Instructor: Sergi Elizalde
Course on canvas.dartmouth.edu.⇗
Syllabus
| Day 1: | Lattice paths, Dyck paths, the reflection principle, rotated paths, recurrences |
| Day 2: | Binomial coefficients, multisets, compositions |
| Day 3: | Integer partitions, set partitions, Stirling numbers of the second kind |
| Day 4: | Permutation statistics: cycles, records, Foata's fundamental bijection |
| Day 5: | Descents, inversions, major index, fixed points, derangements |
| Day 6: | The Principle of Inclusion-exclusion, answers to enumeration questions, generating functions |
| Day 7: | The ring of formal power series, operations with generating functions |
| Day 8: | Linear recurrences and rational generating functions; a non-linear recurrence |
| Day 9: | The symbolic method for unlabeled structures, operations on combinatorial classes |
| Day 10: | Ordinary generating functions for words, compositions, plane trees, Dyck paths |
| Day 11: | Ordinary generating functions for integer partitions and set partitions; the symbolic method for labeled structures |
| Day 12: | The labeled product, sequences and sets of labeled classes; exponential generating functions for labeled rooted trees, set partitions |
| Day 13: | Exponential generating functions for ordered set partitions, permutations, involutions, derangements, labeled graphs. |
| Day 14: | Cycles of labeled classes, the Lagrange inversion formula, Cayley's formula and bijective proof via Prüfer code |
| Day 15: | Partially ordered sets, graded posets, chains |
| Day 16: | Antichains, Sperner's theorem, order matchings using linear algebra |
| Day 17: | Finishing the algebraic proof of Sperner's theorem |
| Day 18: | Lubell's proof of Sperner's theorem, Young's lattice, the hook-length formula |
| Day 19: | Walks in Young's lattice, U and D operators, the RSK correspondence |
| Day 20: | Properties of RSK, the poset of partitions with bounded largest part and bounded number of parts |
| Day 21: | q-binomial coefficients, enumeration under group action |
| Day 22: | Equivalent colorings, group actions |
| Day 23: | Orbits, Burnside's lemma, applications to counting inequivalent colorings |
| Day 24: | Polya's theorem, applications to counting necklaces, dihedral necklaces, Stirling numbers |
| Day 25 - Nov 10: | Presentations |
| Day 26 - Nov 12: |
Presentations Presentations |
| Day 27 - Nov 14: | Presentations |
| Day 28 - Nov 17: | Presentations |