Download An Introduction to Enumeration (Springer Undergraduate by Barry Lewis, Alan Camina PDF

By Barry Lewis, Alan Camina

Written for college kids taking a moment or 3rd 12 months undergraduate path in arithmetic or machine technological know-how, this e-book is the appropriate better half to a direction in enumeration. Enumeration is a department of combinatorics the place the basic subject material is a number of equipment of development formation and counting. An advent to Enumeration offers a complete and useful advent to this topic giving a transparent account of primary effects and an intensive grounding within the use of strong innovations and tools.

Two significant topics run in parallel in the course of the publication, producing capabilities and team conception. the previous subject takes enumerative sequences after which makes use of analytic instruments to find how they're made up. team concept offers a concise creation to teams and illustrates how the speculation can be utilized to count number the variety of symmetries a specific item has. those improve and expand uncomplicated team principles and techniques.

The authors current their fabric via examples which are conscientiously selected to set up key ends up in a traditional environment. the purpose is to gradually construct basic theorems and strategies. This improvement is interspersed with routines that consolidate rules and construct self assurance. a few routines are associated with specific sections whereas others variety throughout a whole bankruptcy. all through, there's an try to current key enumerative rules in a image manner, utilizing diagrams to lead them to instantly obtainable. the improvement assumes a few easy workforce conception, a familiarity with analytic features and their strength sequence growth besides a few easy linear algebra.

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Extra info for An Introduction to Enumeration (Springer Undergraduate Mathematics Series)

Example text

Overall then, we have Lr = Fr−3+2 + Fr−1+2 = Fr−1 + Fr+1 . The sequence that appears here is a sibling sequence to the Fibonacci sequence. 8 Subsets of a circular set. 16 (Lucas sequence) The sequence {Lr }, defined as the sum of pairs of Fibonacci numbers: Lr = Fr−1 + Fr+1 is called the Lucas sequence, and its terms are Lucas numbers. It has the initial terms {Lr } = {2, 1, 3, 4, 7, 11, 18, . } and satisfies the recurrence Lr = Lr−1 + Lr−2 . Note: the Lucas and Fibonacci sequences share the same recurrence relation (you will be asked to prove this in the exercises).

1 The sequence {vr } has the generating function V (z) = 1 + 3z . 1 − 4z Expand the generating function to find an expression for each term of the sequence. 2 Check that Binet’s formulas for the Fibonacci and Lucas sequences do produce integers for the first three terms of each sequence. 3 The sequence {ur } has the generating function U(z) = 2 − 3z . 2 It’s the Denominator that Counts 47 By factorizing the denominator (not necessarily over the integers) find a Binet style formula for the terms of the sequence.

It follows that, sr+1 = 3# (SR ) + 4# (SNR ) = 3 [# (SR ) + # (SNR )] + # (SNR ) = 3sr + # (SNR ) . However # (SNR ) = 3sr−1 since it can be made from a legal stack of size r − 1 followed by a final counter that may be chosen from three pieces: {Y, G, B}. So sr+1 = 3sr + 3sr−1 . 15 = 216. Again, we can continue calculating as many terms of the sequence as desired – though it does become very tedious. 10). 10 Regions created by four lines. 2 Recurrence Relations and Enumeration 31 Suppose we denote the number of regions by Rr .

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