Abstract: MATH/CHEM/COMP 2002, Dubrovnik,
June 2429, 2002

Some primary and
secondary
structures in
combinatorics
Darko Veljan Department of Mathematics, Faculty of Science and Mathematics, University of Zagreb, Bijenička c. 30, HR10002 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 logconcavity (or logconvexity) 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
logconvexity 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 logconcave in the second
variable. 