The term combinatorial Gray code is used to describe an ordering of combinatorial objects so that successive objects differ in a pre-specified way. One example is an ordering of the strings of digits from the set {1,...,R}, called an R-ary Gray code. In 1947, Frank Gray developed an ordering of binary n-strings, called the binary reflected Gray code, that was used in encoding analog signals into strings of binary digits. We will discuss methods of generating many other binary Gray codes.

The term combinatorial Gray code is used to describe an ordering of combinatorial objects so that successive objects differ in a pre-specified way. One example is an ordering of the strings of digits from the set {1,...,R}, called an R-ary Gray code. In 1947, Frank Gray developed an ordering of binary n-strings, called the binary reflected Gray code, that was used in encoding analog signals into strings of binary digits. We will discuss methods of generating many other binary Gray codes.

NP or not NP? That's a million dollar question. In this talk I'll introduce the concept of algorithmic complexity and give some examples. We'll then examine two classes of problems, P and NP, and discuss some ideas about them such as NP completeness. We'll end with a statement of one of the seven millennium problems, that by the end of the talk you should understand.

We'll discuss how one can solve the following problems:

(1) On an n by n chess board after placing k queens on the board how many squares are left unattacked?

(2) How many ways can we place k queens on an n by n board and have EXACTLY u unattcked squares?

(3) On an infinite chess board how many squares can a knight reach in d moves?

(4) How many squares can be reached in d moves and NO LESS by a knight?

In this talk I will review some recent results of Christian Reidys (2008) on counting the number of secondary structures that an abstract single-strand of RNA can obtain ("abstract" means we do not specify an underlying base sequence for the RNA strand). These results rely on the recent observation of Chen et al. (2007) that k-noncrossing partitions of the set {1,...,n} are in one-to-one correspondence with all random walks of length n which remain in the Weyl chamber of dimension k-1 [i.e., the set of all points (x_1,...,x_{k-1}) in Z^{k-1} such that x_1 > ... > x_{k-1} > 0] and that start and end at (k-1,k-2,...,1). The construction of this bijection and, ultimately, the RNA structure counting formula rely on such ideas as vascillating Young tableaux, the RSK algorithm, finite reflection (or Weyl) groups, the reflection principle of Gessel and Zeilberger (1992), and an amazing formula for the exponential generating function of the walks mentioned above due to Grabiner and Magyar (1993). This talk should be self-contained; however, I will only motivate the proofs of the main results through examples, figures, etc., and will not provide details.

In this talk, using as many visuals as possible, I will present twelve ways to count the ways to allocate a finite set A of balls into a finite set B of boxes. To begin, one such way is simply to count the total number of functions from A to B. This corresponds to the arbitrary allocation of labeled balls into labeled boxes. The other options arise as follows. First, we count all allocations, allocations where each box gets at most one ball, or allocations where each box gets at least one ball. Second, we optionally equate allocations which differ only by a relabeling of the balls, or the the boxes, or both. This makes the balls and/or boxes unlabeled and 'indistinguishable' with respect to the enumeration. The resulting table provides an easy summary of a nice portion of elementary combinatorics, and includes the binomial, partition, and Sterling numbers (of the second kind).

Spring Break