Welcome to my research page
I've spent a good chunk of time recently studying descent sets of permutations. Let p be a permutation of n letters. We say that k is a 'descent position' of p if p(k) > p(k+1), k=1,2,...,n-1. The 'descent set' of p is the set of all descent positions. For example, if p=(3,4,2,5,1), then the descent set is {2,4}. For signed permutations, we consider every permutation as beginning with 0, so a descent position of a signed permutation is k such that p(k) > p(k+1), where k=0,1,...,n-1. If p=(-1,3,2,-4,5) then the descent set of p is {0,2,3}. In a general Coxeter group with generating set S={s1,s2,...,sn}, we say that the word w has a descent in position k if the length of w (i.e. shortest representation as a word in elements of S) is greater than the length of w*sk.
 
My jumping off point in this area of research is the study of the augmented descent set of a signed permutation, which is related to the cyclic descent set of an ordinary permutation. With cyclic descents, you consider whether p(n) > p(1). With augmented descents you consider whether p(n) > 0. I have some results in the following paper: Cyclic and augmented descents .
Another thing I'm working on right now is the limiting behavior of groves (do you see a circle emerging below?).


Here's a link to a Java applet for groves on standard initials created by Hal Canary.

I've also written some code for Maple that draws pretty pictures of groves.   Maple code

I gave a talk at Brandeis in March 2003 about asymptotic behavior in groves. Here is my powerpoint presentation.
A slightly updated talk, October 2003: powerpoint presentation

Here is the most recent draft of the arctic circle paper I'm writing with David Speyer:

pdf


Click the pictures to play Burn solitaire. Watch the partitions evolve individually or cumulatively.  (Or below, watch the 'average partition' stabilize)

BURN = (B U L G A R I A N) - (0 0 1 1 1 0 1 1 0)

One unresolved problem I'm working on is a game I call burn solitaire.  It is a modified version of a game called Bulgarian solitaire.  You can play either game with a deck of cards, or a collection of matchsticks, some little blue blocks, or even a group of people. Bulgarian solitaire begins with any arrangement of your deck of cards into piles.  At each step of the game, you remove one card from each pile and put the selected cards into a new pile, noting only the number of cards in each pile. Any game of Bulgarian solitaire will inevitably end in a cycle of states. Burn solitaire is a probabilistic version of Bulgarian solitaire. Instead of removing a card from each pile, we remove a card from each pile with some fixed probability p (in the applets above, p=1/2). Although there is no limiting cycle for burn solitaire, there is still a steady state for the system (we view it as a Markov chain). I have no theorems as yet, but I'm collecting my conjectures on the topic.  Play with the applets!

During fall of 2002 I worked on monomer-dimer tilings with John Baldwin, Nick Anzalone, and Ilya Bronshtein.

Some preliminary results for the monomer-dimer question.  If you don't have Flash Player, you can download it here.

Here is some maple code that I wrote in early investigations.  Save the files below and open them with maple.

For 2 by n rectangles (weights on edges only)

For 1 by n and 2 by n (weights on edges and vertices)

I presented our paper "A Reciprocity Theorem for Monomer-Dimer Coverings," at a conference in Lyon in June 2003.

A fun question (is there some math here?):

Play the game "Pile Up" five times in a row by only pushing the spacebar.  Did you always get past the first level?  The board has height 16 and width 8.  There are five colors of balls, presented randomly in groups of three and dropped down the center of the board.  The balls stack on top of one another in the natural way (some randomness is involved when a ball can fall to the left or right).  If four or more balls of the same color are connected, then they are removed from the board and the remaining balls fall to fill the space.  The first level is completed when a cumulative total of 80 balls are removed from the board.  What is the probability of success when using the 'spacebar' strategy?

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