Alexander Postnikov: 18.212 Algebraic Combinatorics
Instructor: Alexander Postnikov
Time: Monday, Wednesday, Friday 2:00-3:00 pm
Office hours: Monday 3-4 pm or by appointment, Room 2-367
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,
convex polytopes, and other topics ...
Level: advanced undergraduate
Grading: Based on several Problems Sets.
- Catalan numbers, Dyck paths, triangulations, noncrossing set partitions
- symmetric group, statistics on permutations, inversions and major index
- partially ordered sets and lattices, Sperner's and Dilworth's theorems
- q-binomial coefficients, Gaussian coefficients, and Young diagrams
- Young's lattice, tableaux, Schensted's correspondence, RSK
- partitions, Euler's pentagonal theorem, Jacobi triple product
- noncrossing 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...
- Problem Set 1 (due March 04, 2019)
Problem Set 2 (due April 5, 2019)
Problem Set 3 (due May 10, 2019)
Correction: In Problem 5 assume that the n-cube graph is a bidirectected graph
with each edge directed both way. An Eulerian cycle should pass each edge once in each direction. As a bonus question try to think about undirected
version of this problem.
(The students are not required to buy these books.)
The course will more or less cover the textbook:
Algebraic Combinatorics: Walks, Trees, Tableaux, and More
by R. P. Stanley, Springer, 2nd ed, 2018.
Version of 2013 is available as
Enumerative Combinatorics, Vol 1 and Vol 2, by R. P. Stanley,
Cambridge University Press, 2011 and 2001.
Volume 1 is available as
Course in Combinatorics by J. H. van Lint and R. M. Wilson,
Cambridge University Press, 2001.
Lecture Notes by Andrew Lin.
(with links to additional reading materials):
Wed 02/06/2019: Catalan numbers: drunkard's walk problem, generating function,
Catalan numbers (cont'd): formula for C_n, reflection principle, necklaces,
triangulations of polygons, plane binary trees, parenthesizations
Pattern avoidance in permutations, stack- and queue-sortable permutations,
Young diagrams and Young tableaux, the hook-length formula
Frobenius-Young identity, Schensted correspondence,
longest increasing and decreasing subsequences in permutations
Fri 02/15/2019: Proof of the hook-length formula based on
a random hook walk
Mon 02/18/2019: no classes (President's Day)
Tues 02/19/2019 (Monday schedule):
Hook walks (cont'd). Linear extensions of posets. Hook-length-type
formulas for shifted shapes and trees.
- D. E. Knuth: The Art of Computer Programming, Volume 3: Sorting and
Searching, 1973, Section 5.1.4.
Wed 02/20/2019: q-factorials and q-binomial coefficients
Fri 02/22/2019: Grassmannians over finite fields: Gaussian elimination
and row-reduced echelon form
Sets and multisets. Statistics on permutations:
inversions, cycles, descents.
Statistics on permutations (cont'd). Equidistributed statistics.
Major index. Records. Exceedances. Stirling numbers.
Fri 03/01/2019: Stirling numbers (cont'd).
Set-partitions. Rook placements on triangular boards.
Non-crossing and non-nesting set-partitions.
- Mon 03/04/2019:
Problem Set 1 is due.
Increasing binary trees.
3 Pascal-like triangles: Eulerian triangles, Stirling
triangles of 1st and 2nd kind.
Wed 03/06/2019: Discussion of Problem Set 1.
Volunteers can present solutions of some problems.
Try to limit your presentations to 5-10 min.
- Fri 03/08/2019:
Discussion of Problem Set 1 (cont'd).
Posets and lattices. Boolean lattice. Partition lattice. Young's lattice.
Birkhoff's fundamental theorem for finite distributive lattices.
Symmetric chain decompositions.
Sperner's and Dilworth's theorems.
Greene's theorem vs Schensted correspondence.
Up and down operators.
Differential posets (cont'd). Fibonacci lattice.
Unimodality of Gaussian coefficients.
Proof of unimodality of Gaussian coefficients (cont'd).
Theory of partitions. Euler's pentagonal number theorem.
Mon 03/25/2019 - Fri 03/29/2019 no classes (Spring Break)
Mon 04/01/2019: Partition theory (cont'd). Franklin's combinatorial
proof of Euler's pentagonal number theorem.
Jacobi's triple product identity.
Partition theory (cont'd).
Combinatorial proof of Jacobi's triple product identity.
Enumeration of trees. Cayley's formula.
Simple inductive proof of Cayley's formula.
Problem Set 2 is due.
Two combinatorial proofs of Cayley's formula.
Discussion of Problem Set 2.
Discussion of Problem Set 2 (cont'd).
Matrix Tree Theorem.
Spanning trees. Laplacian matrix of a graph.
Reciprocity formula for spanning trees.
Examples: complete graphs, complete bipartite graphs.
Mon 04/15/2019: no classes (Patriots' Day)
Matrix Tree Theorem (cont'd).
Products of graphs. Number of spanning trees
in the hypercube graph.
Oriented incidence matrix.
Fri 04/19/2019: Proof of Matrix Tree Theorem using
Weighted and directed version of Matrix Tree Theorem.
In-trees and out-trees.
Mon 04/22/2019: Proof of Directed Matrix Tree Theorem
based on induction.
Wed 04/24/2019: Proof of Directed Matrix Tree Theorem via
the Involution Principle. Electrical Networks and Kirchhoff's Laws.
Fri 04/26/2019: Electrical networks (cont'd). 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.
Mon 04/29/2019: Eulerian cycles in digraphs and B.E.S.T. theorem.
Wed 05/01/2019: Parking functions.
Tree inversion polynomials.
Problem Set 3 is due.
Last updated: May 2, 2019