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

Abstract

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.