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List all 3-permutations of {1,2,3,4,5}.
The remaining exercises in this section develop another algorithm for generating the permutations of {1,2,3,n}. This algorithm is based on Cantor expansions of integers. Every nonnegative integer less than n! has a unique Cantor expansion
where a,- is a nonnegative integer not exceeding i, for i = 1,2,n -1. The integers a1; a2, an_t are called the Cantor digits of this integer.
Given a permutation of {1,2,..., n}, let fl^1;k =2,3,..., n, be the number of integers less than k that follow k in the permutation. For instance, in the
permutation 43215, at is the number of integers less than 2 that follow 2, so at = 1. Similarly, for this examplea2= 2, a3=3, and a4 = 0. Consider the function from the set of permutations of {1,2,3,..., n} to the set of nonnegative integers less than n! that sends a permutation to the integer that has a„ a2,an_„ defined in this way, as its Cantor digits.
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Chapter 6 Solutions
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
- Prove by induction that n2n.arrow_forward49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove that if is a prime integer, then is a factor of for . (From Exercise of Section, it is known that is an integer.) b. Use induction on to prove that if is a prime integer, then is a factor of .arrow_forward
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