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Generating Functions Contents 1 Enumeration faculty of science MATH 895-4 Fall 2010 department of mathematics Course Schedule LECTURE 2 Generating functions Generating functions Week Date Sections Part/ References Topic/Sections Notes/Speaker from FS2009 Contents1 Sept 7 I.1, I.2, I.3 Combinatorial Symbolic methods Structures 2 14 I.4, I.5, I.6 FS: Part A.1, A.2 Unlabelled structures 1 Enumeration Comtet74 1 3 21 II.1, II.2, II.3 Labelled structures I 1.1 Generating functionsHandout #1 . .1 4 28 II.4, II.5, II.6 (self study) Labelled structures II 2 Manipulating formalCombinatorial power seriesCombinatorial 2 5 Oct 5 III.1, III.2 Asst #1 Due 2.1 Definitions .parameters . .Parameters . .2 FS A.III 6 2.212 SumIV.1, and IV.2 Product(self-study) . .Multivariable . .GFs . .3 2.3 Multiplicative inverse A(x)−1 ...................................3 7 19 IV.3, IV.4 Complex Analysis 2.4 CompositionAnalytic . Methods . .3 FS: Part B: IV, V, VI 8 26 Appendix B4 Singularity Analysis 2.5 DifferentiationIV.5 V.1 . .4 Stanley 99: Ch. 6 9 Nov 2 Asst #2 Due 2.6 Two special series:Handout #1exp, log .....................................Asymptotic methods 4 9 VI.1 (self-study) Sophie 310 The basic toolbox for coefficient extraction 4 12 A.3/ C Introduction to Prob. Mariolys 4 A sampling18 IX.1 of counting sequencesLimit Laws and and generating Comb Marni functions 6 11 4.120 AdvancedIX.2 IdeasRandom . .Structures . .Discrete . Limit . Laws . .Sophie . .6 and Limit Laws FS: Part C Combinatorial 23 IX.3 Mariolys (rotating 12 instances of discrete 1 Enumerationpresentations) 25 IX.4 Continuous Limit Laws Marni Quasi-Powers and Perhaps13 30 theIX.5 most fundamental question about a combinatorialSophie class, is “how many?”. The enumeration problem can be extremely useful forGaussian solving limit other laws problems, and our approach will use generating functions14 Dec 10 since they are so extremelyPresentations powerful. We willAsst only #3 Due just scratch the surface of their potential. Our initial approach will build up a small toolbox of combinatorial constructions • union • cartesian product • sequence • set and multiset Dr. Marni• cycle MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 • pointing and substitution which translate to simple (and not so simple) operations on the generating functions. 1.1 Generating functions We now jump head first into generating functions. Wilf’s book generatingfunctionology is a great first text if you want more details and examples. Definition. The ordinary generating function (OGF) of a sequence (An) is the formal power series 1 X n A(z) = Anz : n=0 We extend this to say that the ogf of a class A is the generating function of its counting sequence An = cardinality(An). Equivalently, we can write the OGF as X A(z) = zjαj: α2A In this context we say that the variable z marks the size of the underlying objects. MARNI MISHNA,SPRING 2011; KAREN YEATS,SPRING 2013 MATH343:APPLIED DISCRETE MATHEMATICS PAGE 1/6 faculty of science MATH 895-4 Fall 2010 department of mathematics Course Schedule LECTURE 2 Generating functions Week Date Sections Part/ References Topic/Sections Notes/Speaker Exercise. Provefrom FS2009 that these two forms for A(z) are equivalent. 1 ForSept example, 7 I.1, I.2, I.3 considerCombinatorial the class of graphsSymbolic methodsH given here Structures 2 14 I.4, I.5, I.6 FS: Part A.1, A.2 Unlabelled structures Comtet74 3 21 II.1, II.2, II.3 Labelled structures I Handout #1 4 28 II.4, II.5, II.6 (self study) Labelled structures II Combinatorial Combinatorial 5 Oct 5 III.1, III.2 Asst #1 Due parameters Parameters If we define size to beFS the A.III number of vertices, then the generating function is simply 6 12 IV.1, IV.2 (self-study) Multivariable GFs H(z) = z4 + z2 + z3 + z4 + z1 + z4 + z3 = z + z2 + 2z3 + 3z4: 7 19 IV.3, IV.4 Analytic Methods Complex Analysis FS: Part B: IV, V, VI 8 26 Appendix B4 Singularity Analysis On the otherIV.5 hand, V.1 if we define size to be the number of edges, then the ogf is Stanley 99: Ch. 6 9 Nov 2 Asst #2 Due Handout #1 Asymptotic methods 4 1 3 6 0 3 2 2 3 4 6 9 VI.1 H(z)(self-study) = z + z + z + z + z + z + Sophiez = 1 + z + z + 2z + z + z : 10 In both12 theseA.3/ casesC we are treating theIntroduction generating to Prob. functionMariolys as some sort of projection in which we replace18 eachIX.1 vertex (edge) by a singleLimit atomic Laws and “ zComb” andMarni forget all detailed information about the object 11 and then20 justIX.2 gather theRandom monomials Structures togetherDiscrete Limit to Laws get theSophie ogf. Thus our examplesand become Limit Laws FS: Part C Combinatorial 23 IX.3 Mariolys (rotating instances of discrete 12 1 presentations) X 1 25 IX.4 Binary words:Continuous LimitW Laws(z) =Marni 2nzn = 1 − 2z Quasi-Powers and n=0 13 30 IX.5 Sophie Gaussian limit laws 1 X n 14 Dec 10 PresentationsPermutations: P (z) =Asst #3 nDue!z n=0 The first of these is a simple geometric sum. The second function does not have any simple expression in terms of standard functions. Indeed it does not converge (as an analytic object) for any z complex z other than 0. However, we can still manipulate it as a purely formal algebraic object. We only care about convergence in that we require that that the coefficient of any finite power of z is finite. This brings us to even more notation: Definition. Let [zn]f(z) denote the coefficient of zn in the power series expansion of f(z) Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY Version of: 11-Dec-09 1 ! n n X n [z ]f(z) = [z ] fnz = fn n=0 2 Manipulating formal power series 2.1 Definitions 2 Let a0; a1; a2;::: be a sequence of real numbers. We call the (possibly infinite) sum a0 + a1x + a2x + k ··· + akx + ::: a formal power series. Here x is a variable, but we do not apriori care if the sum makes sense for any value of x aside from 0. Despite this, we can consider it a function of x and write n it as A(x). We say that ak is the coefficient of x in A(x), and we make use of the shorthand k [x ]A(x) = ak: 2 The sum is said to be formal because we cannot collapse any of the terms. So, if a0 + a1x + a2x = 2 b0 + b1x + b2x , then it must be that a0 = b0, a1 = b1 and a2 = b2. There is a single power series equal to 1: 1 = 1 + 0x + 0x2 + ::: , and a single one equal to 0: 0 + 0x + 0x2 + ::: . The formal power series turns out to be a very convenient way to store a sequence. We will see that we can store infinite sequences with a very small amount of memory using what we know about functions formal power series which arise as Taylor series expansions around 0. MARNI MISHNA,SPRING 2011; KAREN YEATS,SPRING 2013 MATH343:APPLIED DISCRETE MATHEMATICS PAGE 2/6 faculty of science MATH 895-4 Fall 2010 department of mathematics Course Schedule LECTURE 2 Generating functions Week Date Sections Part/ References Topic/Sections Notes/Speaker 2.2 Sumfrom and FS2009 Product A formal1 Sept 7 powerI.1, I.2, seriesI.3 Combinatorial is a mathematical Symbolic object methods which behaves essentially like an infinite polynomial. Structures 2 14 I.4, I.5, I.6 Unlabelled structures P n P We can define additionFS: and Part multiplicationA.1, A.2 of power series. Let A(x) = n≥0 anx and B(x) = n≥0 a = n Comtet74 bnx3 . Then21 II.1, II.2, II.3 Labelled structures I Handout #1 X n 4 28 II.4, II.5, II.6 (self study) A(Labelledx) + B structures(x) := II (an + bn) x n≥0 Combinatorial Combinatorial 5 Oct 5 III.1, III.2 Asst #1 Due parameters Parameters X X n FS A.III A(x) · B(x) := akbn−kx : 6 12 IV.1, IV.2 (self-study) Multivariable GFsn≥0 0≤k≤n n 7 Remark,19 IV.3, we IV.4 have completelyAnalytic Methods specifiedComplex the Analysis coefficient of x for each n, and so the sum and product are well defined. TheyFS: can Partbe B: IV, computed V, VI in finite time given A(x) and B(x). 8 26 Appendix B4 Singularity Analysis IV.5 V.1 Stanley 99: Ch. 6 9 Nov 2 Asst #2 Due Handout #1 Asymptotic−1 methods 2.39 MultiplicativeVI.1 (self-study) inverse A(x) Sophie 10 12 A.3/ C Introduction to Prob. Mariolys P n The multiplicative inverse of a formal power series A(x) is the formal power series C(x) = n≥0 cnx that satisfies18 IX.1 Limit Laws and Comb Marni 11 A(x) · C(x) = 1: 20 IX.2 Random Structures Discrete Limit Laws Sophie and Limit Laws Thus, FS: Part C Combinatorial 23 IX.3 Mariolys (rotating X X n 2 12 instancesakcn −of kdiscretex = 1 + 0x + 0x + :::: presentations) 25 IX.4 n≥0 k≥Continuousn Limit Laws Marni Recall that two formal power seriesQuasi-Powers are equal and if and only all of their coefficients are the same. This 13 30 IX.5 Sophie leads to the system of equations: Gaussian limit laws 14 Dec 10 Presentations Asst #3 Due a0c0 = 1 (1) a1c0 + a0c1 = 0 (2) a2c0 + a1c1 + a0c2 = 0 (3) .
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