By Barry Lewis, Alan Camina

Written for college kids taking a moment or 3rd yr undergraduate path in arithmetic or computing device technological know-how, this e-book is the suitable better half to a direction in enumeration. Enumeration is a department of combinatorics the place the elemental subject material is a number of equipment of development formation and counting. An creation to Enumeration presents a complete and functional creation to this topic giving a transparent account of primary effects and a radical grounding within the use of robust thoughts and tools.

Two significant issues run in parallel in the course of the e-book, producing capabilities and staff conception. the previous subject matter takes enumerative sequences after which makes use of analytic instruments to find how they're made up. workforce conception presents a concise advent to teams and illustrates how the idea can be utilized to count number the variety of symmetries a specific item has. those enhance and expand easy workforce rules and techniques.

The authors current their fabric via examples which are rigorously selected to set up key leads to a average environment. the purpose is to gradually construct primary theorems and strategies. This improvement is interspersed with workouts that consolidate rules and construct self belief. a few workouts are associated with specific sections whereas others variety throughout an entire bankruptcy. all through, there's an try and current key enumerative rules in a photograph means, utilizing diagrams to lead them to instantly obtainable. the improvement assumes a few simple workforce thought, a familiarity with analytic features and their energy sequence enlargement in addition to a few simple linear algebra.

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

**Sample text**

R 0 r 1 r 2 Most of the recurrences that we encountered in the last section can be converted into generating functions by the three-step recipe. A notable exception is the derangement sequence – we must wait for Chapter 7 for that. 26 The recurrence relation satisfied by the Fibonacci sequence {Fr } is Fr = Fr−1 + Fr−2 with the initial values F0 = 0 and F1 = 1. Applying the three-step recipe gives: (i) The recurrence is Fr = Fr−1 + Fr−2 . (ii) Then we sum on the index r with corresponding powers of z: ∑ Fr zr = ∑ Fr−1 zr + ∑ Fr−2 zr .

We then have: z ∑ Rr zr − R0 = z ∑ Rr zr + (1 − z)2 . r 0 r 0 If we let the generating function be R(z) then we have R(z) − 1 = zR(z) + and hence R(z) = z (1 − z)2 1 − z + z2 . (1 − z)3 38 2. 10 Find generating functions for the sequences (i) {vr } = {1, −1, 1, −1, . }; (ii) {ur } = {1, −2, 3, −4, . }; (iii) {or } = {1, 0, 3, 0, 5, 0, . }. 11 1 Integrate the expression 1−z = 1 + z + z2 + · · · and determine the constant of integration by assigning the value z = 0. What is the generating function for the Reciprocal sequence, {0, 1, 12 , 13 , 14 , .

A(z) B(z) where the numerator and denominator are both You may have noticed two intriguing things about the examples already encountered. The first is that the denominator of the generating function (if it is a rational function) has the same “form” as one of the recurrences associated with the corresponding sequence. The second is that a given sequence often has more than one recurrence associated with it. We explore these observations. 13 Using these ideas we may find the recurrence satisfied by the sequence {F0 , F5 , F10 , · · · } consisting of those Fibonacci numbers whose index is a multiple of 5.