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 publicsecret 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 nontrivial 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
[Please ask me for details, if you are interested]

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
RogersRamanujan 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 LagrangeInversion and AlmkvistZeilberger 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 LagrangeInversion and AlmkvistZeilberger to find recurrences for these sequences.
Taken Projects

Study Tetrahedra with integer sides
[Suggested and claimed by Philip Benjamin and Kristen Lew]
See:
problem description,
and
Sastry's article about Brahmagupta Quadrilaterals .

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 Somostype sequences
[Suggested and claimed by Matthew C. Russell]

The socalled
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
selfavoiding 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
SontagZeilberger 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 Cfinite 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 RogersRamanujan type identities, or at least rediscover them
systematically.
[Claimed by Patrick Devlin]
Experimental Math, Spring 2012 main page