Let pow≥2, k≥1, and let a_1,…,a_k,b be positive integers. Write [r]_q:=1+q+…+q^r-1 and, for an integer pow, write [b]_q^pow:=1+q^pow+…+q^pow(b-1). The product q-analog conjecture. If pow| a_i for some 1≤ i≤ k, or b≤1+∑_i=1^k lfloorfraca_ipowrfloor, then the polynomial [a_1]_q…[a_k]_q[b]_q^pow is unimodal. Moreover, if k≤3 or pow≤3, this condition is also necessary. This proposed generalization is motivated by preceding results on products of q-analogs; the stated cases and necessity claim are not established in the source and remain open.

Unimodality conjecture for products of q-analogs

From papers

Let $\pow\geq2$ , $k\geq1$ , and let $a_1,\ldots,a_k,b$ be positive integers. Write $[r]_q:=1+q+\dots+q^{r-1}$ and, for an integer $\pow$ , write $[b]_{q^{\pow}}:=1+q^{\pow}+\dots+q^{\pow(b-1)}$ . The product q-analog conjecture. If $\pow\mid a_i$ for some $1\leq i\leq k$ , or

b\leq1+\sum_{i=1}^k\left\lfloor\frac{a_i}{\pow}\right\rfloor,

then the polynomial

[a_1]_q\dots[a_k]_q[b]_{q^{\pow}}

is unimodal. Moreover, if $k\leq3$ or $\pow\leq3$ , this condition is also necessary. This proposed generalization is motivated by preceding results on products of $q$ -analogs; the stated cases and necessity claim are not established in the source and remain open.

Progress summary

Open

The conjecture was newly formulated and computationally tested in small cases in May 2026, but no proof or disproof has been publicly verified.

Connelly, Ito, Martinez, Shevchenko, and Yang proposed that the product $[a_1]_q\cdots[a_k]_q[b]_{q^r}$ is unimodal under the stated divisibility or inequality condition, and that the condition is necessary when $k\le 3$ or $r\le 3$ . The paper presents both assertions as conjectures.

Known results

Computationally verified for $k\le 5$ , $r\le 6$ , and $\max\{a_i,b\}\le 15$ (Connelly, Ito, Martinez, Shevchenko, and Yang, 2026).
Outside the proposed necessity ranges, the condition is known not to be necessary: $([3]_q)^4[2]_{q^4}$ is unimodal despite violating it (2026).

May 2026 paper

The authors’ paper records Conjecture 5.4 but gives no proof of either sufficiency or necessity, and no verified counterexample within the claimed necessity ranges. The retrieved literature contains no later proof, disproof, or corroborating settlement.

Current status (as of August 2026): The conjecture remains open; only bounded computational checks and examples outside the proposed necessity ranges are established.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

Brendan B. Connelly, Ezekiel Ito, Thomas C. Martinez, Olha Shevchenko and Kacey Yang, “Unimodality of q-Fibonomial coefficients for small cases”, arXiv:2605.12822 (2026).

1 week ago

Infinitely many counterexamples to the claimed necessity, beginning immediately beyond the tested range. The conjecture asserts that if $k\le3$ or $r\le3$ , then unimodality of

F(q)=\prod_{i=1}^{k}[a_i]_q\,[b]_{q^r}

requires either $r\mid a_i$ for some $i$ , or

b\le1+\sum_{i=1}^{k}\left\lfloor\frac{a_i}{r}\right\rfloor.

Set

r=3,\qquad b=2,\qquad k=6,\qquad a_1=a_2=a_3=a_4=a_5=a_6=2.

No $a_i$ is divisible by $3$ , and

1+\sum_{i=1}^6\left\lfloor\frac23\right\rfloor=1<2=b.

Thus both supposedly necessary alternatives fail. Nevertheless,

\begin{aligned} F(q)&=(1+q)^6(1+q^3)\\ &=1+6q+15q^2+21q^3+21q^4+21q^5\\ &\hspace{2em}+21q^6+15q^7+6q^8+q^9, \end{aligned}

whose coefficient sequence

(1,6,15,21,21,21,21,15,6,1)

is symmetric and unimodal. Since $r=3$ , this lies directly within the conjectured necessity range.

More generally, for every $k\ge6$ , take $r=3$ , $b=2$ , and $a_1=\cdots=a_k=2$ . The two numerical alternatives still both fail, while

F(q)=(1+q)^{k-6}\bigl((1+q)^6(1+q^3)\bigr)

remains symmetric and unimodal. Indeed, if $(u_0,\ldots,u_d)$ is symmetric, unimodal, and nonnegative, the coefficients of $(1+q)\sum_j u_jq^j$ are $v_j=u_j+u_{j-1}$ ; they are symmetric and increase up to the midpoint because

v_{j+1}-v_j=u_{j+1}-u_{j-1}\ge0.

Induction proves the claim for every $k\ge6$ .

The authors explicitly tested only $k\le5$ , explaining why this entire infinite family was missed. This disproves the asserted necessity when $r\le3$ ; the separate sufficiency assertion and the $k\le3$ necessity assertion are not decided here.

Source: Connelly, Ito, Martinez, Shevchenko, and Yang, Unimodality of $q$ -Fibonomial coefficients for small cases, Conjecture 5.4.

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