Math 824 (Algebraic Combinatorics)
Fall 2016
General Information 
Lecture Notes 
Problem Sets 
Textbooks 
Final Project 
Links
General Information
 Instructor:
Jeremy Martin
 Email: (the best way to contact me)
 Office: 618 Snow Hall, (785) 8647114
 Office hours: Wednesdays, 34pm, or by appointment
 Lectures:
MWF, 1:001:50pm, 456 Snow
 Prerequisite: Math 724 (Enumerative Combinatorics) or permission of the instructor. Math 790 (Linear Algebra) is not an official requirement but is strongly encouraged.
 KU course number: 66459
 Syllabus
Lecture Notes
Lecture
Notes on Algebraic Combinatorics (one big PDF file; last update
12/7/16)
These notes are licensed under a Creative
Commons AttributionNonCommercialShareAlike 3.0 Unported License.
In short, use them freely but do not sell them or anything derived from
them. If you are a KU graduate student (or for that matter, if you
aren't), do not print out the full set of notes on the department's
printer!
Lecture topics, day by day:
 Mon 8/22: Posets  basic definitions and examples
 Wed 8/22: Young's lattice; simplicial complexes
 Fri 8/24: StanleyReisner rings; f and hvectors; shellability
 Mon 8/29: Polytopes
 Wed 8/31: The fundamental theorem; the BruggesserMani proof of the DehnSommerville relations
 Fri 9/2: Distributive lattices
 Mon 9/5: No class (Labor Day)
 Wed 9/7: Proof of the FTFDL
 Fri 9/9: Modular lattices
 Mon 9/12: Semimodular lattices
 Wed 9/14: Geometric lattices
 Fri 9/16: Matroid closure operators and rank functions
 Mon 9/19: Graphic matroids; independence systems
 Wed 9/21: Equivalence of rank functions and independence systems; matroid complexes; basis systems
 Fri 9/23: Other matroid formulations; representability; the Fano and nonPappus matroids
 Mon 9/26: Direct sum and duality
 Wed 9/28: Deletion and contraction
 Fri 9/30: The Tutte polynomial: examples and proof of equivalence of the closedform and recursive formulas
 Mon 10/3: The Tutte polynomial: recipe formulas and activities
 Wed 10/5: The Tutte polynomial: colorings and orientations
 Fri 10/7: Guest lecture by Hailong Dao
 Mon 10/10: No class (Fall Break)
 Wed 10/12: The incidence algbra and the Möbius function
 Fri 10/14: Möbius inversion
 Mon 10/17: The Möbius algebra; Weisner's theorem
 Wed 10/19: The crosscut theorem; introduction to hyperplane arrangements
 Fri 10/21: Hyperplane arrangements: basic facts, examples, counting regions
 Mon 10/24: Zaslavsky's theorems
 Wed 10/26: The finite field method; supersolvable arrangements
 Fri 10/28: Guest lecture by Marge Bayer: Ehrhart theory
 Mon 10/31: Introduction to representations
 Wed 11/2: Proof of Maschke's theorem; characters and functors
 Fri 11/4: Schur's Lemma and the Fundamental Theorem of Character Theory
 Mon 11/7: Character tables; onedimensional representations and Pontrjagin duality
 Wed 11/9: Representations of the symmetric group; restriction and induction (barely)
 Fri 11/11: Symmetric functions I (Marge Bayer)
 Mon 11/14: Symmetric functions II (Marge Bayer)
 Wed 11/16: Symmetric functions III (Marge Bayer)
 Fri 11/18: Schur functions: proof of symmetry
 Mon 11/21: The Cauchy and dual Cauchy kernels and the Hall inner product
 Wed 11/23: No class  Thanksgiving
 Fri 11/25: No class  Thanksgiving
 Mon 11/28: The RobinsonSchenstedKnuth correspondence; generalized RSK and Schur functions
 Wed 11/30: Restricted and induced representations and Frobenius reciprocity; the Frobenius characteristic map
 Fri 12/2: The hooklength formula
 Mon 12/5: Presentations (Smita and Amanda)
 Wed 12/7: The JacobiTrudi determinant definition of Schur functions
Problem Sets
Problem sets will be assigned approximately biweekly, at least one week before the
deadline.
All solutions must be typeset using LaTeX. Email me your solutions as a PDF file under the name
{yourlastname}{numberofproblemset}.pdf (e.g., "Rota5.pdf"). In general, the
deadline is 11:59pm of the due date.
 Problem Set 1, due Tuesday 9/6
 Problem Set 2, due Friday 9/16
 Problem Set 3, due Monday 10/3: Exercises
3.1, 3.2, 3.6, 3.7, 3.8, and at least one of 3.9 and 3.10
 Problem Set 4, due Friday 10/21: Exercises 4.3, 4.4, 5.1, 5.2
 Problem Set 5, due Friday 11/11: Exercises 5.3, 5.6, 6.2, 6.3, 8.1, 8.2
 Problem Set 6, due Thursday 12/8: Exercises 8.3, 8.4, 9.2, 9.4, 9.5, and at least one of 8.6 or 9.3
Additional resources:
 Here is a header file with useful macros
(last update 8/26/16).
 Here is a sample source
file that uses the header (and will produce the first couple of pages of the lecture notes).
 Packages for making figures include TikZ, Ipe, and xfig.
Textbooks
We will follow the lecture
notes rather than any one specific textbook.
However, the following books may be
helpful (and you should definitely obtain the free downloads). All
these books can be perused in Jeremy's office.
 Richard P. Stanley, Enumerative Combinatorics, volume 1, 2nd ed. (Cambridge, 1997)
(Enumeration; posets and lattices; generating functions)
Buy it from the publisher or
download the free preprint version from Stanley's website
 Richard P. Stanley, Enumerative Combinatorics, volume 2 (Cambridge, 1999)
(More enumeration, including exponential generating functions; symmetric functions)
Buy it from the publisher
 Martin Aigner, A Course in Enumeration (Springer, 2007)
(Enumeration; posets, lattices, and matroids)
 Martin Aigner, Combinatorial Theory (Springer, 1997)
(Enumerative combinatorics, symmetric functions, and matroids)
 Richard P. Stanley, Hyperplane Arrangements
(lecture notes available free online)
 Alexander Schrijver, A Course in Combinatorial
Optimization (lecture notes available free online)
 Thomas Brylawski and James Oxley, The Tutte polynomial and its
applications, Chapter 6 of Matroid applications, N. White, ed.
(Cambridge Univ. Press, 1992)
 Matthias Beck and Raman Sanyal, Combinatorial Reciprocity
Theorems: A Snapshot of Enumerative Combinatorics from a Geometric
Viewpoint (manuscript available free online)
 Bruce
Sagan, The Symmetric Group, 2nd edn. (Springer, 2001)
 Federico
Ardila, Algebraic and geometric methods in enumerative
combinatorics (free book; covers much of the same material as my
notes but from a slightly different perspective, and probably a more
polished presentation)
Final Project
 The assignment. Everyone should meet with Jeremy and choose their paper by October 28.
 The schedule of talks and reviewers is below. If the links to papers don't work, try searching for the paper using MathSciNet
(requires KUID).
Date 
Presenter 
Reviewer 
Paper 
December 5 
Smita Praharaj 
Amanda Wilkens 
A. Knutson and T. Tao
Honeycombs and sums of Hermitian matrices
Notices Amer. Math. Soc. 48, no. 2 (2001), 175186. 
December 5 
Amanda Wilkens 
Smita Praharaj 
B.E. Sagan
Probabilistic proofs of hook length formulas involving trees
Sém. Lothar. Combin. 61A (2009/10), Art. B61Ab, 10 pp.

December 14 
Brandon Caudell 
Connor Smith 
J. Edmonds
Minimum partition of a matroid into independent subsets
J. Res. Nat. Bur. Standards Sect. B 69B (1965), 6772.

December 14 
Joseph Doolittle 
Brandon Caudell 
R.P. Stanley
Two poset polytopes
Discrete Comput. Geom. 1 (1986), 923. 
December 14 
Arturo Jaramillo 
Bibekananda Mishra 
A. Nica and R. Speicher
Noncrossing partitions and free cumulants
From Lectures on the combinatorics of free probability (Cambridge U. Press, 2006), pp. 135194 (mainly pp. 173194) 
December 14 
Bibekananda Mishra 
Joseph Doolittle 
P. Orlik and L. Solomon
Combinatorics and topology of complements of hyperplanes
Invent. Math. 56 (1980), no. 2, 167189. 
December 14 
Connor Smith 
Arturo Jaramillo 
A. Goodall
Fourier analysis on finite Abelian groups: some graphical applications
From Combinatorics, Complexity and Chance: a tribute to Dominic Welsh (G. Grimmett and C. McDiarmid, eds.), Oxford U. Press, 2007, 103129. 
Links
KU links
Software and online resources
Last updated Fri 12/9/16 8:00am