Bijective combinatorics. (2014)

• Record Type: Book
• Further Information: Note: Nicholas A. Loehr.
• Authors: Loehr, Nicholas A
• Contents: Introduction Basic Counting ; Review of Set Theory; Sum Rule; Product Rule; Words, Permutations, and Subsets; Functions; Bijections, Cardinality, and Counting; Subsets, Binary Words, and Compositions; Subsets of a Fixed Size; Anagrams; Lattice Paths; Multisets; Probability; Games of Chance; Conditional Probability and Independence Combinatorial Identities and Recursions ; Generalized Distributive Law; Multinomial and Binomial Theorems; Combinatorial Proofs; Recursions; Recursions for Multisets and Anagrams; Recursions for Lattice Paths; Catalan Recursions; Integer Partitions; Set Partitions; Surjections; Stirling Numbers and Rook Theory; Linear Algebra Review; Stirling Numbers and Polynomials; Combinatorial Proofs of Polynomial Identities Counting Problems in Graph Theory ; Graphs and Digraphs; Walks and Matrices; DAG’s and Nilpotent Matrices; Vertex Degrees; Functional Digraphs; Cycle Structure of Permutations; Counting Rooted Trees; Connectedness and Components; Forests; Trees; Counting Trees; Pruning Maps; Ordered Trees and Terms; Ordered Forests and Lists of Terms; Graph Coloring; Spanning Trees; Matrix-Tree Theorem; Eulerian Tours Inclusion-Exclusion and Related Techniques ; Involutions; The Inclusion-Exclusion Formula; More Proofs of Inclusion-Exclusion; Applications of the Inclusion-Exclusion Formula; Derangements; Coefficients of Chromatic Polynomials; Classical Möbius Inversion; Partially Ordered Sets; Möbius Inversion for Posets; Product Posets Ranking and Unranking ; Ranking, Unranking, and Related Problems; Bijective Sum Rule; Bijective Product Rule; Ranking Words; Ranking Permutations; Ranking Subsets; Ranking Anagrams; Ranking Integer Partitions; Ranking Set Partitions; Ranking Card Hands; Ranking Dyck Paths; Ranking Trees; Successors and Predecessors; Random Selection Counting Weighted Objects ; Weighted Sets; Inversions; Weight-Preserving Bijections; Sum and Product Rules for Weighted Sets; Inversions and Quantum Factorials; Descents and Major Index; Quantum Binomial Coefficients; Quantum Multinomial Coefficients; Foata’s Map; Quantum Catalan Numbers Formal Power Series ; The Ring of Formal Power Series; Finite Products and Powers of Formal Series; Formal Polynomials; Order of Formal Power Series; Formal Limits, Infinite Sums, and Infinite Products; Multiplicative Inverses in K [x ] and K [[x ]]; Formal Laurent Series; Formal Derivatives; Composition of Polynomials; Composition of Formal Power Series; Generalized Binomial Expansion; Generalized Powers of Formal Series; Partial Fraction Expansions; Application to Recursions; Formal Exponentiation and Formal Logarithms; Multivariable Polynomials and Formal Series The Combinatorics of Formal Power Series ; Sum Rule for Infinite Weighted Sets; Product Rule for Infinite Weighted Sets; Generating Functions for Trees; Compositional Inversion Formulas; Generating Functions for Partitions; Partition Bijections; Euler’s Pentagonal Number Theorem; Stirling Numbers of the First Kind; Stirling Numbers of the Second Kind; The Exponential Formula Permutations and Group Actions ; Definition and Examples of Groups ; Basic Properties of Groups; Notation for Permutations; Inversions and Sign; Determinants; Multilinearity and Laplace Expansions; Cauchy-Binet Formula; Subgroups; Automorphism Groups of Graphs; Group Homomorphisms; Group Actions; Permutation Representations; Stable Subsets and Orbits; Cosets; The Size of an Orbit; Conjugacy Classes in Sn ; Applications of the Orbit Size Formula; The Number of Orbits; Pólya’s Formula Tableaux and Symmetric Polynomials ; Partition Diagrams and Skew Shapes; Tableaux; Schur Polynomials; Symmetric Polynomials; Homogeneous Symmetric Polynomials; Symmetry of Schur Polynomials; Orderings on Partitions; Schur Bases; Tableau Insertion; Reverse Insertion; Bumping Comparison Theorem; Pieri Rules; Schur Expansion of hα ; Schur Expansion of eα ; Algebraic Independence ; Power-Sum Symmetric Polynomials; Relations between e’s and h’s ; Generating Functions for e’s and h’s ; Relations between p’s, e’s, and h’s ; Power-Sum Expansion of hn and en ; The Involution ω ; Permutations and Tableaux; Words and Tableaux; Matrices and Tableaux; Cauchy Identities; Dual Bases Abaci and Antisymmetric Polynomials; Abaci and Integer Partitions; Jacobi Triple Product Identity; Ribbons and k -Cores; k -Quotients and Hooks; Antisymmetric Polynomials; Labeled Abaci; Pieri Rule for pk ; Pieri Rule for ek ; Pieri Rule for hk ; Antisymmetric Polynomials and Schur Polynomials; Rim-Hook Tableaux; Abaci and Tableaux; Skew Schur Polynomials; Jacobi-Trudi Formulas; Inverse Kostka Matrix; Schur Expansion of Skew Schur Polynomials; Products of Schur Polynomials Additional Topics; Cyclic Shifting of Paths; Chung-Feller Theorem ; Rook-Equivalence of Ferrers Boards; Parking Functions; Parking Functions and Trees; Möbius Inversion and Field Theory; Quantum Binomial Coefficients and Subspaces; Tangent and Secant Numbers; Tournaments and the Vandermonde Determinant; Hook-Length Formula; Knuth Equivalence; Pfaffians and Perfect Matchings; Domino Tilings of Rectangles Answers and Hints to Selected Exercises Bibliography Index A Summary and Exercises appear at the end of each chapter.
• Edition: 1st
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2014
• Extent: 1 online resource, illustrations (black and white)
• Subjects: 511.62; Combinatorial enumeration problems
• Languages: English
• ISBNs: 9781439848869
• Notes: Note: Includes bibliographical references and index.; Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.