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

TL;DR

This paper introduces a $q$-analog of certain symmetric functions related to partitions, explores its properties, and connects it to classical combinatorial polynomials like tree inversions and parking functions, providing new formulas and recurrences.

Contribution

It presents a novel $q$-analog of symmetric functions $p_{n}^{(r)}$, links it to $q$-Stirling numbers, and derives new recurrence relations and representations for related combinatorial polynomials.

Findings

Defined a $q$-analog of $p_{n}^{(r)}$ with key properties.

Connected the $q$-analog to classical combinatorial polynomials.

Derived new recurrence relations and explicit formulas for $J_{n}^{(r)}$ and their reciprocals.

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.