notes-091008 - MATH 681 1 Notes Combinatorics and Graph...

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MATH 681 Notes Combinatorics and Graph Theory I 1 Recurrence relations, continued continued Linear homogeneous recurrences are only one of several possible ways to describe a sequence as a recurrence. Here are several other situations which may arise. 1.1 Linear nonhomogeneous recurrence relations A linear nonhomogeneous recurrence relation is one which has a linear homogeneous form as well as a nonhomogeneous term. here’s an example: Question 1: How many strings of n numbers from the set { 0 , 1 , 2 , 3 , 4 } are there so that there is at least one “1” and the first “1” occurs before any “0”? Answer 1: Let us call the above number a n , and a string satisfying the above description “valid”. To produce a valid string of length n , we can consider every possible first number in the string as an individual case. A string beginning with “0” is necessarily invalid. If a string begins with “1”, everything that could follow that is valid. Thus, of the 5 n - 1 strings beginning with “1”, all are valid. If a string begins with a “2”, “3”, or “4”, then it is valid if and only if the remaining n - 1 terms form a valid string, which could happen in a n - 1 ways. Thus, a n = 3 a n - 1 + 5 n - 1 . This is quite similar to a first-order linear homogeneous differential equation, except it has that inconvenient 5 n - 1 term. Unfortunately, we don’t have the tools to solve that, yet! In addition, note the initial condition a 0 = 0 , since the string of length zero contains no “1”s and is thus invalid. We could, if we choose, work out a few small values from this: a 1 = 1 , a 2 = 8 , a 3 = 49 , and so forth. We’ll define the class of problems to which this belongs, and discuss solution techniques: Definition 1. A sequence { a n } is given by a linear nonhomogeneous recurrence relation of order k if a n = c 1 a n - 1 + c 2 a n - 2 + c 3 a n - 3 + · · · + c k a n - k + p ( n ) for all n k . The recurrence relation b n = c 1 b n - 1 + c 2 b n - 2 + c 3 b n - 3 + · · · + c k b n - k is referred to as the associated linear homogeneous recurrence relation One result is as easy to show for LNRRs as for LHRRs; the following can be proven as a very slight variation of the similar proof for LHRRs. Proposition 1. A sequence is uniquely determined by an LNRR of order k and the initial values a 0 , a 1 , a 2 , . . . , a k - 1 . However, there is one very important difference between LNRRs and LHRRs: linear combinations of LNRR-satisfying sequences do no, in general, satisfy the LNRR. Howver, we do have the result: Proposition 2. If { a n } satisfies an LNRR, and { b n } satisfies the associated LHRR, then { a n + b n } satisfies the LNRR. Proof. We know that a n = c 1 a n - 1 + c 2 a n - 2 + c 3 a n - 3 + · · · + c k a n - k + p ( n ) Page 1 of 5 October 1, 2009
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MATH 681 Notes Combinatorics and Graph Theory I and b n = c 1 b n - 1 + c 2 b n - 2 + c 3 b n - 3 + · · · + c k b n - k Adding these two equations will give ( a n + b n ) = c 1 ( a n - 1 + b n - 1 ) + c 2 ( a n - 2 + b n - 2 ) + · · · + c k ( a n - k + b n - k ) + p ( n ) and thus { a n + b n } satisfies the LNRR.
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