| 1 | Catalan numbers: drunkard's walk problem, generating function, recurrence relation (PDF) |
| 2 | Catalan numbers (cont.): formula for Cn, reflection principle, necklaces, triangulations of polygons, plane binary trees, parenthesizations (PDF) |
| 3 | Pattern avoidance in permutations. Stack- and queue-sortable permutations. Young diagrams and Young tableaux. Hook-length formula (PDF) |
| 4 | Frobenius-Young identity. Schensted correspondence. Longest increasing and decreasing subsequences in permutations (PDF) |
| 5 | Proof of the hook-length formula based on a random hook walk (PDF) |
| 6 |
 Hook walks (cont.). Linear extensions of posets. Hook-length-type formulas for shifted shapes and trees (PDF)
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| 7 | q-factorials and q-binomial coefficients (PDF) |
| 8 | Grassmannians over finite fields: Gaussian elimination and row-reduced echelon form (PDF) |
| 9 | Sets and multisets. Statistics on permutations: inversions, cycles, descents (PDF) |
| 10 | Statistics on permutations (cont.). Equidistributed statistics. Major index. Records. Exceedances. Stirling numbers (PDF) |
| 11 | Stirling numbers (cont.). Set-partitions. Rook placements on triangular boards. Non-crossing and non-nesting set-partitions (PDF) |
| 12 | Eulerian numbers. Increasing binary trees. 3 Pascal-like triangles: Eulerian triangles, Stirling triangles of 1st and 2nd kind (PDF) |
| 13 | Discussion of problem set 1 |
| 14 | Discussion of problem set 1 (cont.) |
| 15 | Posets and lattices. Boolean lattice. Partition lattice. Young's lattice (PDF) |
| 16 | Distributive lattices. Birkhoff's fundamental theorem for finite distributive lattices (PDF) |
| 17 | Sperner's property. Symmetric chain decompositions. Sperner's and Dilworth's theorems. Greene's theorem (PDF) |
| 18 | Greene's theorem vs Schensted correspondence. Up and down operators. Differential posets (PDF) |
| 19 | Differential posets (cont.). Fibonacci lattice. Unimodality of Gaussian coefficients (PDF) |
| 20 |
Proof of unimodality of Gaussian coefficients (cont.). Theory of partitions. Euler's pentagonal number theorem (PDF)
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| 21 |
Partition theory (cont.). Franklin's combinatorial proof of Euler's pentagonal number theorem. Jacobi's triple product identity (PDF)
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| 22 | Partition theory (cont.). Combinatorial proof of Jacobi's triple product identity. Enumeration of trees. Cayley's formula. Simple inductive proof of Cayley's formula (PDF) |
| 23 | Two combinatorial proofs of Cayley's formula (PDF) |
| 24 | Discussion of problem set 2 |
| 25 | Discussion of problem set 2 (cont.) |
| 26 | Matrix Tree Theorem. Spanning trees. Laplacian matrix of a graph. Reciprocity formula for spanning trees. Examples: complete graphs, complete bipartite graphs (PDF) |
| 27 | Matrix Tree Theorem (cont.). Products of graphs. Number of spanning trees in the hypercube graph. Oriented incidence matrix (PDF) |
| 28 | Proof of Matrix Tree Theorem using Cauchy-Binnet formula. Weighted and directed version of Matrix Tree Theorem. In-trees and out-trees (PDF) |
| 29 | Proof of Directed Matrix Tree Theorem based on induction (PDF) |
| 30 | Proof of Directed Matrix Tree Theorem via the Involution Principle. Electrical Networks and Kirchhoff's Laws (PDF) |
| 31 | Electrical networks (cont.). Kirchhoff's matrix. Relations of electrical networks with Matrix Tree Theorem and spanning trees. Series-parallel connections. Probabilistic interpretation of the electrical potential in terms of random walks on graphs (PDF) |
| 32 | Eulerian cycles in digraphs and B.E.S.T. theorem (PDF) |
| 33 | Parking functions and tree inversion polynomials (PDF) |
| 34 | Pascal triangle. Bernoulli numbers. Riemann zeta function. Binary trees. Catalan numbers (PDF) |
| 35 | Lindstrom-Gessel-Viennot lemma and Lindstrom Theorem (PDF) |
| 36 | Proof of Lindstrom-Gessel-Viennot lemma. Weighted version of Lindstrom-Gessel-Viennot lemma. Plane partitions (PDF) |
| 37 | MacMahon Proposition. Rhombus tiling. Domino tiling. Kasteleyn Theorem. Temperley Theorem (PDF) |
| 38 | Discussion of problem set 3 |
| 39 | Plane partitions. Symmetric polynomials. Semi-standard Young tableau. Schur polynomial. Jacobi-Trudi Theorem. Hook-content formula (PDF) |
