Lec 1, 8/23 Wed, Sec 1.1:
Course overview, general principles of enumeration, counting of words & subsets,
binomial theorem, multisets/compositions.
Lec 2, 8/25 Fri, Sec 1.2:
Lattice paths, basic identities, extended binomial coefficient,
summing polynomials, Delannoy numbers.
Lec 3, 8/28 Mon, Sec 1.3:
Counting graphs and trees, multinomial coefficients (trees by degrees,
Fermat's Little Theorem), Ballot problem.
Lec 4, 8/30 Wed, Sec 1.3-2.1:
Central binomial convolution,
Catalan numbers (generalization, bijections, recurrence),
Fibonacci numbers and 1,2-lists).
Lec 5, 9/1 Fri, Sec 2.1-2:
Derangements, recurrences in two indices (distribution problems,
Delannoy numbers), characteristic equation method (through Fibonacci solution).
Lec 6, 9/6 Wed, Sec 2.2:
Characteristic equation method (repeated roots, inhomogeneous terms),
generating function method (linear constant coefficients, Catalan solution).
Lec 7, 9/8 Fri, Sec 2.3-3.1:
Substitution method (Tower of Hanoi, derangements, Stirling's approximation),
generating functions (sum/product operations, multisets, restricted
multiplicity).
Lec 8, 9/11 Mon, Sec 3.1:
Generating functions in two variables (subsets), permutation statistics (OGFs
by #inversions, #cycles), Eulerian numbers (Worpitzky's Identity by barred
permutations, A(n,k) formula by substitution).
Lec 9, 9/13 Wed, Sec 3.2:
Generating function manipulations (differentiation, evaluation at special
values, shifting index, summing initial coefficients),
summation by convolutions, Snake Oil method.
Lec 10, 9/15 Fri, Sec 3.3:
Exponential generating functions. Products of EGFs (words),
examples and applications of EGFs (flags on poles, restricted words,
Stirling numbers, binomial inversion, derangements).
Lec 11, 9/18 Mon, Sec 3.3-4:
The Exponential Formula (graphs, partitions, permutations, recurrence),
Lagrange Inversion Formula (statement and application to trees),
partitions of integers (basic generating functions).
Lec 12, 9/20 Wed, Sec 3.4:
Asymptotic number of partitions,
Combinatorics of partitions (Ferrers diagrams, conjugation, Fallon's Identity,
classes of triangles, Euler's Identity).
Lec 13, 9/22 Fri, Sec 4.1:
Inclusion-exclusion (basic formula, totient application,
Stirling numbers, derangements, identities, skipped Eulerian numbers)
Lec 14, 9/25 Mon, Sec 4.1:
Permutations with restricted positions (rook polynomials), OGF by number of
properties, probleme des menages (brief summary).
Signed involutions (inclusion-exclusion formula, partitions into distinct
odd parts (skipped).
Lec 15, 9/27 Wed, Sec 4.1-2:
disjoint-path systems in digraphs, application to disjoint-path systems of
lattice paths and rhombus tilings.
Examples for counting under symmetry, Lagrange's Theorem,
Burnside's Lemma.
Lec 16, 9/29 Fri, Sec 4.2-3:
Cycle indices, symmetries of cube, pattern inventory (Polya's Theorem),
application to isomorphism classes of graphs.
Young tableaux (brief presentation of Hook-length formula,
RSK correspondence, and consequences of RSK correspondence).
Chapter 5, First Concepts for Graphs, for background reading
Lec 17, 10/2 Mon, Sec 6.1:
Bipartite Matching (Hall's Theorem, Marriage Theorem, Birkhoff-von Neumann
Theorem, transveral with "large" minimum (ranchers/farmers))
Lec 18, 10/4 Wed, Sec 6.1-2:
Min/max relations (Ore's defect formula, Konig-Egervary Theorem,
Gallai's Theorem, Konig's Other Theorem),
General Matching (Augmenting paths, Tutte's 1-Factor Condition,
start Berge-Tutte Formula).
Lec 19, 10/6 Fri, Sec 6.2-7.1:
Factors in graphs (finish Berge-Tutte Formula, 1-factors in regular graphs,
Petersen's 2-Factor Theorem (via Eulerian circuit and Hall's Theorem),
reduction of f-factor to 1-factor in blowup),
Connectivity (definitions, Harary graphs).
Lec 20, 10/9 Mon, Sec 7.1:
Connectivity under cartesian product, edge-connectivity definitions, Whitney's
Theorem, edge-connectivity for diameter 2 (Plesnik), ....bonds and blocks
skipped.
Lec 21, 10/11 Wed, Sec 7.2:
k-Connected Graphs (Independent x,y-paths, linkage and blocking sets,
Pym's Theorem, Menger's Theorems (8 versions), Ford-Fulkerson CSDR (postponed),
Expansion and Fan Lemmas, cycles through specified vertices), ear decomposition
and Robbins' Theorem.
Lec 22, 10/13 Fri, Sec 7.3:
Spanning cycles (necessary condition, Ore & Dirac conditions,
closure, statement of Chvatal condition, Chvatal example,
Chvatal-Erdos Theorem),
circumference (brief mention of Bondy's Lemma and Fan's Theorem).
Lec 23, 10/16 Mon, Sec 8.1:
Vertex coloring: examples, easy bounds, greedy coloring, degree bounds, Minty
Theorem, interval graphs.
Lec 24, 10/18 Wed, Sec 8.1-2:
Triangle-free graphs & Mycielski's construction,
color-critical graphs (minimum degree, edge-connectivity),
list coloring (examples and degree-choosability).
Lec 25, 10/20 Fri, Sec 8.2-3:
List extension of Brooks' Theorem,
edge-coloring (complete graphs, Petersen graph, bipartite graphs),
Anderson-Goldberg generalization of Vizing's Theorem.
Lec 26, 10/23 Mon, Sec 9.1:
Planar graphs and their duals, cycles vs bonds, bipartite plane graphs,
Euler's Formula and edge bound.
Lec 27, 10/25 Wed, Sec 9.2:
Regular polyhedra, Kuratowski's Theorem and convex embeddings,
6-coloring of planar graphs
Lec 28, 10/27 Fri, Sec 9.3:
Coloring of planar graphs (5-colorability, 5-choosability, Kempe),
discharging (approach to 4CT, list edge-coloring of planar graphs),
(skipped Tait's Theorem, Grinberg's Theorem).
Lec 29, 10/30 Mon, Sec 10.1:
Applications of pigeonhole principle (divisible pairs, paths in cubes,
domino tilings, monotone sublists, increasing trails).
Lec 30, 11/1 Wed, Sec 10.2:
Ramsey's Theorem and applications (convex m-gons, table storage).
Lec 31, 11/3 Fri, Sec 10.3:
Ramsey numbers, graph Ramsey theory (tree vs complete graph),
Schur's Theorem, Van der Waerden Theorem (statement and example)
Lec 32, 11/6 Mon, Sec 12.1:
Partially ordered sets (definitions and examples, comparability graphs and
cover graphs, Dilworth's Theorem,
equivalence of Dilworth and Konig-Egervary, relation to PGT).
Lec 33, 11/8 Wed, Sec 12.2:
graded posets, symmetric chain decompositions for subsets and products,
bracketing decomposition, application to monotone Boolean functions).
Lec 34, 11/10 Fri, Sec 12.3:
LYM posets (Sperner's Theorem via LYM, equivalence with regular covering and
normalized matching, LYM and symmetric unimodal rank-sizes => symmetric chain
decomposition, CSDR from Menger, statement of log-concavity & product result).
Lec 35, 11/13 Mon, Sec 14.1:
existence arguments (Ramsey number, 2-colorability of k-uniform
hypergraphs), pigeonhole property of expectation (linearity and indicator
variables, Caro-Wei bound on independence number, application of Caro-Wei to
Turan's Theorem), brief sketch of Binet Formula for Fibonacci numbers by
conditional probability.
Lec 36, 11/15 Wed, Sec 14.2:
Deletion method (Ramsey numbers, dominating sets, sketch for large girth and
chromatic number), Local Lemma (symmetric version, sketch of Ramsey number
application).
Lec 37, 11/17 Fri, Sec 14.2-3:
Local Lemma applications (Ramsey number, list coloring).
Random graph models, almost-always properties,
connectedness of the random graph.
Markov's Inequality and connectedness, notion of threshold function.
Lec 38, 11/27 Mon, Sec 14.3:
Random graphs:
diameter 2, Second moment method,
threshold functions for disappearance of isolated vertices
and appearance of balanced graphs.
Lec 39, 11/29 Wed, Sec 14.3:
comments on sharp thresholds, graph evolution,
connectivity/cliques/coloring of random graphs;
lower bound for crossing number of graphs (Chapter 16).
Lec 40, 12/01 Fri, Sec 17.1:
Latin squares (orthogonal families, Moore-MacNeish construction),
block designs (examples, elementary constraints on parameters).
Lec 41, 12/04 Mon, Sec 17.1:
Fisher's Inequality, symmetric designs (Bose), discussion of
Bruck-Chowla-Ryser, Hadamard matrices (restriction on order, relation to
designs, application to bipartite Ramsey problem).
Lec 42, 12/06 Wed, Sec 17.2:
projective planes (elementary properties, relation to designs and Latin
squares, application to extremal problems).
Lec 43, 12/08 Fri, Sec 17.2-3:
difference sets and multipliers, Steiner triple systems.