Last Update: Oct. 24, 2020

The Class should be divided into up to 8 teams, each with a team leader. Each team can contain from 1 to 4 students (including the team leaders, ideal size: 3). The team leader will be in charge of coordinating the various contributions, and managing the extension of the Maple package that accompany the project, that I started but should be extended considerably. The team leader would also be in charge of comminicating the output files, fairly large "data-bases" that would generate new sequences, and study them. The minimum requirement would be listed below, but it is hoped that students can study the data and make interesting conjectures and perhaps even theorems. There should also be a write-up, that my, potentially, lead to a paper in the arxiv, and possibly a "real" journal. This "polishing up" can be done after the class is over. But the preliminary version of

• The extended Maple package

• The output file(s) consisting of many integer sequences (in the style of the OEIS)

• The write-up (a .pdf) summarizing the approach, describing the purpose of the project, and describing any global trends (either conjectured or proved)

For examples (from previous graduate classes See example from 2019, another example (of a different kind). See also an example from 2018 and another example from 2018.

Suggested Projects

• Extend and use the Maple package ComboProject1.txt

  Its purpose is to generate and investigate integer sequences counting the number of HAMILTONIAN CYCLES for interesting graphs that come from Chess. Generalizing and Extending Euler's Knight's tour

• Extend and use the Maple package ComboProject2.txt

  Its purpose is to create a database of sequences enumerating Lattice Walks to the diagonal in the 2-Dimensional Manhattan Lattice for many sets of atomic steps and also counting those walks that stay in x ≥ y . It also finds their recurrences, growth rates, critical exponents, asymptotics, and congruence properties The final output is a list of lists arranged in LEXICGORAPHIC ORDER

• Extend and use the Maple package ComboProject3.txt

  Its purpose is to create a database of sequences enumerating` Lattice Walks to the diagonal in the 3-Dimensional Manhattan Lattice for many sets of atomic steps` and also counting those walks that stay in x>=y>=z It also finds their recurrences, growth rates, critical exponents, asymptotics, and congruence properties. The final output is a list of lists arranged in LEXICGORAPHIC ORDER.

• Extend and use the Maple package ComboProject4.txt

  Its purpose is to generate a database of all binomial coefficients sum of the form

  Sum (binomial(n,k)*binomial(a1*n+b1*k,k),k=0..n)*x^k for all non-trivial a1, b1,x <=K for some fixed K

  Also

  Sum (binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k)*x^k,k=0..n)

  for all non-trivial a1, b1,a2,b2,x <=K for some fixed K

  Also

  Sum (binomial(n,k)*binomial(a1*n+b1*k,k)*binomial(a2*n+b2*k,k),k=0..n)*binomial(a3*n+b3*n)*x^k,k=0..n) for all non-trivial a1, b1,a2,b2,a3,b3,x <=K for some fixed K

  It gives databases with the beginning part of each sequence, the recurrence (generated by the Zeilberger algorithm), And growh constants and critical exponents. It also tries to investigate congruence properties in the style of Ira Gessel's nice paper.

• Extend and use the Maple package ComboProject5.txt

• Extend and use the Maple package ComboProject6.txt

You are welcome to suggest other projects.

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