  We study monomial ideals, always locally given by a monomial, like a
reasonable first step to estimate in general the number of monoidal
transformations of Villamayor's algorithm of resolution of singularities. The
resolution of a monomial ideal $<X_1^{a_1}\cdot ... \cdot X_n^{a_n}>$ is
interesting due to its identification with the particular toric problem $<Z^c-
X_1^{a_1}\cdot ... \cdot X_n^{a_n}>$.
  In the special case, when all the exponents $a_i$ are greater than or equal
to the critical value $c$, we construct the largest branch of the resolution
tree which provides an upper bound involving partial sums of Catalan numbers.
This case will be called ``minimal codimensional case''. Partial sums of
Catalan numbers (starting $1,2,5,...$) are $1,3,8,22,...$ These partial sums
are well known in Combinatorics and count the number of paths starting from the
root in all ordered trees with $n+1$ edges. Catalan numbers appear in many
combinatorial problems, counting the number of ways to insert $n$ pairs of
parenthesis in a word of $n+1$ letters, plane trees with $n+1$ vertices, $...
$, etc.
  The non minimal case, when there exists some exponent $a_{i_0}$ smaller than
$c$, will be called ``case of higher codimension''. In this case, still
unresolved, we give an example to state the foremost troubles.
  Computation of examples has been helpful in both cases to study the behaviour
of the resolution invariant. Computations have been made in Singular (see
\cite{sing}) using the \emph{desing} package by G. Bodn\'ar and J. Schicho, see
\cite{lib}.
