Combinatorics

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

By Barry Lewis, Alan Camina

Written for college students taking a moment or 3rd 12 months undergraduate path in arithmetic or machine technological know-how, this booklet is the right spouse to a path in enumeration. Enumeration is a department of combinatorics the place the elemental material is quite a few tools of development formation and counting. An creation to Enumeration offers a entire and useful creation to this topic giving a transparent account of primary effects and an intensive grounding within the use of robust options and tools.

Two significant subject matters run in parallel during the publication, producing features and workforce thought. the previous subject takes enumerative sequences after which makes use of analytic instruments to find how they're made up. staff thought offers a concise creation to teams and illustrates how the idea can be utilized to count number the variety of symmetries a specific item has. those improve and expand easy workforce principles and techniques.

The authors current their fabric via examples which are rigorously selected to set up key ends up in a normal environment. the purpose is to steadily construct primary theorems and methods. This improvement is interspersed with routines that consolidate principles and construct self belief. a few workouts are associated with specific sections whereas others variety throughout a whole bankruptcy. all through, there's an try and current key enumerative principles in a photograph method, utilizing diagrams to lead them to instantly obtainable. the improvement assumes a few simple staff conception, a familiarity with analytic features and their energy sequence growth in addition to a few easy linear algebra.

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

Example text

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).

If we want the recurrence to be true for r = 0 then we must set s0 = 1. The sequence continues (each term being the sum of the two previous terms) {sr } = {1, 2, 3, 5, 8, . } and each term is also a Fibonacci number, this time displaced two places: sr = Fr+2 . Time to define the Fibonacci sequence. 13 (Fibonacci sequence) The Fibonacci sequence {Fr } satisfies the recurrence Fr = Fr−1 + Fr−2 with the initial terms F0 = 0 and F1 = 1. The Fibonacci sequence has the initial terms {Fr } = {0, 1, 1, 2, 3, 5, 8, .

But there are some surprises in store. 48 3. 11 Consider the sequence {ar } with the generating function 3z ∑ ar zr = 1 − 3z + 2z2 . r 0 The denominator factorizes 3z ∑ ar zr = (1 − z)(1 − 2z) r 0 and if we take the second factor over to the left-hand side, we have (1 − 2z) ∑ ar zr = r 0 3z = 3z ∑ zr (1 − z) r 0 and then comparing coefficients of zr gives ar − 2ar−1 = 3 which is the recurrence ar = 2ar−1 + 3. 2r−1 . Of course, had we taken the whole denominator across, we would have obtained the recurrence ar − 3ar−1 + 2ar−2 = 0 for r 2.

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