My research interests are mostly in algebraic combinatorics, usually these are combinatoric questions which are motivated by specific algebras and algebra constructions.

Some of the problems are simple enough to state in terms of combinatorics alone that don't require a lot of background knowledge to understand what the questions are, but in order to make progress you need the algebra to understand the combinatorics and the combinatorics to understand the algebra.



I posted a problem that I came across in my research about studying the poset of Dyck paths ordered by the relation: D1 <= D2 if D1 is below D2. I worked with graduate student Jennifer Woodcock who learned some combinatorics and wrote the following research paper.
Properties of the Poset of Dyck Paths Ordered by Inclusion


Some continuation of this work (which I propose for a student who is interested in taking up a combinatorics project) will be to find formulas for the number of chains, antichains, maximum/maximal antichains.

Jennifer was persistant and pushed through some additional results on the problem of finding the number of saturated chains in this poset.



The affine reflection group of a given Lie type can be visualized as a tiling of n-dimensional space. The affine Grassmannian can be visualized as the tiling of the space that is between certain hyperplanes (those that correspond to the reflections of the finite reflection group). Recently Brant Jones and Chris Hanusa gave many combinatorial objects which are in bijection with this tiling for types B/C/D (type A was already fairly well known). Realistically we can only 'draw' this picture in 2-dimensions. I wanted to be able to visualize the affine grassmannian in 3-dimensions. For type A_3 this is a tiling of a region of 3-space with pyramids. A summer student, Nik Cook worked to draw this picture in Google sketchup. He did an awesome job, but I am a few steps away from being able to send this to a 3d-printer.

Report by Nik Cook on constructing the finite and affine reflection group of type A_3
His Google sketchup files 4 layers of the affine Grassmannian of type A_3,
hollowed out plain permutahedron,
hollowed out fancy permutahedron.

I really want to continue this project and build the same objects for B_3, C_3 and D_3 as well as any other types that can be visualized (e.g. affine B_2/D_2, affine B_3/D_3, etc.)



The algebra of symmetric functions is generated by an infinite set of elements h1, h2, h3,.... which commute with each other (i.e. hi * hj = hj * hi). This algebra has been studied extensively but it has a younger cousin, NSym, called non-commutative symmetric functions. This algebra is generated by h1, h2, h3,... but now hi * hj is not equal to hj *hi. We would like to find a basis of this new algebra with the following properties:

1) the commutative image is the fundamental basis of the symmetric functions (called the Schur functions)
2) there are three operations on this space called the internal product, external product and the coproduct. These operations should have non-negative coefficients when expanded in this basis.

Find the basis. Do these conditions uniquely define it? Do the analogous conditions uniquely define the Schur function basis in the ring of the symmetric functions?

Here is a tutorial on symmetric functions and also on the use of sage. I made a few of these videos, others were made by Jennifer Light it is meant to be a tutorial for the basics of this algebra.



In the ring of symmetric functions there is also an internal product. I have looked at this internal product for s_dd * s_\la when \la is a two row or hook shape. I would like to know what happens for \la a partition of length <= 4 with all even parts or all odd parts. I would also be interested to know what happens for \la a rectangular partition or a self conjugate partition.



The algebra of symmetric functions in non-commuting variables, NCSym, has a basis which is indexed by set partitions. The subset of non-crossing partitions has Catalan number of elements. For which bases of NCSym does the linear span of b_A for A a non-crossing partition form a Hopf subalgebra?



Pattern avoidance in Coxeter words. Which patterns are compatible with the Coxeter relations (i.e. which patterns have the property that w avoids pattern iff u avoids pattern for all reduced words w and u) for the symmetric group and how do we enumerate the permutations whose reduced words avoid these patterns. Generalize these ideas to all Coxeter groups.



Find a combinatorial description for the symmetric function coefficients < s_{\lambda}[ s_n ], s_\mu >



Some weird conjectures
The coefficient of s_321 in the k^th Kronecker product of s_411 is 22...22 (in base 10) where the digit 2 appears k-1 times. e.g. < s_411 * s_411, s_321 > = 2, < s_411 * s_411 * s_411, s_321 > = 22, etc.



I know that I haven't defined many of the terms in the problems above, but this short list of research ideas can be expanded on and will take the time to explain the background and direction ideas if you are interested in working on one of them.