Lesson 24: Generating Permutations

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Lesson 24 Generating Permutations

• Objectives
• Generate all permutations of a set
• Generate all subsets of a set
• Generate all r combinations of a set

• Outline
• Permutations of a set
• Subsets
• HW due May 4

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Generating Permutations

• Given a particular set, we wish to generate all
  possible permutations in an orderly way
• This can be used to check solutions to problems
  involving sets
• Example Traveling salesman
• Every set can has a one-to-one correspondence
  with the set 1, 2, 3, , n

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Permutations

• Permutations of a set can be generated in
  lexicographic order.
• Example 321546 precedes 321456

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Example 1

• Place the following permutations of 1, 2, 3, 4,
  5, 6 in lexicographic order

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Generating permutations

• Find the least significant sequence of descending
  digits
• 7625341
• Swap the preceding number with the next greatest
  number in the descending sequence
• Put the rest of the numbers in ascending sequence

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Example 2

• 362541
• 263541
• 264531 (next 5 terms)

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Problem 1

• Find the next number in the permutation sequence
  following 17386542

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Example 3

• Find the permutation of 1, 2, 3, 4, 5, 6
  immediately before 261345 in lexicographic order.

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Example 4

• If the permutations of 1,2,3,4,5,6 are put in
  lexicographic order, with 123456 in position 1,
  123465 in position 2, etc., find the permutation
  in position 362.

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Example 5

• If the permutations of 1,2,3,4,5 are put in
  lexicographic order, in what position is the
  permutation 41253?

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Subsets

• To generate all subsets of a set, express
  membership by a 1 in a bit string.