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

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Presentation transcript:

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.

2. Exponential Generating Functions The Compositional Formula For classes A, B, and J: If every B -structure can be decomposed uniquely as a finite set Y of pairwise disjoint A -structures, together with a J -structure on Y, then

2. Exponential Generating Functions Example: Let K be the class of connected endofunctions. Let R be the class of rooted trees.

2. Exponential Generating Functions Example: Let K be the class of connected endofunctions. Let R be the class of rooted trees. Since it follows that

2. Exponential Generating Functions Example: Let K be the class of connected endofunctions. Let R be the class of rooted trees. Since it follows that

2. Exponential Generating Functions Sum of classes A and B An structure on X is either a red A -structure or a green B -structure on X.

2. Exponential Generating Functions An structure on X

2. Exponential Generating Functions An structure on X

2. Exponential Generating Functions Sum of classes A and B The exp.gen.fn of is

2. Exponential Generating Functions Product of classes A and B An structure on X is an A -structure on S and a B -structure on X\S (for some subset S of X).

2. Exponential Generating Functions An structure on X

2. Exponential Generating Functions Product of classes A and B The exp.gen.fn of is

3. Counting Rooted Trees A rooted tree is equivalent to a root vertex and a set of disjoint rooted (sub-)trees.

3. Counting Rooted Trees

From we deduce that

3. Counting Rooted Trees From we deduce that LIFT applies with F(u)=u and G(u)=exp(u):

3. Counting Rooted Trees From we deduce that LIFT applies with F(u)=u and G(u)=exp(u):

3. Counting Rooted Trees From we deduce that LIFT applies with F(u)=u and G(u)=exp(u): Therefore

5. Nested Set Systems A nested set system is a pair in which X is a finite set and is a set of subsets of X such that if and then either or or. Let N be the class of nested set systems. What is # N _n?

5. Nested Set Systems A nested set system with vertex-set X.

5. Nested Set Systems Let N be the class of nested set systems. What is # N _n?

5. Nested Set Systems Let N be the class of nested set systems. What is # N _n? We’ll use the bivariate generating function

5. Nested Set Systems Let N be the class of nested set systems. What is # N _n? We’ll use the bivariate generating function This is an exp.gen.fn in the indeterminate x and records in the exponent of y.

5. Nested Set Systems A nested set system is proper if it does not contain any sets of size zero or one. Let M be the class of proper nested set systems

5. Nested Set Systems A proper nested set system

5. Nested Set Systems

A proper nested set system is equivalent to a set of disjoint blobs – each blob is a singleton or a “cell”.

5. Nested Set Systems A cell is a proper nested set system for which -- Let Q be the class of cells.

5. Nested Set Systems A proper nested set system is equivalent to a set of disjoint blobs – each blob is a singleton or a “cell”.

5. Nested Set Systems

The “protoplasm” of a cell is a proper nested set system that is not empty, not a singleton, and not a cell. 5. Nested Set Systems

The “protoplasm” of a cell is a proper nested set system that is not empty, not a singleton, and not a cell. 5. Nested Set Systems

Let and

5. Nested Set Systems

Therefore, the number of nested set systems on the vertex-set {1,2,…,n} is

n, ~ #N_n (up to k = 500) (k = 500 term of the series) 0, , e-46 1, , e-43 2, , e-40 3, , e-38 4, , e-35 5, , e-32 6, , e-30 7, , e-27 8, , e-24 9, , e-22 10, , e-19 11, , e-16 12, , e-14 13, , e-11 14, , e-8 15, , e-5 16, , e-3 17, , , , , , Nested Set Systems

References A. Joyal Une theorie combinatoire des series formelles Adv. in Math. 42 (1981), I.P. Goulden, D.M. Jackson “Combinatorial Enumeration” John Wiley & Sons, New York, F. Bergeron, G. Labelle, P. Leroux “Combinatorial Species and Tree-like Structures” Cambridge U.P., Cambridge, R.P. Stanley “Enumerative Combinatorics, volume II” Cambridge U.P., Cambridge, 1999.