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

Vincent Brugidou

TL;DR

The paper constructs a natural $q$-analogue of the symmetric-function family $p_n^{(r)}$ via a generating-function framework and relates it to the classical form through $q$-Stirling numbers. Specialization in the $q$-deformed exponential $E_{xq}(t)$ produces the polynomials $J_n^{(r)}$, which satisfy a new row-wise linear recurrence and admit explicit determinant and combinatorial forest interpretations; their reciprocals enumerate parking functions. The $J_n^{(r)}$ provide positive-coefficient, monic polynomials with rich combinatorial representations, including connections to inversion enumerators for rooted forests and to functional digraph counts via Katz-type formulas. Overall, the work links symmetric-function theory, $q$-combinatorics, and parking-function/forest statistics, yielding new recurrences, explicit formulas, and diverse representations with potential applications in enumerative combinatorics.

Abstract

Let the symmetric functions be defined for the pair of integers $\left( n,r\right) $, $n\geq r\geq 1$, by $p_{n}^{\left( r\right) }=\sum m_{λ}$ where $m_{λ}$ are the monomial symmetric functions, the sum being over the partitions $λ$ 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.

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

TL;DR

The paper constructs a natural -analogue of the symmetric-function family via a generating-function framework and relates it to the classical form through -Stirling numbers. Specialization in the -deformed exponential produces the polynomials , which satisfy a new row-wise linear recurrence and admit explicit determinant and combinatorial forest interpretations; their reciprocals enumerate parking functions. The provide positive-coefficient, monic polynomials with rich combinatorial representations, including connections to inversion enumerators for rooted forests and to functional digraph counts via Katz-type formulas. Overall, the work links symmetric-function theory, -combinatorics, and parking-function/forest statistics, yielding new recurrences, explicit formulas, and diverse representations with potential applications in enumerative combinatorics.

Abstract

Let the symmetric functions be defined for the pair of integers , , by where are the monomial symmetric functions, the sum being over the partitions of the integer with length . We introduce by a generating function, a -analog of and give some of its properties. This -analog is related to its the classical form using the -Stirling numbers. We also start with the same procedure the study of a -analog of . By specialization of this -analog in the series , we recover in a purely formal waya class of polynomials 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 . The form of this recurrence is also given for the reciprocal polynomials of , known to be the sum enumerators of parking functions. Explicit formulas for and their reciprocals are deduced, leading inversely to new representations of these polynomials as forest statistics.
Paper Structure (11 sections, 13 theorems, 109 equations, 1 figure)

This paper contains 11 sections, 13 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition of $\left[ p_{n}^{(r)}\right]$
  4. Some properties of $p_{n}^{\left( r\right) }$
  5. Definition of $\left[ p_{n}^{\left( r\right) }\right]$
  6. Relations between $\left[ p_{n}^{(r)}\right]$ and $p_{n}^{(r)}$
  7. Extension to $p,q$-analogue
  8. The polynomials $J_{n}^{\left( r\right) }$ and their linear recurrence
  9. Reciprocal polynomials
  10. An explicit formula for the $J_{n}^{(r)}$
  11. Level statistics on forests

Key Result

Proposition 3.2

For $n\geq r\geq 1$, we have

Figures (1)

  • Figure 1: A weighted forest $F\in \mathcal{F}_{13,R}$ with $R=\left\{ 7,11,2\right\}$. The label of each vertex is in black, its weight is in red. $u_{i}=\left| V_{i}\right|$ for $0\leq i\leq 3.$

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • Theorem 4.1
  • Corollary 4.2
  • Definition 5.1
  • Proposition 5.2
  • Lemma 6.1
  • Lemma 6.2
  • Theorem 6.3
  • ...and 11 more