CSM Week 1: Introductory Cross-Disciplinary Seminar

1. CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo

2. CSM Week 1: Introductory Cross-Disciplinary Seminar Combinatorial Enumeration Dave Wagner University of Waterloo I. The Lagrange Implicit Function Theorem and Exponential Generating Functions

3. 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

4. I. LIFT and Exponential Generating Functions 1. The Lagrange Implicit Function Theorem

5. I. LIFT and Exponential Generating Functions 1. The Lagrange Implicit Function Theorem 2. Exponential Generating Functions

6. I. LIFT and Exponential Generating Functions 1. The Lagrange Implicit Function Theorem 2. Exponential Generating Functions 3. There are rooted trees (two ways)

7. 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

8. 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

9. 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

10. 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

12. The human mind has never invented a labor-saving device equal to algebra. -- J. Willard Gibbs

14. 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

15. 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

16. 1. The Lagrange Implicit Function Theorem (b) For this formal power series with the constant term is zero:

17. 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

18. 1. The Lagrange Implicit Function Theorem Proofs: (i) Complex analysis (Cauchy residue formula) [requires K=C and nonzero radii of convergence]

19. 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]

20. 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)

21. 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)

22. 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)

24. 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.

25. 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:

26. 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:

27. 2. Exponential Generating Functions Minimal requirements on a class of structures: * depends only on * If then

28. 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)

29. 2. Exponential Generating Functions Example: the class of endofunctions is the set of all functions Exponential generating function (no particularly useful formula)

30. 2. Exponential Generating Functions Example: the class of permutations is the set of permutations on the set Exponential generating function

31. 2. Exponential Generating Functions Example: the class of cyclic permutations is the set of cyclic perm.s on the set Exponential generating function

32. 2. Exponential Generating Functions Example: the class of (finite) sets (“ensembles”) is the set of ways in which is a set. Exponential generating function

33. 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.

34. 2. Exponential Generating Functions Notice that

35. 2. Exponential Generating Functions Notice that That is…

36. 2. Exponential Generating Functions Notice that That is… This suggests a relation among classes:

37. 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.

38. 2. Exponential Generating Functions

39. 2. Exponential Generating Functions A permutation is equivalent to a (finite unordered) set of (pairwise disjoint) cyclic permutations.

40. 2. Exponential Generating Functions

41. 2. Exponential Generating Functions An endofunction is equivalent to a set of disjoint connected endofunctions.

42. 2. Exponential Generating Functions A connected endofunction is equivalent to a cyclic permutation of rooted trees.

43. 2. Exponential Generating Functions A connected endofunction is equivalent to a cyclic permutation of rooted trees.

44. 2. Exponential Generating Functions A rooted tree is equivalent to a root vertex and a set of disjoint rooted (sub-)trees.

45. 2. Exponential Generating Functions A rooted tree is equivalent to a root vertex and a set of disjoint rooted (sub-)trees.

46. 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

47. 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

48. 2. Exponential Generating Functions Example: Let Q be the class of connected graphs.

49. 2. Exponential Generating Functions Example: Let Q be the class of connected graphs. Since it follows that

50. 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.