A.Omelchenko  Combinatorics of cascade diagrams

We introduce {\it cascade diagram representation\/} for projections of $k$-tangles (tangles with
$k$ legs). The enumerating problem for cascade diagrams is closely related with the problem of
enumeration of paths in so called Motzkin triangle with multiplicities. The generating function
for numbers $a_{n,k}$ of cascade diagrams with $n$ crossings and $2k$ legs is obtained.
Asymptotic behavior of the number of cascade diagrams at large number of crossings $n$ is
studied. Some generalizations of the Motzkin numbers and the Dick numbers are considered.

This is a joint work with A.Bogdanov, V.Meshkov, and M.Petrov.