  We recover Gessel's determinantal formula for the generating function of
permutations with no ascending subsequence of length m+1. The starting point of
our proof is the recursive construction of these permutations by insertion of
the largest entry. This construction is of course extremely simple. The cost of
this simplicity is that we need to take into account in the enumeration m-1
additional parameters --- namely, the positions of the leftmost increasing
subsequences of length i, for i=2,...,m. This yields for the generating
function a functional equation with m-1 "catalytic" variables, and the heart of
the paper is the solution of this equation. We perform a similar task for
involutions with no descending subsequence of length m+1, constructed
recursively by adding a cycle containing the largest entry. We refine this
result by keeping track of the number of fixed points. In passing, we prove
that the ordinary generating functions of these families of permutations can be
expressed as constant terms of rational series.
