--- title: Andrews' Conjecture and q-Series Positivity url: https://www.emergentmind.com/topics/andrews-conjecture type: topic --- # Andrews' Conjecture and q-Series Positivity 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 \(6n-1\) [2508.10871]. ## 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 \(3\) with no consecutive multiples of \(3\). Let \(C(m,n)\) be the set of partitions of \(n\) into \(m\) odd parts, each occurring at most twice, and let \(\mathcal{D}(m,n)\) be the set of Schur partitions \(\pi\) of \(n\) where the number of parts plus the number of even parts of \(\pi\) is \(m\). Andrews’ refinement states \[ |C(m,n)|=|\mathcal{D}(m,n)|. \tag{1.1} \] This refinement has the generating function form \[ \sum_{m,n\geq0}D(m,n)x^mq^n=\prod_{i=1}^\infty\bigl(1+xq^{2i-1}+x^2q^{4i-2}\bigr), \] where \(D(m,n)=|\mathcal{D}(m,n)|\). The right-hand side already displays the basic finite factors that later appear in the factorization theory of the Alladi–Schur polynomials. For \(N\geq 0\), the Alladi–Schur polynomials are defined by the partial generating functions \[ d_N(x)=\sum_{m,n\geq0}D_N(m,n)x^mq^n, \] where \(D_N(m,n)=|\mathcal{D}_N(m,n)|\) and \(\mathcal{D}_N(m,n)\) is the set of partitions in \(\mathcal{D}(m,n)\) with parts \(\leq N\). Thus \(d_N(x)\) 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 [2508.10871]. ## 2. Factorization and the formulation of the conjecture A central role is played by the finite product polynomials \[ p_n(x)=\prod_{i=1}^n\bigl(1+xq^{2i-1}+x^2q^{4i-2}\bigr)\qquad (n\geq 1). \] Andrews proved that \[ p_n(x)\mid d_{6n-1}(x)\quad\text{and}\quad \frac{d_{6n-1}(x)}{p_n(x)}=\sum_{i=0}^nc(n,i)x^i, \tag{1.2} \] where the \(c(n,i)\) are polynomials in \(q\). More generally, Andrews’ factorization theorem implies \[ p_{\bigl\lfloor\frac{n+4}{6}\bigr\rfloor}(x)\mid d_n(x)\quad\text{for }n\not\equiv 3\pmod6, \qquad p_{\bigl\lfloor\frac{n-2}{6}\bigr\rfloor}(x)\mid d_n(x)\quad\text{for }n\equiv 3\pmod6, \] or equivalently, \[ p_{\left\lceil\frac{n+3\chi_o(n)}{6}\right\rceil -\chi_o(n)}(x)\mid d_n(x), \] where \(\chi_o\) is the indicator function for odd integers. This leads to the quotient family \[ \mathscr{d}_n(x)=\frac{d_n(x)}{p_{\left\lceil\frac{n+3\chi_o(n)}{6}\right\rceil-\chi_o(n)}(x)}. \tag{1.3} \] For the special case \(n=6N-1\), equation (1.2) becomes \[ \mathscr{d}_{6N-1}(x)=\sum_{i=0}^N c(N,i)x^i. \] Andrews’ Conjecture is then: > For all \(n\) and \(j\), \(c(n,j)\) has nonnegative coefficients. Equivalently, for each \(n\), the coefficients in the \(x\)-expansion of \(d_{6n-1}(x)/p_n(x)\) are polynomials in \(q\) with all coefficients nonnegative. The later theorem shows that this conjecture is subsumed by a stronger positivity statement for every \(\mathscr{d}_n(x)\), not only those with index \(6n-1\) [2508.10871]. ## 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 \[ d_N(x)=d_{N-1}(x)+x^{1+\chi_2(N)}q^{N}d_{N-3-\chi_3(N)}(x), \tag{2.2} \] where \(\chi_n\) denotes the indicator function for the set of multiples of \(n\). After dividing by the appropriate factor \(p_n(x)\), the quotient family satisfies recurrences of simpler positivity type. For \(N\geq 3\), \[ \mathscr{d}_{2N}(x)=\mathscr{d}_{2N-3}(xq^2). \tag{2.1} \] For odd indices \(2N-1\geq 5\), there are two cases: \[ \mathscr{d}_{2N-1}(x)=\mathscr{d}_{2N-2}(x)+xq^{2N-1}\mathscr{d}_{2N-4}(x) \quad\text{if }2N-1\not\equiv0\pmod3, \tag{2.3} \] and, with \(\mathscr{d}_{-1}(x)=1\), \[ \mathscr{d}_{6N+3}(x)=\lambda_{N+1}(x)\mathscr{d}_{6N+2}(x)+xq^{6N+3}\mathscr{d}_{6N-1}(x) \quad\text{if }2N-1\equiv0\pmod3, \tag{2.4} \] where \[ \lambda_N(x)=1+xq^{2N-1}+x^2q^{4N-2}. \] The paper also derives additional even-indexed and consolidated odd-indexed relations: \[ \lambda_{\left\lceil\frac{2N}{6}\right\rceil}(x)\mathscr{d}_{2N}(x)=\mathscr{d}_{2N-1}(x)+x^2q^{2N}\mathscr{d}_{2N-3}(x) \quad\text{if }N\not\equiv 0\pmod{3}, \tag{3.1} \] \[ \mathscr{d}_{6N}(x)=\mathscr{d}_{6N-1}(x)+x^2q^{6N}\mathscr{d}_{6N-4}(x) \quad\text{if }N\equiv 0\pmod{3}, \tag{3.2} \] and \[ \mathscr{d}_{2N-1}(x)=\lambda_{\left\lceil \frac{2N+2}{6}\right\rceil}(x)\mathscr{d}_{2N-4}(xq^2)+xq^{2N-1}(1-xq)\mathscr{d}_{2N-7}(xq^2). \tag{3.3} \] These recurrences are significant because they express each quotient polynomial in terms of earlier quotient polynomials using only shifts in \(q\), multiplication by manifestly nonnegative monomials, or multiplication by \(\lambda_N(x)\), which itself has nonnegative coefficients. This suggests a natural inductive route to coefficient-wise nonnegativity [2508.10871]. ## 4. Resolution of the conjecture The main theorem states: \[ \text{For }n\geq1,\ \mathscr{d}_{n}(x)\text{ is a polynomial in }x\text{ and }q\text{ with nonnegative integer coefficients.} \] Hence Andrews’ Conjecture is true. The proof proceeds by induction, beginning with explicit base cases: \[ \mathscr{d}_{1}(x)=1+xq,\quad \mathscr{d}_{2}(x)=1,\quad \mathscr{d}_{3}(x)=1+x(q+q^3)+x^2q^2, \] \[ \mathscr{d}_{4}(x)=1+xq^3,\quad \mathscr{d}_{5}(x)=1+x(q^3+q^5),\quad \mathscr{d}_{6}(x)=1+x(q^3+q^5)+x^2q^6. \] Each of these is manifestly in \(\mathbb{Z}_{\ge 0}[x,q]\). The inductive step separates even and odd indices. For even indices, equation (2.1), \[ \mathscr{d}_{2N}(x)=\mathscr{d}_{2N-3}(xq^2), \] preserves nonnegativity because it is only a graded shift in \(q\). For odd indices not divisible by \(3\), equation (2.3), \[ \mathscr{d}_{2N-1}(x)=\mathscr{d}_{2N-2}(x)+xq^{2N-1}\mathscr{d}_{2N-4}(x), \] writes the polynomial as a sum of two nonnegative polynomials. For odd indices divisible by \(3\), equation (2.4), \[ \mathscr{d}_{6N+3}(x)=\lambda_{N+1}(x)\mathscr{d}_{6N+2}(x)+xq^{6N+3}\mathscr{d}_{6N-1}(x), \] is again manifestly nonnegative. The original conjecture follows immediately for the special indices \(6n-1\), since \[ \mathscr{d}_{6n-1}(x)=\sum_{i=0}^n c(n,i)x^i \] and all coefficients of \(\mathscr{d}_{6n-1}(x)\) are nonnegative. Therefore every \(c(n,i)\) is a polynomial in \(q\) with nonnegative coefficients [2508.10871]. ## 5. Coefficient families and stronger consequences The theorem is stronger than the original conjecture because it gives positivity for every quotient polynomial \(\mathscr{d}_n(x)\), not only for the subsequence indexed by \(6n-1\). Writing \[ \mathscr{d}_n(x)=\sum_{i\geq0}\mathscr{c}(n,i)x^i, \tag{3.4} \] the recurrences induce structural relations among the coefficients \(\mathscr{c}(n,i)\). From (2.1), \[ \mathscr{c}(2N,i)=\mathscr{c}(2N-3,i)\,q^{2i}. \] The paper also gives explicit relations between \(\mathscr{c}(n,i)\) and the original conjectural coefficients \(c(n,j)\): \[ \mathscr{c}(6N,j)=c(N,j)+q^{6N}c(N-1,j-2)\,q^{2(j-2)}, \tag{3.5} \] \[ \mathscr{c}(6N-1,j)=c(N,j), \tag{3.6} \] \[ \mathscr{c}(6N-2,j)=c(N,j)-q^{6N-1}c(N-1,j-1)\,q^{2(j-1)}, \tag{3.7} \] \[ \mathscr{c}(6N-3,j)=q^{-2j}c(N,j)+q^{6N-4}c(N-1,j-2), \tag{3.8} \] \[ \mathscr{c}(6N-4,j)=c(N-1,j)\,q^{2j}, \tag{3.9} \] \[ \mathscr{c}(6N-5,j)=q^{-2j}c(N,j)-q^{6N-3}c(N-1,j-1). \tag{3.10} \] These formulas transfer positivity information from \(\mathscr{d}_n(x)\) to \(c(n,j)\) and show that coefficient-wise nonnegativity is only one aspect of a more rigid structure. The resulting corollary proves, for \(0