 5 JONES 2 1. Assign a number to each letter of the keyword. 1 2 3 4 5 JONES 2. Write the cycle, placing the number in alphabetic order E J N O S (4,1,3,2,5) 3. Encoded message: FGTAHDTFBONGEHETTLHTLNSOIOCIEX Divide the number of letters 30 by 5, because JONES has 5 letters, to discover there are 6 rows. 30 letters divided by 5 = 6. Every 6th letter, put a hash mark to determine the letters for each column. Column 4 Column 1 Column 3 Column 2 Column5 FGTAHD | TFBONG | EHETTL | HTLNSO | IOCIEX

4. Write the letters, six at a time, starting in column 4, followed by the next set of six letters in column 2, followed by the next set of xix letter in column 3, and the next set of 6 letter ins column 2, and the final six letters in column 5. 1 2 3 4 5 JON E S THE F I FTH GO BLE T C ONT A I NST H E GOL DX

Test the waters 1 2 3 4 5 6 7 8 9 10 11 12 13 S T E E E A T H S T T W R The numbers refer to the position of the letters in a message. (1,2,3) (4,7) (5,10,11) (6,8,12,13,9) First we see we have 1,2,3,4,5,6,7,8,9,10,11,12,13 (our top numbers for our permutations) Next we see 1R2,2R3,3R1 4R7,7R4 5R10,10R11,11R5 6R8,8R12,12R13,13R9,9R6

(1,2,3) (4,7) Result = (5,10,11) (6,8,12,13,9)

We have (1,2), (2,3), (3,1), (4,7), (5,10),(6,8), (7,4), (8,12), (9,6), (10,11), (11,5), (12,13), (13,9) Inverse permutation (2,1),(3,2),(1,3),(7,4), (10,5), (8,6), (4,7),(12,8),(6,9), (11,10), (5,11), (13,13),(9,13)

(2,1),(3,2),(1,3),(7,4), (10,5), (8,6), (4,7),(12,8),(6,9), (11,10), (5,11), (13,13),(9,13) 1 2 3 4 5 6 7 8 9 10 11 12 13 S T E E E A T H S T T W R T E S T T H E W A T E R S 1 2 3 4 5 6 7 8 9 10 11 12 13 (2,1) The second letter goes to position 1 (3,2) The 3rd letter goes to position 2 (1,3) The 1st letter goes to position 3 (7,4) The 7th letter goes to position 4 (10,5) The 10th letter goes to position 5 (8,6) The 8th letter goes to position 6 (4,7) The 4th letter goes to position 7 (12,8) The 12th letter goes to position 8 (6,9) The 6th letter goes to position 9 (11,10) The 11th letter goes to position 10 (5,11) The 5th letter goes to position 11 (13,12) The 13th letter goes to position 12 (9,13) The 9th letter goes to position 13