Winter 2011
· Instructor: Sergi Elizalde
· Lectures: MWF 11:15  12:20 in Kemeny 004
· Xperiod: Tu 12:00  12:50
· Office Hours: M 10:0011:00, F 1:403:30
· Office: Kemeny 332
· Email:
· Phone: 6468191
Announcements
See below for the tentative schedule of student
presentations.
The final exam will be handed out on Wednesday, Mar 9, and
due on Monday, Mar 14.
Homework
·
Problem Set 1. Due
on Wednesday, 1/19/11.
·
Problem Set 2. Due
on Friday, 2/4/11.
·
Problem Set 3. Due
on Friday, 2/18/11.
·
Problem Set 4. Due
on Friday, 3/4/11.
Recommended texts
Even though there is no official textbook, much of the material will be taken from the second edition of Richard Stanley’s Enumerative Combinatorics, Volume 1. An online version of this new (not yet published) edition can be downloaded here.
Other topics will be taken from:
Tentative syllabus
Generating functions [FS Ch
III; EC 1.1 and Ch 46; Wilf’s Generatingfunctionology,
available here.]
·
How to count; formal power series
[EC 1.1, Chapter 4 of these notes]
·
The symbolic method for unlabeled
structures; ordinary generating functions; triangulations, lattice paths,
compositions, plane trees [FS Ch I, Chapter 5 of these notes]
·
The symbolic method for labeled
structures; exponential generating functions; the Lagrange inversion formula;
labeled trees, set partitions, permutations, involutions, derangements, Stirling numbers [FS Ch II, EC Ch 5, Chapter 6 of these notes]
·
Examples of bivariate
generating functions; statistics on Dyck paths
[Notes]
·
Rational generating functions [EC
4.1]
·
Polynomials and quasipolynomials
[EC 4.34.4]
·
Algebraic generating functions [EC
6.16.2]
·
Dfinite generating functions [EC
6.4]
Permutations [EC Ch 1, Bo Ch
13]
·
Cycles, lefttoright maxima,
inversions [EC 1.3]
·
Descents, Eulerian
numbers, major index, excedances [EC 1.4]
·
Geometric representations of
permutations, increasing trees [EC 1.5]
·
Alternating permutations, Euler
numbers [EC 1.6]
·
Permutations of multisets,
qbinomial coefficients [EC 1.7]
Partitions and tableaux [EC
1.8, BS 26, EC Ch 7]

Generating functions for partitions
[EC 1.8, BS 2]

Euler’s pentagonal theorem [EC 1.8]

Standard Young tableaux, the
Hooklength formula [BS 4]

The RSK algorithm [EC 7.11, 7.13, Bo
7.1]

Increasing and decreasing
subsequences [BS 6]
Introduction to pattern
avoidance [Bo Ch 4]

Enumerative results, Wilfequivalence classes, patterns of length 3 and 4 [Bo
4.14.4]

Asymptotic behavior, the StanleyWilf conjecture [Bo 4.5]
Student presentations
·
Fri,
2/25 – Scott, Möbius function and set partititons
·
Mon,
2/28 – Jonathan, Viennot’s geometric version of RSK
·
Mon,
2/28 – Dan, Connections with topology
·
Wed, 3/2 – Zach, Connections of
Young tableaux and RSK with representation theory
·
Wed,
3/2 – Kassie, Minmax trees and the cdindex
·
Fri, 3/4 – Megan, Proof of the
hooklength formula
·
Fri, 3/4 – Sarah, A combinatorial
proof of the Lagrange Inversion Formula
·
Mon, 3/7 – Carter, Proof of StanleyWilf conjecture
·
Mon, 3/7 – Nathan, The
transfermatrix method
Homework, exams, and grading
The course grade will be based on the homework (30%), a presentation in class (20%), and a takehome final exam (50%).
Collaboration is encouraged on the homework, but you are not allowed to copy someone else's work. The solutions must be written individually. You have to mention on your problem set the names of the students that you worked with, and also if you used some books or articles that are not on the above list.
Possible student presentations
Here is a list of suggested topics for student presentations. Please let me know once you have chosen one. If you have some other topic in mind that you would like to talk about, feel free to discuss it with me. Each student should plan a 30min presentation, more or less.

Proof
of the hooklength formula [Novelli, Pak, Stoyanovskii, A direct bijective
proof of the hooklength formula, Disc.
Math. Comp Sci. 1 (1997), 5367] OR [Greene, Nijenhuis,
Wilf, A probabilistic proof of a formula for the
number of Young tableaux of a given shape, Adv.
in Math. 31 (1979) 104–109] → taken by Megan

Connections with representation theory [BS 5] → taken by Zach

Reduced
decompositions [BS 7]

Möbius function and set partititons [BS
10]
→ taken by Scott

Hyperplane arrangements [BS 11]

Face
numbers of polytopes [BS 12]

Connections
with topology [BS 13] →
taken by Danny

Proof
of StanleyWilf conjecture [Bo 4.5] [Marcus, Tardos, Excluded permutation matrices and the StanleyWilf conjecture, J. Combin. Theory Ser. A 107 (2004),
153–160] → taken by Carter

Viennot’s
geometric version of RSK [Section 3.6 of Sagan, The Symmetric Group, Springer] → taken by
Jonathan

Combinatorial
proof(s) of the Lagrange Inversion Formula [EC Theorem 5.4.2 (second and/or
third proof). The proofs make some references to previous results in the
chapter] → taken by
Sarah

Bijective
proof of the symmetry of the joint distribution of the inversion number and the
major index on permutations [Foata, Schutzenberger, Major index and inversion
number of permutations, Math. Nach.
83 (1978), 143159] [EC1 pg. 4142 gives an indirect proof]

Minmax
trees and the cdindex
[EC 1.6.3] →
taken by Kassie

The
transfermatrix method [EC 4.7] →
taken by Nathan
Students with
disabilities: Students
with learning, physical, or psychiatric disabilities enrolled in this course
that may need disabilityrelated classroom accommodations are encouraged to
make an office appointment to see me before the end of the second week of the
term. All discussions will remain confidential, although the Student
Disability Services office may be consulted to discuss appropriate implementation of any accommodation requested.