  Let the symmetric functions be defined for the pair of integers $\left(
n,r\right) $, $n\geq r\geq 1$, by $p_{n}^{\left( r\right) }=\sum m_{\lambda }$
where $m_{\lambda }$ are the monomial symmetric functions, the sum being over
the partitions $\lambda $ of the integer $n$ with length $r$. We introduce by a
generating function, a $q$-analog of $p_{n}^{\left( r\right) }$ and give some
of its properties. This $q$-analog is related to its the classical form using
the $q$-Stirling numbers. We also start with the same procedure the study of a
$p,q$-analog of $p_{n}^{\left( r\right) }$.
  By specialization of this $q$-analog in the series $\sum\nolimits_{n=0}^{
\infty }q^{\binom{n}{2}}t^{n}/n!$, we recover in a purely formal way$\ $a class
of polynomials $J_{n}^{\left( r\right) }$ historically introduced as
combinatorial enumerators, in particular of tree inversions. This also results
in a new linear recurrence for those polynomials whose triangular table can be
constructed, row by row, from the initial conditions $ J_{r}^{\left( r\right)
}=1$. The form of this recurrence is also given for the reciprocal polynomials
of $J_{n}^{\left( r\right) }$, known to be the sum enumerators of parking
functions. Explicit formulas for $J_{n}^{\left( r\right) }$ and their
reciprocals are deduced, leading inversely to new representations of these
polynomials as forest statistics.
