Items related to Combinatorial Methods
Synopsis
I. Counting and Enumeration on a Set.- A. Introduction.- 1. Set Generating Functions.- 2. Numerical Generating Functions.- Examples.- Fibonacci Numbers.- B. Counting with Restrictions - Techniques.- 1. Inclusion - Exclusion Principle.- The Euler Function.- Rencontres, Derangement or Montmort Problem.- The Menage Problem.- 2. Permutations with Restricted Position. The Master Theorem.- Exercises.- Example.- Rencontre Problem.- Menage Problem.- 3. Extension of the Master Theorem.- C. Partitions, Compositions and Decompositions.- 1. Permutation Counting as a Partition Problem.- a) Counting with allowed transitions.- b) Counting with prohibited transitions.- 2. Classification of Partitions.- a) Distribution of unlabeled objects: Compositions.- b) Distribution of unlabeled objects: Partitions.- 3. Ramsey's Theorem.- Example.- 4. Distribution of Labeled Objects.- a) Distinguishable boxes.- b) Collections of pairs - graph theory.- c) Indistinguishable boxes (and labeled objects).- d) Partially labeled graphs - The Polya Theorem.- Examples.- Proof of Polya's Theorem.- Examples.- Exercises.- e) Counting unrooted (free) unlabeled graphs.- Dissimilarity Theorem.- Example.- II. Counting and Enumeration on a Regular Lattice.- A. Random Walk on Lattices.- 1. Regular Cubic Lattices.- Examples.- 2. General Lattices.- i) Nearest neighbor random walk on a face centered cubic lattice.- ii) Nearest neighbor random walk on a body centered cubic lattice.- B. One Dimensional Lattices.- 1. The Ballot Problem.- Example.- 2. One Dimensional Lattice Gas.- C. Two Dimensional Lattices.- 1. Counting Figures on a Lattice, General Algebraic Approach.- 2. The Dimer Problem - Transfer Matrix Method.- Exercises.- 3. The Dimer Problem - Pfaffian Method.- Exercises.- 4. The Dimer Problem - First Permanent Method.- 5. The Dimer Problem - Second Permanent Method.- D. Counting Patterns on Two Dimensional Lattices.- 1. The Ice Problem - Introduction.- 2. Square Ice - The Transfer Matrix Method.- 3. Square Ice - Exact Solution.- 4. Other Hydrogen Bonded Models - Dimer Solution.- E. The Ising Model.- 1. Introduction.- 2. Estimates of the Curie Temperature.- 3. Combinatorial Solution of the Ising Model.- 4. Other Combinatorial Solutions.- 5. Spin Correlations.
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