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)
02/27/2019 update: 5 bonus problems are added
(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.
(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.
Up and down operators.
Differential posets. Fibonacci lattice.
Unimodality of Gaussian coefficients.
Sperner's and Dilworth's theorems.
Symmetric chain decompositions.
Mon 03/25/2019 - Fri 03/29/2019 no classes (Spring Break)
Mon 04/15/2019: no classes (Patriots' Day)
Last updated: February 27, 2019