Research Experiences in Algebraic Combinatorics

Kezia's Reach Webpage

Research:

This semester I worked with Harvard student, Trevor Bass, to determine a combinatorial model for rectangular and octagonal figures embedded within the aztec diamond. We were able to determine a recurrence relation to determine the number of tilings of Aztec octagons using Kuo's condensation. A draft of the paper we are working can be read here

I will be working on the p-adic properties of enumeration of domino tilings.

History
Mathematicians John, Sachs and Zernitz found a formula for the exponent of two in the number of domino tilings of an m x n rectangular checkerboard based on the prime factorization of that number. This work was continued in 1961 by In 1961 Temperley and Fisher, and Kasteleyn. Pachter later showed that the number of tilings of a 2n x 2n square is always 2^n by an odd perfect square. Using this information Cohn was able to show that the number of tilings of a 2n x 2n board shows 2-adic continuity.

Work Plan
I will be working with a group to continue and extend Cohn's work. We shall look at 3-adic properties of domino tilings of an m x n board and then we will extend it to look at p-adic properties. The plan for this semester is:

1. The first few weeks will be spent becoming familiar with the techniques needed to approach such a question. This includes learning about domino tilings, p-adic theory and reading Cohn's paper. Pachter and Kasteleyn's papers will be studied to grasp their methods in order to use them in our research.
2. Then for the remainder of the semester, work will be done to attempt to prove Professor James Propp's conjecture that N(2n,2n) is divisible by 3 whenever n is congruent to 2 mod 5.
3. Finally work will be done to show 2-adic continuity of N(n, 2n) as a function of m and to prove that N(n, 2n) is congruent to 1 modulo 4.

I am working with:
Siddique Khan
John Gonzales
Amanda Beeson
Trevor Bass

Our Reach research has slightly changed direction. Eric Kuo wrote a math paper describing how condensation can be used to determine certain properties of an Aztec diamond. We shall use Kuo's condensation, specifically, how the octahedral recurrence, to determine the number of tiling of an m x n rectangle embedded in the Aztec Diamond. We shall also look at the relationship between the weighting the faces of the diamond and weighting the edges of the diamond, to determine why the both give the same answer even though they have different starting conditions.

Weekly summary of work done:

Total hours to date: 74 hrs

Contact Information

Kezia Charles
kezia@mit.edu
617-225-8419
IM: keziac