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This course covers the applications of algebra to combinatorics.
0
1
English
English [CC]
FREE
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Description
Topics include enumeration methods, permutations, partitions, partially ordered sets and lattices, Young tableaux, graph theory, matrix tree theorem, electrical networks, convex polytopes, and more.
Course content
- Catalan numbers: drunkard’s walk problem, generating function, recurrence relation Unlimited
- Catalan numbers (cont.) Unlimited
- Pattern avoidance in permutations Unlimited
- Frobenius-Young identity. Schensted correspondence. Unlimited
- Proof of the hook-length formula based on a random hook walk Unlimited
- Hook walks (cont.). Linear extensions of posets Unlimited
- q-factorials and q-binomial coefficients Unlimited
- Grassmannians over finite fields: Gaussian elimination and row-reduced echelon form Unlimited
- Sets and multisets. Statistics on permutations: inversions, cycles, descents Unlimited
- Statistics on permutations (cont.). Unlimited
- Stirling numbers (cont.) Unlimited
- Eulerian numbers Unlimited
- Posets and lattices. Boolean lattice. Partition lattice. Young’s lattice Unlimited
- Distributive lattices. Birkhoff’s fundamental theorem for finite distributive lattices Unlimited
- Sperner’s property. Symmetric chain decompositions. Unlimited
- Greene’s theorem vs Schensted correspondence. Up and down operators. Differential posets Unlimited
- Differential posets (cont.). Fibonacci lattice Unlimited
- Proof of unimodality of Gaussian coefficients (cont.). Unlimited
- Partition theory (cont.). Unlimited
- Partition theory (cont.). Combinatorial proof of Jacobi’s triple product identity. Unlimited
- Two combinatorial proofs of Cayley’s formula Unlimited
- Matrix Tree Theorem. Spanning trees. Unlimited
- Matrix Tree Theorem (cont.). Unlimited
- Proof of Matrix Tree Theorem using Cauchy-Binnet formula. Unlimited
- Proof of Directed Matrix Tree Theorem based on induction Unlimited
- Proof of Directed Matrix Tree Theorem via the Involution Principle. Unlimited
- Electrical networks (cont.). Unlimited
- Eulerian cycles in digraphs and B.E.S.T. theorem Unlimited
- Parking functions and tree inversion polynomials Unlimited
- Pascal triangle. Bernoulli numbers. Unlimited
- Lindstrom-Gessel-Viennot lemma and Lindstrom Theorem Unlimited
- Proof of Lindstrom-Gessel-Viennot lemma. Unlimited
- MacMahon Proposition. Rhombus tiling. Domino tiling. Unlimited
- Plane partitions. Symmetric polynomials Unlimited
Instructor
Massachusetts Institute of Technology
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1916
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