Instructors: Sergi Elizalde
Course on canvas.dartmouth.edu.⇗
Here is a tentative list of topics that will be covered, together with the corresponding references.
|Main reference||Other references|
|Catalan numbers. Sets and multisets. Compositions.||[dM] Chapter 1||[Aig] Sections 1.1-1.2
[EC1] Section 1.2
|Integer and set partitions. Stirling numbers. Permutations.||[dM] Chapter 3||[Aig] Sections 1.3-1.5
[vLW] Chapter 13
[EC1] Section 1.3.
|Inclusion-Exclusion.||[dM] Chapter 2||[Aig] Section 5.1
[EC1] Sections 2.1-2.3
[vLW] Chapter 10
|Recurrences. Formal power series.||[dM] Chapter 4||[Aig] Sections 2.1, 2.2, 3.1
[Wf] Section 2.1
[vLW] Chapter 14
|The symbolic method. Unlabeled structures.
Ordinary generating functions.
|[dM] Chapter 5||[FS] Chapter 1
[Wf] Section 2.2
|Labeled structures. Exponential generating functions.||[dM] Chapter 6||[Aig] Section 3.3
[FS] Chapter 2
[Wf] Section 2.3
|Topics in algebraic combinatorics|
|Enumeration under group action.
Counting orbits using the Burnside Lemma. Polya's theorem.
|[St] Chapter 7||[Aig] Sections 6.1-6.3|
|Partially ordered sets. Chains and antichains.
|[St] Chapter 4|
|Young tableaux. The hook-length formula. Operators on Young's lattice.||[St] Chapter 8||[EC2] Section 7.11
[BS] Section 4
This is a more accurate description of the topics covered each day so far:
|Day 1:||Lattice paths, Dyck paths, the reflection principle, rotated paths, recurrences|
|Day 2:||Binomial coefficients, multisets, compositions, partitions|
|Day 3:||More partitions, set partitions, Stirling numbers, cycles in permutations|
|Day 4:||Permutation statistics: cycles, records, descents, inversions|
|Day 5:||Inversion table, major index, fixed points, derangements; inclusion-exclusion|
|Day 6:||More 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, partitions, plane trees|
|Day 11:||Binary trees, set partitions; the symbolic method for labeled structures|
|Day 12:||The labeled product, sequences and sets of labeled classes; labeled rooted trees, set partitions|
|Day 13:||Cycles of labeled classes; surjective maps, labeled graphs, permutations, involutions; multivariate generating functions|
|Day 14:||The Lagrange inversion formula, Cayley's formula and Prufer code|
|Day 15:||Group actions, orbits, equivalent colorings|
|Day 16:||[Justin] Enumeration under group action|
|Day 17:||[Justin] Enumeration under group action|
|Day 18:||[Justin] Enumeration under group action|
|Day 19:||Applications of Polya's theorem: necklaces, dihedral necklaces, Stirling numbers|
|Day 20:||Partially ordered sets, graded posets, chains, antichains|
|Day 21:||Sperner's theorem, order matchings using linear algebra|
|Day 22:||Finishing the algebraic proof of Sperner's theorem; Lubell's proof|
|Day 23 -Nov 1:||
Juan's presentation: A general partition theorem.
|Day 24 -Nov 3:||
Avery and Sophie's presentation: Walks in graphs.
|Day 25 -Nov 6:||
Lizzie's presentation: The inversion number and the major index.
|Day 26 -Nov 8:
||Sara and Emma's presentation: Two proofs of the hook-length formula|
|Day 27 -Nov 10:||Doug and Zach's presentation: Random walks in Zd|
|Day 28 -Nov 13:||
Amir's presentation: A combinatorial proof of the Lagrange inversion formula
Shikhin's presentation: The RSK correspondence
The syllabus page shows a table-oriented view of the course schedule, and the basics of course grading. You can add any other comments, notes, or thoughts you have about the course structure, course policies or anything else.
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