## Information About the Final Project

A research paper is due at the end of the term. It should be on a topic related to course material, unless permission is given otherwise. The paper is not expected to contain original research, but it should have some element of novelty, such as unifying the material from several sources, simplifying a part of a proof, creating interesting examples, etc. Unless it consists primarily of original research, it should contain at least two references. There is no required length, but around 7-10 LaTeX pages would be reasonable. You are encouraged to, but not required to, type your paper using LaTeX.

Reference on acknowledging sources (PDF)

Reference on writing a mathematical paper (PDF)

Samples of student papers from the MIT Undergraduate *Journal of Mathematics*, June 1999

## Possible Topics and References

Below is a list of some possible topics, with references at the end. The topics are listed roughly in the order that we encountered the related material in class. You may also consider other possible topics not listed below:

- Identities Involving Fibonacci and Lucas Numbers. Many possible sources including papers from Arthur Benjamin's Web site or the book [BQ03].
- The transfer-matrix method (an application of counting walks in graphs to other problems in enumerative combinatorics. See [Sta86, section 4.7] for basic information. Further references are cited in [Sta86, pp. 260-264], and the exercises to chapter 4 of [Sta86] contains some further examples.
- The relation between properties of a graph and the eigen values of its adjacency or Laplacian matrix. Many possibilities, see [CDS95] or [Lo93, Section 11].
- More on the Radon Transform, see [DG85]. Also a generalization of the Radon transform to Cayley Graphs and connections to character theory.
- Mixing Times for Random Walks and the Cheeger Inequality. Some links from Fan Chung Graham's Web site.
- Describe the theory of a special kind of Posets. Numerous possibilities, some of which are Binomial Posets, Distributive Lattices, Face Posets of polytopes, or Posets arising from hyperplane arrangements. A good starting point is section 3 of [Sta99]. Also see Richard Stanley's lecture notes from Park City 2004.
- Partition identities, such as the Rogers-Ramanujan identities. Some basic information appears in section 2 of [BS]. A more extensive reference is [An76].
- There are various proofs of the Hook Length Formula in the literature. One possible project would be to study one of these in depth, another would be to compare and contrast two different proofs. Some of the proofs include the probabilistic approach of Greene-Nijenhuis-Wilf [GNW79] mentioned in class, a bijective proof of Novelli-Pak-Stoyanovskii [NPS97], or a recent algebraic proof by Bandlow. (PDF)
- The full RSK algorithm for semi-standard Young tableaux, including proofs of the Symmetry Theory and other applications. See chapter 7 of [Sta99], chapter 3 of [Sag01], or part I of [F97].
- Variations of RSK and including RSK applied to Oscillating Tableaux (see for example by C. Krattenthaler). Another possibility is connection to balanced tableaux, see the next topic.
- Reduced decompositions of permutations. These concern expressing a permutation as a product of a minimal number of adjacent transpositions (i, i +1). This topic also is related to variations of RSK (balanced tableaux) and applications of the nil Coxeter algebra [FS94].
- Differential Posets, extension of material from Section 8 of the lecture notes. Also see [Sta88] and [Sta90].
- Skew-schur Functions. Many possibilities, including questions like how to write a skew-schur function as the sum of schur functions. Related to the problem of writing the product of schur functions as a sum of schur functions and Littlewood-Richardson coefficients. See chapter 7 of [Sta99] or chapter 4 of [Sag01].
- Algebraic Generating Functions. See chapter 6 [Sta99] for a good starting place. Project could also explore different combinatorial interpretation of Catalan numbers beyond what we have seen in class.
**The Following topics are related to future course material:** - Aspects of representation theory of S
_{n}. Many connections, possible topics include the Murganham-Nakayama rule, the Branching rule. See chapter 4 of [Sag01]. Also can discuss connections to representation theory of GL_{n}such as Schur-Weyl duality. - MacMahon Master Theorem. A very important combinatorial result. See [KP07] for recent discussion of variants.
- Zeta functions of algebraic curves. For those interested in connections to number theory or algebraic geometry, a beautiful subject where techniques of generating functions and symmetric functions can be used to study point-counting on curves.
- Generalizations of Polya's theorem, especially de Brujin's theorem where there is a group also acting on the set of colors. A nice survey is due to de Brujin [deB64]. Many further results appear in [HP73].
- Tree enumeration. Alargetopic, see for instance [Sta99, chapter 5] or [Mo70] for the labeled case. For counting unlabeled trees using Polya's theorem, see for instance chapter 4 of [HP73].
- The Chromatic Polynomial, the Tutte Polynomial, the Jones Polynomial of Knot theory, or connections to a cyclic orientations of a graph. See for example chapters 15-16 of [GR01].
- Study further aspects of chip-firing games on graphs, possibly including the abelian sandpile model, the computation of critical groups of graphs, and G-parking functions. See for example chapters 13-14 of [GR01].
- Alternatively, a study of classical parking functions and possibly their connections to (q, t)-Catalan numbers is another possible topic.
- Basics on Polytopes or specific examples such as Transportation and Birkhoff Polytopes, Cylic Polytopes, Associahedra, or Permutohedra. See for example [Zieg95]. Can also look at Ehrhart Reciprocity, see for example chapter 4 of [Sta86].
- Tilings. There are many results involving algebra. See also sections 7 and 8 of [BS]. The reference (PS - 1.6MB) contains many further interesting results on tilings. See Conway or Thurston for related material.
- Topics from Statisitical Mechanics such as Matching Enumeration, Pfaffians, the Ising Model, or aspects of the proof of the Alternating Sign Matrix Conjecture.
- Introduction to Cluster Algebras. Cluster algebras are quickly becoming a huge subject. Possible topics include descriptions of quivers and quiver mutation, classification of cluster algebras of finite type, or of small rank.
- The Gessel-Viennot theory of evaluation of determinants based on the combinatorics of involutions. A brief introduction appears in [Sta86, section 2.7]. Further references are [GV85] and [S-W, section 4.5] (Stanton-White -Constructive Combinatorics). Chapter 4 of this latter reference contains some further applications of involutions. Krattenthaler [Kra99] which also contains beautiful determinantal identities connected to lattice paths. There is also a recent generalization of Gessel-Viennot theory due to K. Talaska [Tal08].

## Reference List

[An76] = Andrews, George E. The *Theory of Partitions*. Reading, Mass: Addison-Wesley Pub. Co., Advanced Book Program, 1977. ISBN: 9780201135015.

[BQ03] = Benjamin, Arthur, and Jennifer J. Quinn. *Proofs That Really Count: The Art of Combinatorial Proof*. Dolciani mathematical expositions, no. 27. Washington, DC: Mathematical Association of America, 2003. ISBN: 9780883853337.

[CDS95] = Cvetković, Dragoš M., Michael Doob, and Horst Sachs. *Spectra of Graphs: Theory and Application*. Heidelberg, Germany: Johann Ambrosius Barth Verlag, 1995. ISBN: 9783335004073.

[BS] = Bjorner, A., and R. Stanley. "A Combinatorial Miscellany." (PDF)

[deB64] = De Bruijn, N. C. *Polya's Theory of Counting in Applied Combinatorial Mathematics*. Edited by E. F. Beckenback. New York, NY: John Wiley and Sons, Inc., 1964, pp. 144-184.

[DG85] = Diaconis, P., and R. L. Graham. "The Radon transform on ." *Pacific J Math* 118 (1985): 323-345.

[FS94] = Fomin, S., and R. P. Stanley. "Schubert Polynomials and the Nilcoxeter Algebra." *Advances in Mathematics* 103, no. 2 (1994): 196-207.

[F97] = Fulton, William. *Young Tableaux: With Applications to Representation Theory and Geometry*. London Mathematical Society student texts, 35. Cambridge, England: Cambridge University Press, 1996. ISBN: 9780521567244.

[GV85] = Gessel, I., and G. Viennot. "Binomial Determinants, Paths, and Hook Length Formulae." *Advances in Mathematics* 58, no. 3 (1985): 300-321.

[GNW79] = Greene, C., and A. Nijenhuis, and H. Wilf. "A Probabilistic Proof of a Formula for the Number of Young Tableaux of a Given Shape. *Advances in Mathematics* 31 (1979): 104-109. (PDF)

[GR01] = Godsil, C. D., and Gordon Royle. *Algebraic Graph Theory*. Graduate texts in mathematics, 207. New York, NY: Springer, 2001. ISBN: 9780387952413.

[HP73] = Harary, Frank, and Edgar M. Palmer. *Graphical Enumeration*. New York, NY: Academic Press, 1973. ISBN: 9780123242457.

[KP07] = Konvalinka, Matjaz, and Igor Pak. "Non-Commutative Extensions of the MacMahon Master Theorem." *Advances in Mathematics* 216, no. 1 (2007): 29.

[Kra99] = Krattenthaler, C. "Advanced Determinant Calculus." *Séminaire Lotharingien Combin* 42 ("The Andrews Festschrift"), Article B42q (1999): 67.

[Lo93] = Lovász, László. *Combinatorial Problems and Exercises*. Amsterdam, The Netherlands: North Holland, 1993. ISBN: 9780444815040.

[Mo70] = Moon, John W. *Counting Labelled Trees*. Canadian mathematical monographs, no. 1 [Montreal]: Canadian Mathematical Congress, 1970.

[NPS97] = Novelli, Jean-Christophe, Igor Pak, and Alexander V. Stoyanovskii. "A Direct Bijective Proof of the Hook-Length Formula." *Discrete Mathematics and Theoretical Computing* 1, no. 1 (1997): 53.

[Sag01] = Sagan, Bruce Eli. *The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions*. Graduate texts in mathematics, 203. New York, NY: Springer, 2001. ISBN: 9780387950679.

[Sta86] = Stanley, Richard P. *Enumerative Combinatorics*. Cambridge studies in advanced mathematics, 49, 62. Cambridge, England: Cambridge University Press, 1997. ISBN: 9780521553094.

[Sta99] = ———. *Enumerative Combinatorics*. Vol. 2. Cambridge, MA: Cambridge Univ Press, 1999. ISBN: 9780521560696.

[Sta88] = ———. "Differential Posets." *Journal of the American Mathematical Society* 1, no. 4 (October 1988): 919-961.

[Sta90] = ———. "Variations of Differential Posets." In *Invariant Theory and Tableaux, The IMA Volumes in Mathematics and Its Applications*. Vol. 19. Edited by D. Stanton. New York, NY: Springer, 1990, pp. 145-165.

[Tal08] = Talaska, K. "A Formula for Plücker Coordinates Associated with a Planar Network." *International Mathematics Research Notices*, Vol. 2008, Article ID rnn081.

[Zieg95] = Ziegler, Günter M. *Lectures on Polytopes*. Graduate texts in mathematics, 152. New York, NY: Springer-Verlag, 1994. ISBN: 9783540943655.