Abstract:

With the recent invention of DNA microarray technology, it has become possible to measure the expression levels of thousands of genes at a given moment. The challenge is now to identify regulatory genes and their complex interactions. Extracting pertinent information, or identifying pattern, in this morass of data presents a considerable mathematical problem.

In the first half of the talk I will give a mathematical formulation of the problem which is free of hypotheses. Then I will give a general approach for the blind identification of significant genes and explain how this leads to several interesting problems in enumerative combinatorics.

In the second half of the talk I will focus on just one such problem, which is the enumeration of permutations with given up-down signature. Given a permutation on $N$ letters, its up-down signature $\sigma$ is defined to be its sequence of rises and falls. It is a classic counting problem to establish the number $C(\sigma)$ of permutations with given up-down signature. I will introduce a universal polynomial generating series for the up-down numbers $C(\sigma)$ and use it to derive some new results on the asymptotic and arithmetic properties of the up-down numbers.