# Suggested Projects for Math 640: EXPERIMENTAL MATHEMATICS, Spring 2012

http://www.math.rutgers.edu/~zeilberg/EM12/SugestedProjects.html

First posted: March 21, 2012

Last Update: May 6, 2012

Added May 6, 2012: See the FINAL PROJECTS

Added May 6, 2012: the page below is preserved for historical reasons. You must have committed to a project by April 4, 2012. First draft due May 4, 2012, 11:59:59pm . It should be in the same public-secret directory where your homework resides. As with the homework, please indicate whether you give me permission to post it for the whole world to see.

Feel free to propose your own project! As long as it uses Maple (or even Mathematica, or SAGE) in a non-trivial way it is fine. It does not have to do anything with sequences

NOTE: Any project can be done by n people, with 1 ≤ n ≤ 2 .

## Untaken Projects

• Create a database of all sequences defined by binomial coefficients identities of a given bounded, "complexity", and in the process let the compute discover (old and new) binomial coefficient identities that the computer can automatically prove using the celebrated Zeilberger algorithm.
• Create a database of sequences enumerating integer sequences of various restrictions (the most interesting of which should go to the OEIS) and in the process search for partition identities of Odd=Distinct and Rogers-Ramanujan type, hopefully discovering new ones.
• Create a database (the most interesting of which should go to the OEIS) of sequences counting ordered trees with various conditions of number and children AND (possibly also forbidden subtrees). Use Lagrange-Inversion and Almkvist-Zeilberger to find recurrences for these sequences.
• Create a database (the most interesting of which should go to the OEIS) of sequences counting rooted labelled trees of bounded degree with various conditions of number and children AND Use Lagrange-Inversion and Almkvist-Zeilberger to find recurrences for these sequences.

## Taken Projects

• Study Tetrahedra with integer sides
[Suggested and claimed by Philip Benjamin and Kristen Lew]

• Apply Experimental Mathematics to deep questions in Number Theory [Suggested and claimed by John Miller]

• Systematically discover and try and prove (if possible automatically) Generalized Somos-type sequences [Suggested and claimed by Matthew C. Russell]

• The so-called alternating permutations may be described as permutations that avoid the "patterns" {uu, dd}. Write a Maple package to enumerate permutations avoiding an arbitrary set of patterns (e.g. {udu,uuu}) and that conjectures (and who knows? even proves) exponential generating functions in the style of André's celebrated sec(x)+tan(x) .
[Claimed by Alex Gentul and Ben Miles]

• Create a database of sequences counting self-avoiding walks avoiding in addition certain sets as subpaths.
[Claimed by Neil Lutz]

• Let ex(n,H) be the maximum number of edges in a n vertex graph that does not contain H as a subgraph (i.e. ex(n,H)+1 is the minimum number of edges that guarantees H is a subgraph). Compute seq(ex(n,H),n=1..N) for various graphs H.
[Suggested and claimed by John Kim]

• Extend the Sontag-Zeilberger article "A Symbolic Computation Approach to a Problem Involving Multivariate Poisson Distributions" that relied so heavily on WZ Theory (cf. "An algorithmic proof theory for hypergeometric (ordinary and "q") multisum/integral identities").
[Suggested and claimed by Michael Burkhardt]

• Explore systematically C-finite representations of sequences enumerating wide classes of forbidden patterns, with various alphabets.
[Suggested and Claimed by Charles Wolf]

• Study binomial coefficients for Macdonald polynomials
[Suggested and Claimed by Yusra Naqvi]

• Create a database of sequences related to linguistic extensions of Szemerédi's theorem.
[Claimed by Nathaniel Shar]

• Create a database of sequences enumerating integer partitions with various restrictions, trying to discover empirically new Rogers-Ramanujan type identities, or at least rediscover them systematically.
[Claimed by Patrick Devlin]

Experimental Math, Spring 2012 main page