A q-analog of certain symmetric functions and one of its specializations

Let the symmetric functions be defined for the pair of integers $(n,r)$, $n\ge r\ge 1$, 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 ${\sum }_{n=0}^{\mathrm{\infty }}{q}^{\left(\genfrac{}{}{0px}{}{n}{2}\right)}{t}^n\mathrm{/}n!$, 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.