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