Suggested Projects for Math 640: EXPERIMENTAL MATHEMATICS, Spring 2008

Last Update: April 14, 2008.

Feel free to propose your own project!

The suggestions below are very brief. Feel free to ask me more. Once you pick a project, I'll tell you more about it.

Untaken Projects

1. Use Goulden-Jackson to systematically study ALL generating functions for ALL possible sets of mistakes (of bounded lengths and size of course). Find "champions", in asymptotic size of avoiding. Also use Goulden-Jackson to compute higher moments and correlations.
2. Study the literature on isomer enumeration, and try to extend it using Maple and Polya theory (implemented in Maple).
3. Use Experimental mathematics to find Symmetric Chain Decompositions of some lattices.
4. Study one-dimensional self-avoiding walks with an arbitrary set of steps (Note: if the set of steps is {-1,1}, then it is trivial). In other words, given a set of positive and negative integers, S, what can you say about the sequence

   a(n,t):=number of lists [v₁,v₂, ..., v_t], such that for each i, the partial sums
   v₁+v₂+ ... +v_i
   are never 0, and v₁+v₂+ ... +v_t=n.

   In particular, what can you say about the enumerating sequence of all t-step self-avoiding walks, i.e.
   A(t):= Σ_{t*min(S) ≤ n ≤t*max(S) n}   a(n,t)
   

5. Try to find Rogers-Ramanujan-style theorems empirically.
6. Investigate moments and possibly other statistical properties of the random variable "number of Young Tableaux of a given shape". Also investigate empirically, and possibly discover new ones, the identities in Guoniu Han's paper,

Taken Projects

• Use the Matrix Tree Theorem to find sequences for the number of spanning trees of infinite families of graphs. Try to conjecture expressions for them or their generating functions (automatically of course!). [Paul Raff]
  Read Paul Raff's great paper.
  
• Interface Goulden-Jackson with natural language (e.g. English), see how far it predicts the actual behavior, by analysing texts that you can easily download form the internet.
  You may use the List of English words, in Maple format.
  [Beth Kupin and Debbie Yuster]
  Here is Beth Kupin and Debbie Yuster's awesome Final Project.
• Interface Goulden-Jackson with "natural" language, i.e. the language of nature, i.e. DNA. Look up, in google scholar and MathSciNet, the few applications of Goulden-Jackson already in biology, and extend them. [Debbie Yuster, in collaboration with Beth Kupin]
• Interface Goulden-Jakson with Dyck words. [Andrew Baxter]
• Investigate solid partitions [Brian Nakamura].
  Here is Brian Nakamura's Final Project.
• Investigate 3D Young Tableaux [Emilie Hogan]
  Here is Emilie Hogan's Maple Package and Emilie Hogan's writeup