18.212      Algebraic Combinatorics         MIT Spring 2023

Instructor: Alexander Postnikov

Time: Monday, Wednesday, Friday 1:00-2:00 pm

Place: MIT, Room 1-390

Office hours: By appointment

Description: Applications of algebra to combinatorics and vise versa. We will discuss enumeration methods, permutations, partitions, partially ordered sets and lattices, Young tableaux, graph theory, matrix tree theorem, electrical networks, random walks, convex polytopes, and other topics.

Course Level: Advanced Undergraduate.

Keywords: Catalan numbers, Dyck paths, triangulations, non-crossing set partitions, symmetric group, statistics on permutations, inversions and major index, partially ordered sets and lattices, Sperner's and Dilworth's theorems, Young diagrams, Young's lattice, Gaussian q-binomial coefficients, standard Young tableaux (STY), Robinson-Schensted-Knuth (RSK) correspondence, partitions, Euler's pentagonal theorem, Jacobi triple product, non-crossing paths, Lindstrom-Gessel-Viennot lemma, spanning trees, parking functions, Prufer codes, matrix-tree theorem, electrical networks, random walks on graphs, graph colorings, chromatic polynomial, Mobius function, continued fractions, enumeration under group action, Burnside's lemma, Polya theory, transportation and Birkhoff polytopes, cyclic polytopes, permutohedra, domino tilings, matching enumeration, Pfaffians, Ising model.

Grading: Based on several Problems Sets.

Problem Sets:

Lectures: (We include links to the lecture notes from previous years.)

  1. Mon, Feb 6. The Catalan numbers: Dyck paths, recurrence relation, and exact formula. Notes

  2. Wed, Feb 8. The Catalan numbers (cont'd): reflection method and cyclic shifts. Notes

  3. Fri, Feb 10. The Catalan numbers (cont'd): combinatorial interpretations (binary trees, plane trees, triangulations of polygons, non-crossing and non-nesting matchings, etc). Notes

  4. Mon, Feb 13. The Narayana numbers. Integer partitions. Young diagrams and standard Young tableaux. The hook length formula. Notes (pages 1-7) and the last page of these notes

  5. Wed, Feb 15. Probabilistic "hook walk" proof of the hook length formula. Notes (pages 8-17)

  6. Fri, Feb 17. Other hook-length-type formulas: Tree hook length formula and "broken leg" hook length for shifted shapes. Bijection between increasing binary trees and permutations. Alternating permutations. Notes (pages 1 and 7-9)

  7. Tue, Feb 21. Set partitions. Arc diagrams and rook placements. Non-crossing and non-nesting set partitions. Peaks and valleys. Notes (pages 5-16)

  8. Wed, Feb 22. Cycles in permutations. The Stirling numbers of the first kind. Two Stirling triangles. The Stirling numbers (of both kinds) as matrix coefficients in change of bases matrices. Notes (pages 12-21), and these Notes (pages 9-15)

  9. Fri, Feb 24. The Stirling numbers of both kinds (cont'd). Statistics on permutations.

  10. Mon, Feb 27. Statistics on permutations: inversions, descents, cycles, major index, records, exceedances. Notes (pages 2-8)

  11. Wed, Mar 1. The Eulerian numbers and the Eulerian triangle. Sperner's Theorem. Posets. Chains and antichains. Notes (page 16) and these Notes (pages 1-5)

  12. Fri, Mar 3. Ranked posets. Rank unimodality and Sperner property of posets. Symmetric chain decompositions (SCD). Products of posets. Notes (pages 6-12)

  13. Mon, Mar 6. Construction of a SCD for a product of chains. Dilworth's, Mirsky's, and Greene's theorems for posets. Posets associated to permutations. Increasing and decreasing subsequences in permutations. Notes (pages 1-4)

  14. Wed, Mar 8. Ramsey's and Erdos-Szekeres theorems. Graphs vs posets vs permutations. Lattices: Boolean lattice, partition lattice, Young's lattice. Notes (pages 4-9)

  15. Fri, Mar 10. The lattice of linear subspaces in a vector space over a finite field. q-analogs: q-numbers, q-factorials, Gaussian q-binomial coefficients. q-Pascal's triangle. Notes (pages 1-5)

  16. Mon, Mar 13. q-binomial coefficients (cont'd). Grassmannians over finite fields and Gaussian elimination. Young diagrams as reduced row-echelon forms of matrices. Notes (pages 6-10)

  17. Wed, Mar 15. The Robinson-Schensted-(Knuth) correspondence (RSK). Increasing and descreasing subsequences in permutations. Notes (pages 1-7)

  18. Fri, Mar 17. RSK (cont'd). Symmetry of RSK. Involutions. Pattern avoidance in permutations. 321-avoiding permutations and Dyck paths. Up and down operators on Young's lattice. Notes (pages 7-16)

  19. Mon, Mar 20. Differential posets. Up and down operators. Notes (pages 1-14)

  20. Wed, Mar 22. Fibonacci lattice. Notes (pages 14-19). Partition theory: generating functions, partitions with odd and distinct parts. Notes (pages 1-5)

  21. Fri, Mar 24. Partition theory (cont'd). Partitions with odd distinct parts. Partition identities. Notes (pages 6-8)

  22. Mon, April 3. Partition theory (cont'd). Euler's Pentagonal Theorem. Notes (pages 8-15)

  23. Wed, Apr 5. Partition theory (cont'd). Jacobi's triple product formula. Notes (pages 1-9)

  24. Fri, Apr 7. Counting labelled trees. Cayley's formula. Notes (pages 1-10)

  25. Mon, Apr 10. Two bijective proofs of Cayley's formula: Prufer's coding and Egecioglu-Remmel bijection. Notes (pages 10-14) and Notes (pages 1-7)

  26. Wed, Apr 12. Spanning trees in graphs. Adjacency matrix. Incidence matrix. Laplacian matrix. Matrix Tree Theorem. Notes (pages 8-15)

  27. Fri, Apr 14. Proof of the Matrix Tree Theorem. Oriented incidence matrix. Cauchy-Binet formula. Notes (pages 1-10).

  28. Wed, Apr 19. Matrix-tree theorem (cont'd). Examples: The complete graph and the hypercube graph. Notes (pages 11-12) and Notes (pages 1-9)

  29. Fri, Apr 21. Electrical (resistor) networks. Kirchhoff's and Ohm's laws. Kirchhoff matrix = weighted Laplacian matrix. Notes (pages 1-10)

  30. Mon, Apr 24. Electrical networks (cont'd). Effective resistance. Random walks on graphs. Notes (pages 11-16) and Notes (pages 6-11).

  31. Wed, Apr 26. Chip firing game. Notes

  32. Fri, Apr 28. Parking functions. Notes (pages 1-10)

  33. Mon, May 1. Chip firing game. Notes

  34. Wed, May 3. Lindstrom's lemma (aka the Gessel-Viennot method). Notes

  35. Fri, May 5. Extreme points of subsets in R^n. The Birkhoff polytope.

  36. Mon. May 8. The Arnold-Euler-Bernoulli triangle. Alternating permutations. Andre's theorem. Tangent and secant numbers. Notes

  37. Fri, May 12. Eulerian cycles. BEST Theorem. Notes

    Mon, May 15. Rhomus tilings and domino tilings. Notes and these Notes.

Additional reading materials: (The students are not required to buy these books.)

[AC]  Algebraic Combinatorics: Walks, Trees, Tableaux, and More by R. P. Stanley, Springer: PDF

[EC1] Enumerative Combinatorics Vol 1 by R. P. Stanley, Cambridge University Press.

Last updated:   May 15, 2023