9/11: Took minutes for the meeting
9/15 - 9/20: Unsuccessfully tried to find a relationship between the perfect matchings of k-by-2n and k-by-4n rectangles.
9/21: Found a recurrence for the sums of the perfect matchings that stems from removal of a fixed corner vertex (and another vertex) in subsequent straight snakes
9/25: Fix four vertices of a graph G. Found a relationship between perfect matchings of a graph G, the four graphs obtained by the four permutations of removal of 2 vertices (provided that one cannot remove vertices of the same "color", and the graph obtained by removing the four vertices.
9/29: Found and proved (by induction) a formula for the Fibonacci polynomials given by F1(x,y)=y, F2(x,y)=x^2+y^2 and the recurrence relation Fn(x,y)=yF[n-1](x,y) + (x^2)F[n-2](x,y)
9/30: Convinced myself that the relationship found on 9/25 with square snakes also works with hexagonal snakes. Tried to assign weights to hexagonal snakes in accordance with their corresponding square snakes. Failed.
10/7: Worked with transfer matrices A=(1 1) and B=(0 1) that correspond to square snakes.
(1 0) (1 1)
10/14: Conjectured and proved a recurrence for 2 x 2 x n snakes. (i.e. cube snakes that travel in a straight line). For the sketch of the proof, see here.
10/31: Found online that the number of perfect matchings of a 2 x 2 x n "cube" snake is a perfect square when we have an odd number of cubes and twice a perfect square when we have even numbers of cubes. Attempting to find a bijection between ordered pairs of 3 x 2n domino tilings and perfect matchings of an odd number of cubes.
11/04: Worked on creating a four-variable generating function for the perfect matchings of "cube" snakes to see if it would give more insight as to how to formulate a bijection between ordered pairs of 3 x 2n domino tilings and perfect matchings of 2 x 2 x 2n "cube" snakes. Also, considering if the fact that 3-D snakes moving in two planes will have the same number of perfect matchings as the sum of the squares of the perfect matchings of the corresponding 2-D graphs. For details about this fact, see here..
11/05: Its really early in the morning, but I just couldnt put the work down, so I stayed up past bedtime, but it looks like I found the four variable generating funcion including weights of a cube (where z is a horizontal weight, x is a "depth" weight and y is a vertical weight.
11/15: Created a new one dimensional graph whose perfect matchings with every additional extension follows a sequence 0,1,2,6,8,32,40,168,208,880,1088,4608,5696,24128,... so that the a(n)th term where n is even (starting with a(0)) gives the sequence 0,2,6,32,168,880,4608,24128... which, when I looked up this sequence in OLEIS told me for that at least this many terms could be given by the formula b(n) = (1/2)*sum[(n,k)*F(3*k)] where F(3*k) is the 3kth fibonacci number indexed by F(0)=0, F(1)=F(2)=1,... Interesting. Combinatorics is fun!
12/04: Sam, Carl and I have been working on coefficients of successive iterations of Newton's Method for arbitrary quadratics, and have begun to see why all of the numbers are so highly composite. I make the following two claims with only numerical evidence to back them up, but they just seem too pretty to not be true.
Claim 1: After iterating Newton's method m times for ax^2 + bx + c , the coefficient of x^k in the denominator will be binomial( 2^m , k ) times some polynomial in a,b,c. The interesting thing is that binomial( 2^m , k ) always seems to be the gcd of the coefficients of the polynomial in a,b,c.
Claim 2: After iterating Newton's method m times for ax^2 + bx + c , the coefficients x^n for all m contains only terms a^r * b^s * c^t, where r+s+t = 2^m - 1.
1/20 - 4/21: Ive been working with Emilie on a family of new quadratic sequences that generate integers for the whole semester that are closely analogous to the Dana Scott Recurrence and a little more distantly analogous to the Somos sequences. Weve been working toward extracting some sort of combinatorial object that these sequences could count the perfect matching of. The article (which is still in progress can be viewed at its latest state here.
4/25: I have written up work on the weighted matchings of the cube snake (2x2xn) from first semester. See the pdf file here.
