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
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