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## CSM Week 1: Introductory Cross-Disciplinary Seminar

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**CSM Week 1: Introductory Cross-Disciplinary Seminar**Combinatorial Enumeration Dave Wagner University of Waterloo**CSM Week 1: Introductory Cross-Disciplinary Seminar**Combinatorial Enumeration Dave Wagner University of Waterloo I. The Lagrange Implicit Function Theorem and Exponential Generating Functions**CSM Week 1: Introductory Cross-Disciplinary Seminar**Combinatorial Enumeration Dave Wagner University of Waterloo I. The Lagrange Implicit Function Theorem and Exponential Generating Functions II. A Smorgasbord of Combinatorial Identities**I. LIFT and Exponential Generating Functions**1. The Lagrange Implicit Function Theorem**I. LIFT and Exponential Generating Functions**1. The Lagrange Implicit Function Theorem 2. Exponential Generating Functions**I. LIFT and Exponential Generating Functions**1. The Lagrange Implicit Function Theorem 2. Exponential Generating Functions 3. There are rooted trees (two ways)**I. LIFT and Exponential Generating Functions**1. The Lagrange Implicit Function Theorem 2. Exponential Generating Functions 3. There are rooted trees (two ways) 4. Combinatorial proof (sketch) of LIFT**I. LIFT and Exponential Generating Functions**1. The Lagrange Implicit Function Theorem 2. Exponential Generating Functions 3. There are rooted trees (two ways) 4. Combinatorial proof (sketch) of LIFT 5. Nested set systems**I. LIFT and Exponential Generating Functions**1. The Lagrange Implicit Function Theorem 2. Exponential Generating Functions 3. There are rooted trees (two ways) 4. Combinatorial proof (sketch) of LIFT 5. Nested set systems 6. Multivariate Lagrange**I. LIFT and Exponential Generating Functions**1. The Lagrange Implicit Function Theorem 2. Exponential Generating Functions 3. There are rooted trees (two ways) 4. Combinatorial proof (sketch) of LIFT 5. Nested set systems 6. Multivariate Lagrange**The human mind has never invented a labor-saving device**equal to algebra. -- J. Willard Gibbs**1. The Lagrange Implicit Function Theorem**K: a commutative ring that contains the rational numbers. F(u) and G(u): formal power series in K[[u]]: Assume that**1. The Lagrange Implicit Function Theorem**K: a commutative ring that contains the rational numbers. F(u) and G(u): formal power series in K[[u]]: Assume that (a) There is a unique formal power series R(x) in K[[x]] such that**1. The Lagrange Implicit Function Theorem**(b) For this formal power series with the constant term is zero:**1. The Lagrange Implicit Function Theorem**(b) For this formal power series with the constant term is zero: For all n>=1 the coefficient of x^n in F(R(x)) is**1. The Lagrange Implicit Function Theorem**Proofs: (i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of convergence]**1. The Lagrange Implicit Function Theorem**Proofs: (i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of convergence] (ii) Algebraic (formal calculus, formal residue operator) [requires g_0 to be invertible in K]**1. The Lagrange Implicit Function Theorem**Proofs: (i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of convergence] (ii) Algebraic (formal calculus, formal residue operator) [requires g_0 to be invertible in K] (iii) Combinatorial (bijective correspondence)**1. The Lagrange Implicit Function Theorem**Proofs: (i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of convergence] (ii) Algebraic (formal calculus, formal residue operator) [requires g_0 to be invertible in K] (iii) Combinatorial (bijective correspondence)**1. The Lagrange Implicit Function Theorem**Proofs: (i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of convergence] (ii) Algebraic (formal calculus, formal residue operator) [requires g_0 to be invertible in K] (iii) Combinatorial (bijective correspondence)**2. Exponential Generating Functions**A class of structures associates to each finite set another finite set -- this is the set of A-type structures supported on the set X.**2. Exponential Generating Functions**A class of structures associates to each finite set another finite set -- this is the set of A-type structures supported on the set X. Simplified notation:**2. Exponential Generating Functions**A class of structures associates to each finite set another finite set -- this is the set of A-type structures supported on the set X. Simplified notation: Exponential generating function:**2. Exponential Generating Functions**Minimal requirements on a class of structures: * depends only on * If then**2. Exponential Generating Functions**Example: the class of (simple) graphs is the set of graphs with vertex-set Exponential generating function (no particularly useful formula)**2. Exponential Generating Functions**Example: the class of endofunctions is the set of all functions Exponential generating function (no particularly useful formula)**2. Exponential Generating Functions**Example: the class of permutations is the set of permutations on the set Exponential generating function**2. Exponential Generating Functions**Example: the class of cyclic permutations is the set of cyclic perm.s on the set Exponential generating function**2. Exponential Generating Functions**Example: the class of (finite) sets (“ensembles”) is the set of ways in which is a set. Exponential generating function**2. Exponential Generating Functions**Example: the class of sets of size k is the set of ways in which is a k-element set. Exponential generating function Especially important: the case k=1 of singletons…. has exp.gen.fn x.**2. Exponential Generating Functions**Notice that**2. Exponential Generating Functions**Notice that That is…**2. Exponential Generating Functions**Notice that That is… This suggests a relation among classes:**2. Exponential Generating Functions**Notice that That is… This suggests a relation among classes: A permutation is equivalent to a (finite unordered) set of (pairwise disjoint) cyclic permutations.**2. Exponential Generating Functions**A permutation is equivalent to a (finite unordered) set of (pairwise disjoint) cyclic permutations.**2. Exponential Generating Functions**An endofunction is equivalent to a set of disjoint connected endofunctions.**2. Exponential Generating Functions**A connected endofunction is equivalent to a cyclic permutation of rooted trees.**2. Exponential Generating Functions**A connected endofunction is equivalent to a cyclic permutation of rooted trees.**2. Exponential Generating Functions**A rooted tree is equivalent to a root vertex and a set of disjoint rooted (sub-)trees.**2. Exponential Generating Functions**A rooted tree is equivalent to a root vertex and a set of disjoint rooted (sub-)trees.**2. Exponential Generating Functions**The Exponential/Logarithmic Formula For classes A and B, If every B-structure can be decomposed uniquely as a finite set of pairwise disjoint A-structures, then**2. Exponential Generating Functions**The Exponential/Logarithmic Formula For classes A and B, If every B-structure can be decomposed uniquely as a finite set of pairwise disjoint A-structures, then and hence**2. Exponential Generating Functions**Example: Let Q be the class of connected graphs.**2. Exponential Generating Functions**Example: Let Q be the class of connected graphs. Since it follows that**2. Exponential Generating Functions**Example: Let Q be the class of connected graphs. Since it follows that More informatively, records the number of edges in the exponent of y.