**Martin Klazar**

Charles University, Prague

If is a closed set of finite permutations (i.e., a lower ideal in
where is the set of all finite permutations
and is the standard containment ordering) and
denotes the set of all -permutations in , then the counting function
is subject to various dichotomies and restrictions
forbidding many functions to have this form; this was shown (*Electronic
J. of Combinatorics*, 2003) by T. Kaiser and M. Klazar. For example, either
for all or is eventually constant,
or--another dichotomy--either for all with a
constant or for all , where
are the Fibonacci numbers.

In my talk I will present generalizations and extensions of these results to other classes of objects (like those mentioned in the title) and other containments, and I will discuss a general approach to obtain them uniformly as instances of a general metaresult.

2005-05-23