Suggested Projects for Math 640: EXPERIMENTAL MATHEMATICS, Spring 2012

First posted: March 21, 2012

Last Update: May 6, 2012

Added May 6, 2012: See the FINAL PROJECTS

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]

  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 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]