Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Generating permutations and combinations is useful and sometimes important.
Note: We can place any n-set into a 1-1 correspondence with the first n natural
numbers. All permutations can be listed using {1, 2, …, n} instead of the actual
set elements. There are n! possible permutations.
Definition: In the lexicographic (or dictionary) ordering, the permutation of
{1,2,…,n} a 1 a 2 …a n precedes b 1 b 2 …b n if and only if a i ; b i , for all 1;i;n.
Examples:
• 5 elements. The permutation 21435 precedes 21543.
• Given 362541, then 364125 is the next permutation lexicographically.
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Algorithm: Generate the next permutation in lexicographic order.
procedure next_perm(a 1 a 2 …a n : a i ,{1,2,…,n} and distinct)
j := n – 1
while a j > a j+1
j := j – 1
{j is the largest subscript with a j < a j+1 }
k := n
while a j > a k
k := k – 1
{a k is the smallest integer greater than a j to the right of a j }
Swap a j and a k
r := n, s := j+1
while r > s
Swap a r and a s
r := r – 1, s:= s + 1
{This puts the tail end of the permutation after the j th
increasing order}
position in
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