Bergeron's coefficient-wise Gaussian-polynomial inequality in the non-divisibility regime, with its unimodality strengthening

For integers $0<a\le b<c\le d$ with $ad=bc$, F. Bergeron conjectured that the difference $\binom{b+c}{b}_q-\binom{a+d}{a}_q$ of two Gaussian polynomials (q-binomial coefficients) has non-negative coefficients, and Zanello strengthened this to unimodality. The conjecture is the principal specialization of Vessenes' Schur-positivity generalization of the Foulkes conjecture. A 2026 breakthrough of Raicu--Sam--Weyman--Yang on the Foulkes--Howe map, combined with corollaries of Gangl--Gutiérrez--Szwej, settles both statements whenever the minimum entry $a$ divides either of the two middle entries $b$ or $c$ (including the entire beta-family $(c,d)=(\beta a,\beta b)$ posed by Amdeberhan--Callan), and small $a\le 3$ was known earlier. The open problem is the remaining non-divisibility regime — quadruples with $a$ dividing neither $b$ nor $c$, starting at $(a,b,c,d)=(4,6,10,15)$; Amdeberhan--Callan's fully open auxiliary inequality on products and reciprocals of Gaussian polynomials (their Conjecture 4) remains a closely related open question from the same source.

Background

A Gaussian polynomial (or $q$-binomial coefficient) is the rational function $\binom{n}{k}_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}$, where $[m]!_q=\prod_{j=1}^{m}\frac{1-q^j}{1-q}$; it is in fact a polynomial in $q$ with non-negative, symmetric, unimodal coefficients (a sequence is unimodal if it never strictly increases after a strict decrease). It counts, for example, binary words of length $n$ with $k$ zeros weighted by the number of inversions. For polynomials $f,g$ one writes $f\ge g$ when $f-g$ has non-negative coefficients. The classical Foulkes conjecture predicts an inclusion of symmetric-power representations $\mathrm{Sym}^m(\mathrm{Sym}^n V)\subseteq \mathrm{Sym}^n(\mathrm{Sym}^m V)$ for $m\le n$; Vessenes generalized it, conjecturing that $(h_b\circ h_c)-(h_a\circ h_d)$ is Schur positive (expandable with non-negative integer coefficients in the Schur basis of symmetric functions) whenever $a\le b<c\le d$ and $ad=bc$, where $h_n\circ h_k$ denotes plethysm — substitution of one complete homogeneous symmetric function into another — which corresponds to an embedding $S^a(S^d V)\subseteq S^b(S^c V)$ of $GL(V)$-modules. Since $(h_n\circ h_k)(1,q)=\binom{n+k}{k}_q$, the principal specialization of Vessenes' conjecture implies, as underlined by F. Bergeron in his q-Foulkes program, that $\binom{b+c}{b}_q-\binom{a+d}{a}_q\ge 0$ for all $0<a\le b<c\le d$ with $ad=bc$ (this is Conjecture 1 of Amdeberhan--Callan). Both Gaussian polynomials here are symmetric of the same degree $ad=bc$, so the difference is symmetric, and Zanello conjectured the stronger statement that its coefficients are unimodal. Before 2023 the inequality was known for $a=2$ (Abdesselam--Chipalkatti, via surjectivity of a Foulkes--Howe-type map between plethystic compositions of symmetric powers) and for $a\le 3$ (Vessenes, refined by Zanello via Zeilberger's KOH theorem, a computer-algebraic telescoping method for hypergeometric sums). Amdeberhan--Callan proved the three-parameter beta-family case $(c,d)=(\beta a,\beta b)$ for $\beta=2$ using Wilf--Zeilberger certificate identities (the WZ method, in which a hypergeometric identity is certified by an explicit rational telescoping function). In February 2026, Raicu--Sam--Weyman--Yang proved that the Foulkes--Howe map $\alpha_k:\mathrm{Sym}^k(\mathrm{Sym}^{ab}U)\to \mathrm{Sym}^{ak}(\mathrm{Sym}^b U)$ of $\mathrm{SL}_2$-representations is injective for $k\le b$ and surjective for $k\ge b$, which yields Vessenes' conjecture for $\mathrm{SL}_2(\mathbb{C})$ whenever the minimum entry $a$ divides either of the two middle entries $b$ or $c$; Gangl--Gutiérrez--Szwej observed that the character of a $\mathrm{GL}_2$-complement of the resulting embedding is exactly the difference of the two Gaussian polynomials, so Bergeron's inequality and Zanello's unimodality both hold under this divisibility condition. This resolves the entire beta-family (since $a\mid c=\beta a$), but the method breaks when $a$ divides neither $b$ nor $c$, a regime the authors of both papers themselves flag as open. Amdeberhan--Callan also posed (their Conjecture 4) an auxiliary inequality for $0\le k\le a<b$, namely $\binom{a}{k}_q\binom{a+b}{b-k}_q\ge \binom{b}{k}_q\binom{a+b}{a-k}_q$, equivalently $\binom{a}{k}_q\binom{b}{k}_q\binom{a+b}{b}_q\big(\binom{a+k}{k}_q^{-1}-\binom{b+k}{k}_q^{-1}\big)\ge 0$, as a codicil of their calculations toward the beta-Conjecture; it remains open.

Problem Statement

Prove or disprove Bergeron's coefficient-wise inequality for the remaining admissible quadruples: for all integers $0<a\le b<c\le d$ with $ad=bc$ and $a$ dividing neither $b$ nor $c$ (equivalently, for all admissible quadruples, the cases $a\mid b$ or $a\mid c$ being established in the literature), is it true that every coefficient of $\binom{b+c}{b}_q-\binom{a+d}{a}_q$ is non-negative? Settle likewise the stronger form: are the coefficients of this symmetric polynomial non-negative and unimodal (Zanello's strengthening)? The two questions carry one scientific objective — establish the coefficient-wise dominance $\binom{b+c}{b}_q\ge\binom{a+d}{a}_q$ and its unimodality refinement for every equal-area quadruple — with the unimodality claim being strictly stronger than the non-negativity claim on the same domain: the difference is a symmetric polynomial, and a symmetric polynomial with non-negative coefficients need not be unimodal, so the unimodality claim neither reduces to nor follows from the non-negativity claim. A concrete minimal test instance is $(a,b,c,d)=(4,6,10,15)$. The 2026 divisibility results cover exactly the cases $a\mid b$ or $a\mid c$ in this labeling; in particular, quadruples in which $a$ divides $d$ but neither $b$ nor $c$ (for example $(6,9,16,24)$) are not covered by them and belong to the open regime. Amdeberhan--Callan's Conjecture 4, stated in Background, remains open as a separate auxiliary question of the source paper and is not itself part of the objective here.

References

1. Tewodros Amdeberhan, David Callan, Gaussian inequality, Rocky Mountain Journal of Mathematics 54 (2024), DOI 10.1216/rmj.2024.54.1527, preprint arXiv:2304.03395 (2023), https://arxiv.org/abs/2304.03395
2. Fabrizio Zanello, On Bergeron's Positivity Problem for q-Binomial Coefficients, Electronic Journal of Combinatorics 25(2) (2018) P2.17, DOI 10.37236/7358, arXiv:1709.06187, https://arxiv.org/abs/1709.06187
3. François Bergeron, A q-Analog of Foulkes' Conjecture, Electronic Journal of Combinatorics 24(1) (2017) P1.38, DOI 10.37236/6004, https://doi.org/10.37236/6004
4. Rebecca Vessenes, Generalized Foulkes' Conjecture and tableaux construction, Journal of Algebra 277(2) (2004) 579-614, DOI 10.1016/j.jalgebra.2004.01.017, https://doi.org/10.1016/j.jalgebra.2004.01.017
5. Abdelmalek Abdesselam, Jaydeep Chipalkatti, Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture, Advances in Mathematics 208(2) (2007) 491-520, DOI 10.1016/j.aim.2006.03.003, https://doi.org/10.1016/j.aim.2006.03.003, preprint arXiv:math/0411110 (2004), https://arxiv.org/abs/math/0411110
6. Claudiu Raicu, Steven V. Sam, Jerzy Weyman, Fuxiang Yang, Powers of binary forms and derived Hermite reciprocity, arXiv:2602.15175 (2026), https://arxiv.org/abs/2602.15175
7. Moritz Gangl, Álvaro Gutiérrez, Michał Szwej, On the generalised Foulkes conjecture for SL_2(C) under divisibility conditions, arXiv:2507.06220 (2025, revised 2026), https://arxiv.org/abs/2507.06220
8. Tewodros Amdeberhan, Matthias Beck, Bergeron's conjecture & a tale of two binomial coefficients, arXiv:2607.04050 (2026), https://arxiv.org/abs/2607.04050