Abstract: MATH/CHEM/COMP 2002, Dubrovnik, June 24-29, 2002

Some primary and secondary

structures in combinatorics

Darko Veljan

Department of  Mathematics, Faculty of Science and  Mathematics, University of Zagreb, Bijenička c. 30, HR-10002 Zagreb, Croatia

A standard undergraduate course in combinatorics usually covers basic properties of some special combinatorial numbers such as binomial coefficients, Stirling numbers,(sometimes) Eulerian numbers, Fibonacci and Catalan numbers, and optionally a few more. Problems on global behaviour of sequences of these numbers, in particular log-concavity (or log-convexity) are hardly mentioned.

In this talk we will try:

a) to refresh quickly this curriculum with some bijective proofs in the standard (primary) part and

b) to add some "concrete" material to it concerning Dyck and Motzkin paths and numbers, Narayana numbers and especially secondary structures which come from biology.

In some sense, secondary structures include all this new stuff. Proofs dealing with this new material still stay in the realms of elementary combinatorics, linear algebra and calculus. An emphasis will be given to "calculus proofs" of log-convexity results for combinatorial numbers from both primary and secondary combinatorics. For example, we shall show by using partial derivatives that the Stirling partition numbers are log-concave in the second variable.