Andrews' Conjecture and q-Series Positivity

• Andrews’ Conjecture establishes that the quotient coefficients from the factorization of Alladi–Schur polynomials are nonnegative polynomials in q.
• It refines the Alladi–Schur theorem by using Schur partitions and generating functions expressed as finite product polynomials, leading to structured recursive formulations.
• Recursive identities and explicit relations among coefficient families validate strong inductive positivity and suggest further combinatorial applications.

Andrews’ Conjecture, in the sense settled in “On a nonnegativity conjecture of Andrews,” is the assertion that certain quotient coefficients arising from the factorization of the Alladi–Schur polynomials are coefficient-wise nonnegative as polynomials in $q$. More precisely, if

$\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$

then Andrews conjectured that for all $n$ and $j$, the polynomial $c(n,j)$ has nonnegative coefficients. The conjecture is resolved by proving the stronger statement that an entire quotient family $\mathscr{d}_n(x)$ has nonnegative integer coefficients in both $x$ and $q$, thereby extending positivity well beyond the special indices $6n-1$ (Alamoudi, 14 Aug 2025).

1. Schur partitions and the refined Alladi–Schur framework

The conjecture belongs to the combinatorial and $q$-series framework surrounding the Alladi–Schur theorem and Andrews’ refinement of it. A Schur partition is a partition into parts that differ by at least $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$0 with no consecutive multiples of $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$1. Let $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$2 be the set of partitions of $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$3 into $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$4 odd parts, each occurring at most twice, and let $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$5 be the set of Schur partitions $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$6 of $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$7 where the number of parts plus the number of even parts of $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$8 is $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$9. Andrews’ refinement states

$n$0

This refinement has the generating function form

$n$1

where $n$2. The right-hand side already displays the basic finite factors that later appear in the factorization theory of the Alladi–Schur polynomials.

For $n$3, the Alladi–Schur polynomials are defined by the partial generating functions

$n$4

where $n$5 and $n$6 is the set of partitions in $n$7 with parts $n$8. Thus $n$9 is a bounded-part analogue of the infinite product generating function, and Andrews’ conjecture concerns the structure of certain quotients obtained from these finite polynomials (Alamoudi, 14 Aug 2025).

2. Factorization and the formulation of the conjecture

A central role is played by the finite product polynomials

$j$0

Andrews proved that

$j$1

where the $j$2 are polynomials in $j$3.

More generally, Andrews’ factorization theorem implies

$j$4

or equivalently,

$j$5

where $j$6 is the indicator function for odd integers.

This leads to the quotient family

$j$7

For the special case $j$8, equation (1.2) becomes

$j$9

Andrews’ Conjecture is then:

For all $c(n,j)$0 and $c(n,j)$1, $c(n,j)$2 has nonnegative coefficients.

Equivalently, for each $c(n,j)$3, the coefficients in the $c(n,j)$4-expansion of $c(n,j)$5 are polynomials in $c(n,j)$6 with all coefficients nonnegative. The later theorem shows that this conjecture is subsumed by a stronger positivity statement for every $c(n,j)$7, not only those with index $c(n,j)$8 (Alamoudi, 14 Aug 2025).

3. Recursive structure of the quotient family

The proof of nonnegativity is built from recursive identities. At the level of the original Alladi–Schur polynomials, the standard technique of removing the largest part gives

$c(n,j)$9

where $\mathscr{d}_n(x)$0 denotes the indicator function for the set of multiples of $\mathscr{d}_n(x)$1.

After dividing by the appropriate factor $\mathscr{d}_n(x)$2, the quotient family satisfies recurrences of simpler positivity type. For $\mathscr{d}_n(x)$3,

$\mathscr{d}_n(x)$4

For odd indices $\mathscr{d}_n(x)$5, there are two cases: $\mathscr{d}_n(x)$6 and, with $\mathscr{d}_n(x)$7,

$\mathscr{d}_n(x)$8

where

$\mathscr{d}_n(x)$9

The paper also derives additional even-indexed and consolidated odd-indexed relations: $x$0

$x$1

and

$x$2

These recurrences are significant because they express each quotient polynomial in terms of earlier quotient polynomials using only shifts in $x$3, multiplication by manifestly nonnegative monomials, or multiplication by $x$4, which itself has nonnegative coefficients. This suggests a natural inductive route to coefficient-wise nonnegativity (Alamoudi, 14 Aug 2025).

4. Resolution of the conjecture

The main theorem states:

$x$5

Hence Andrews’ Conjecture is true.

The proof proceeds by induction, beginning with explicit base cases: $x$6

$x$7

Each of these is manifestly in $x$8.

The inductive step separates even and odd indices. For even indices, equation (2.1),

$x$9

preserves nonnegativity because it is only a graded shift in $q$0. For odd indices not divisible by $q$1, equation (2.3),

$q$2

writes the polynomial as a sum of two nonnegative polynomials. For odd indices divisible by $q$3, equation (2.4),

$q$4

is again manifestly nonnegative.

The original conjecture follows immediately for the special indices $q$5, since

$q$6

and all coefficients of $q$7 are nonnegative. Therefore every $q$8 is a polynomial in $q$9 with nonnegative coefficients (Alamoudi, 14 Aug 2025).

5. Coefficient families and stronger consequences

The theorem is stronger than the original conjecture because it gives positivity for every quotient polynomial $6n-1$0, not only for the subsequence indexed by $6n-1$1. Writing

$6n-1$2

the recurrences induce structural relations among the coefficients $6n-1$3. From (2.1),

$6n-1$4

The paper also gives explicit relations between $6n-1$5 and the original conjectural coefficients $6n-1$6: $6n-1$7

$6n-1$8

$6n-1$9

$q$0

$q$1

$q$2

These formulas transfer positivity information from $q$3 to $q$4 and show that coefficient-wise nonnegativity is only one aspect of a more rigid structure. The resulting corollary proves, for $q$5,

$q$6

$q$7

and

$q$8

Here $q$9 for $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$00 and $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$01 means $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$02 for all $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$03. The divisibility statement

$\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$04

is especially notable: it sharpens nonnegativity into a strong lower bound on the $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$05-valuation. The paper derives these conclusions by combining Theorem 1 with the recurrences and with Andrews’ identities

$\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$06

and

$\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$07

A plausible implication is that the quotient family $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$08 is the more natural positivity object, with the coefficients $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$09 occupying only one residue class in a larger recursive system (Alamoudi, 14 Aug 2025).

6. Historical position, examples, and possible extensions

The historical background begins with the Alladi–Schur theorem, communicated by K. Alladi to G. Andrews, asserting that the number of partitions of $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$10 into odd parts, each occurring at most twice, equals the number of Schur partitions of $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$11. Andrews’ refinement augments this by the parameter $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$12, recording the number of parts plus the number of even parts, and gives the two-variable identity in (1.1). In subsequent work Andrews introduced the polynomials $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$13, established their factorization by the finite products $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$14, and isolated the coefficient polynomials $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$15, which led to the nonnegativity conjecture. The paper under discussion settles that conjecture and cites a bijective proof of Andrews’ refinement by Y. Alamoudi (Alamoudi, 2024, Alamoudi, 14 Aug 2025).

The small-index examples already illustrate the theorem: $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$16

$\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$17

$\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$18

$\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$19

Each example lies in $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$20, and together they display the patterned shifts and residue-class dependence later encoded by the recurrences.

The paper does not list specific new open problems, but it notes directions that may naturally follow from the strengthened theorem. These include sharpening inequalities for $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$21, investigating unimodality or log-concavity in $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$22 for fixed $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$23, and exploring further total positivity phenomena for the arrays $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$24 and $\frac{d_{6n-1}(x)}{p_n(x)}=\sum_{j=0}^n c(n,j)x^j,$25. This suggests that the resolution of Andrews’ Conjecture is not merely a terminal positivity result, but part of a broader structural theory for factorized truncations of the Alladi–Schur generating functions (Alamoudi, 14 Aug 2025).