### 18.212      ALGEBRAIC COMBINATORICS         MIT Spring 2021

Instructor: Alexander Postnikov

Time: Monday, Wednesday, Friday 2:00-3:00 pm EST       on-line

All lectures will be in real time on zoom. The zoom passcode is the first 2-digit Catalan number, written as a number.

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.

Topics:

• 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, Schensted's correspondence, RSK
• partitions, Euler's pentagonal theorem, Jacobi triple product
• non-crossing paths, Lindstrom lemma (aka Gessel-Viennot method)
• spanning trees, parking functions, and Prufer codes
• matrix-tree theorem, electrical networks, random walks on graphs
• graph colorings, chromatic polynomial, Mobius function
• lattice paths and 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
• and more...
Grading: Based on several Problems Sets.

Problem Sets:

Lectures:

The lecture notes will appear on this page shortly before the lectures. The registered students can view the recordings of lectures on canvas.

1. Wed, Feb 17. The Catalan numbers: Dyck paths, recurrence relation, and exact formula.
Notes   Video

2. Fri, Feb 19. The Catalan numbers (cont'd): drunkard's walk, reflection principle, cyclic shifts.
Notes   Video

3. Mon, Feb 22. The Catalan numbers (cont'd): combinatorial interpretations (plane trees, triangulations of polygons, non-crossing and non-nesting arc diagrams, etc). Stack and queue sorting. Pattern avoidance for permutations.
Notes   Video

4. Wed, Feb 24. Partitions of integers and Young diagrams. Standard Young tableaux. The hook length formula. Probabilistic "hook walk" proof.
Notes   Video

5. Fri, Feb 26. Integer partions vs set partitions. The Bell numbers and the Stirling numbers of the second kind. Rook placements. Non-crossing and non-nesting set partitions. Peaks and valleys. The Narayana numbers.
Notes   Video

6. Mon, Mar 1. Exponential generating functions. The exponential formula. The Bell numbers and the Stirling numbers of the second kind.
Notes   Video

7. Wed, Mar 3. Statistics on permutations: inversions, descents, cycles, major index, records, exceedances. The Eulerian numbers. The Stirling numbers of the first kind. The Stirling numbers as the coefficients in the change of bases matrices. Three "Pascal-like" triangles: The Stirling triangles of both kinds and the Eulerian triangle.
Notes   Video

8. Fri, Mar 5. Sperner's theorem. Posets. Chains and antichains. The Boolean lattice. Symmetric chain decompositions (SCD).
Notes   Video

9. Tue, Mar 9. Dilworth's, Mirsky's, and Greene's theorems. Increasing and decreasing subsequences in permutations. Erdos-Szekeres theorem. Lattices. Examples of lattices: The Boolean lattice, Young's lattice, and the partition lattice.
Notes   Video

10. Wed, Mar 10. Order ideals. Distributive lattices. Birkhoff's fundamental theorem on finite distributive lattices. Linear extenstions of posets and saturated chains.
Notes   Video

11. Fri, Mar 12. q-analogs. The q-factorials and the Gaussian q-binomial coefficients. Gaussian elimination and Grassmannians over finite fields. Permutations of multisets.
Notes   Video

12. Mon, Mar 15. The Robinson-Schensted-(Knuth) correspondence (RSK). Increasing and descreasing subsequences in permutations. 321-avoiding permutations and Dyck paths. Up and down operators on Young's lattice.
Notes   Video

13. Wed, Mar 17. Differential posets. Up and down operators. The Fibonacci lattice.
Notes   Video

14. Fri, Mar 19. Differential posets and rook placements. Oscillating tableaux. Perfect matchings and involutions.
Notes   Video

15. Wed, Mar 24. Partition theory. Generating functions. Partitions with odd and distinct parts. Self-conjugate partitions. Euler's Pentagonal Theorem.
Notes   Video

16. Fri, Mar 26. Partition theory (cont'd). Jacobi triple product. Euler's and Gauss' identities. Gaussian q-binomial coefficients again.
Notes   Video

17. Mon, Mar 29. Sylvester's proof of unimodality of Gaussian q-binomial coefficients.
Notes   Video

18. Wed, Mar 31. Counting labelled trees. Cayley's formula: algebraic proof by Renyi and bijective proof by Egecioglu-Remmel.
Notes   Video

19. Fri, Apr 2. Two bijective proofs of Cayley's formula. Prufer coding and decoding. Spanning trees in graphs. Spanning trees in the complete bipartite graph K_{m,n}. Laplacian matrix (a.k.a Kirchhoff matrix). Kirchhoff's Matrix Tree Theorem.
Notes   Video

20. Mon, Apr 5. Proof of the Matrix Tree Theorem based on the Cauchy-Binet formula. Examples: Cayley's formula and the number of spanning trees in the d-hypercube.
Notes   Video

21. Wed, Apr 7. Eigenvalues of the adjacency matrix vs eigenvalues of the Laplacian. Products of graphs. Spanning trees of the hypercube.
Notes   Video

22. Fri, Apr 9. Reciprocity formula for spanning trees. Extensions of the MTT: weighted version and directed version. Arborescences and cofactors of the Laplacian matrix.
Notes   Video

23. Mon, Apr 12. Two proofs of the directed matrix tree theorem: (a) by induction, (b) by involution principle. Increasing trees. Abel's binomial formula.
Notes   Video

24. Wed, Apr 14. Electrical networks. Kirchhoff's and Ohm's laws. The Kirchhoff's matrix. Resistances of networks via spanning trees. Series-parallel networks.
Notes   Video

25. Wed, Apr 15. Electrical networks (cont'd). Inverse boundary problem. Y-Delta transforms. Random walks on graphs and electrical networks. Interpretations of the Cayley number n^{n-2}: spanning trees, parking functions, regions of the Shi arrangement, volume of the permutohedron.
Notes   Video

26. Wed, Apr 21. Parking functions. Labelled Dyck paths. Generalized parking functions.
Notes   Video

27. Fri, Apr 23. Chip firing game and the Abelian sandpile model.
Notes   Video

28. Mon, Apr 26. The Abelian sandpile model (cont'd). Avalanche operators, recurrent configurations, the sandpile group.
Notes   Video

29. Wed, Apr 28. G-parking functions. Directed Eulerian cycles. BEST Theorem.
Notes   Video

30. Fri. Apr 30. Inversions in trees. Tree inversion polynomials and parking functions. Recurrence relations for the inversion polynomials. Alternating permutations.
Notes   Video

31. Mon. May 3. Alternating permutations (cont'd). Tangent and secant numbers. The Euler-Bernoulli triangle. The bijection between (complete) increasing binary trees and (alternating) permutations. Motzkin paths.
Notes   Video

32. Wed. May 5. Enumeration of weighted Motzkin and Dyck paths. The Francon-Viennot bijection. Continued fractions and Flajolet's fundamental lemma. Increasing 012-trees.
Notes   Video

33. Mon, May 10. Lattice paths and continued fractions (cont'd). Lindstrom's lemma aka the Gessel-Viennot method.
Notes   Video

34. Wed, May 12. Counting plane partitions: determinantal formula and MacMahon's formula. Rhombus tilings, pseudo-line arrangements, and perfect matchings. The arctic circle phenomenon.
Notes   Video

35. Fri, May 14. Colorings of graphs. The chromatic polynomial. Acyclic orientations. Chordal graphs.
Notes   Video

36. Mon, May 17. The Tutte polynomial.
Notes   Video

37. Wed, May 19. Domino tilings. Enumeration of perfect matchings. Kasteleyn's theorem. Permanents and determinants. Domino tilings and spanning trees. Temperley's theorem.
Notes   Video

Recommended Textbooks: (The students are not required to buy these books.)

The course will more or less cover the textbook:
[AC]  Algebraic Combinatorics: Walks, Trees, Tableaux, and More by R. P. Stanley, Springer: PDF