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On a nonnegativity conjecture of Andrews

Yazan Alamoudi

TL;DR

This paper settles Andrews' nonnegativity conjecture for the Alladi-Schur polynomial refinements by introducing the quotient $\mathscr{d}_n(x)=\frac{d_n(x)}{p_{\lceil\frac{n+3\chi_o(n)}{6}\rceil-\chi_o(n)}(x)}$ and proving it has nonnegative coefficients in $x$ and $q$. The author provides two key recurrences for $\mathscr{d}_n(x)$, proves the result by strong induction, and then derives explicit relations for the coefficient polynomials $\mathscr{c}(n,i)$ in terms of Andrews' $c(n,i)$, yielding additional bounds and divisibility properties. The work extends the framework with further recurrences, refined combinatorial interpretations, and a connection to the generating function refining the Alladi-Schur theorem. Overall, the results confirm Andrews' conjecture and deepen understanding of the structure of Alladi-Schur–type polynomials and their coefficients in partition-related settings.

Abstract

I settle a conjecture of Andrews related to the Alladi-Schur polynomials. In addition, I give further relations and implications to two families of polynomials related to the Alladi-Schur polynomials.

On a nonnegativity conjecture of Andrews

TL;DR

This paper settles Andrews' nonnegativity conjecture for the Alladi-Schur polynomial refinements by introducing the quotient and proving it has nonnegative coefficients in and . The author provides two key recurrences for , proves the result by strong induction, and then derives explicit relations for the coefficient polynomials in terms of Andrews' , yielding additional bounds and divisibility properties. The work extends the framework with further recurrences, refined combinatorial interpretations, and a connection to the generating function refining the Alladi-Schur theorem. Overall, the results confirm Andrews' conjecture and deepen understanding of the structure of Alladi-Schur–type polynomials and their coefficients in partition-related settings.

Abstract

I settle a conjecture of Andrews related to the Alladi-Schur polynomials. In addition, I give further relations and implications to two families of polynomials related to the Alladi-Schur polynomials.

Paper Structure

This paper contains 3 sections, 7 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. Further relations and remarks related to the quotients $\mathscr{d}_{n}(x)$

Key Result

Theorem 1

For $n\geq1$, $\mathscr{d}_{n}$ is a polynomial in $x$ and $q$ with nonnegative integer coefficients. Hence, Andrews' conjecture is true.

Theorems & Definitions (16)

  • Conjecture : Andrews' conjecture stated in And2
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Theorem 1
  • Lemma 3
  • proof
  • Remark
  • ...and 6 more