26 problems

Two-parameter Borwein positivity conjecture. The polynomials $A_{L,m}(q)$ for $0\leq m\leq L$ and $C_{L,m}(q)$ for $0\leq m\leq L-1$ have nonnegative coefficients.

Refined Borwein coefficient-positivity conjecture. The polynomials $\mathcal{A}_{L,M}(q)$ and $\mathcal{C}_{L,M}(q)$ have nonnegative coefficients.

For each integer $k\geq1$, define … Here $A(q)\geq_qB(q)$ denotes coefficientwise comparison, so $U_k(q)\geq_q0$ means that all coefficients of $U_k(q)$ are nonnegative. The $U_k$…

For $r\in\mathbb{Q}^+$ and an integer $k\geq 1$, define the integer trace of the generalized $q$-binomial coefficient by retaining its integer exponents, and write … For power seri…

Gatzweiler–Krattenthaler conjecture. If this quotient is a polynomial, then it has non-negative integer coefficients. The conjecture is a positivity assertion for polynomial quotie…

For $k,n\in\mathbb{N}$ with $k\ge n+1$, let $G_{k,n}(q)$ be defined by … Write $A(q)\succeq0$ when all coefficients of $A(q)$ are nonnegative. Monotonicity conjecture for…

For $k,n\in\mathbb{N}_0$, define $G_{k,n}(q)$ by … Here $\begin{bmatrix}a\b\end{bmatrix}_q$ denotes the $q$-binomial coefficient, and $A(q)\succeq0$ means that all coefficients of…

For $n\geqslant k\geqslant0$, define the quantum binomial by … where $[n]=(q^{-n}-q^n)/(q^{-1}-q)$ and $[0]=0$. A sequence of Laurent polynomials is strongly Schur log-concave if e…

Let $R_n^{(1,3)}$ be the $(1,3)$-bosonic-fermionic coinvariant ring, let $\operatorname{Frob}(R_n^{(1,3)};q;u,v,w)$ denote its Frobenius series, and let $s_{(1^n)}$ denote the Schu…

Let $n$ be a positive integer and let $r$ be an arbitrary integer. Write $[n]=(1-q^n)/(1-q)$ for the $q$-integer and let $\Phi_n(q)$ denote the $n$th cyclotomic polynomial. For int…

Let $n$ be a positive odd integer, let $q$ be an indeterminate, let $\Phi_n(q)$ be the $n$th cyclotomic polynomial, and use $(x;q)_k=\prod_{j=0}^{k-1}(1-xq^j)$ and…

Conjectured three-rowed formulas. The two displayed identities hold.

Let $G_s(n,k)$ be the recurrence obtained by summing over partitions of $k$ having size at most $s$, with $q$ an indeterminate and $n,k,s$ in the domain where this recurrence is de…

Let $k$, $m$, and $b$ be integers with $k\ge 2$, $m\ge k$, and, for $k\ge 3$, $k-2\le b\le \frac{km-4k+4}{k-2}$, where $k(m-b)$ is even. Define … Here $\binom{m}{k}_q$ denotes the…

Let $a,b,c,d$ be positive integers such that $a$ is the smallest and $ad=bc$. The $q$-binomial coefficient is … Thus $\binom{m+n}{m}_q$ is a symmetric polynomial in $q$. Bergeron's…

General $q$-congruence conjecture. One has

The four-factor $q$-congruence conjecture. The following four congruences hold:

Multivariate positivity conjecture. is a polynomial in $q$ with nonnegative integer coefficients. This is stated as a generalization of the q-Narayana positivity conjecture; no res…

Let $m$ be a positive integer, let $n_1,\ldots,n_m$ be positive integers, and set $n_{m+1}=n_1$. Let $j$ be an integer and let $r$ be a non-negative integer. The cyclic q-Narayana-…

Let $n$ and $r$ be positive integers, and let $a$ be a non-negative integer. Define the $q$-Narayana numbers by … Let $E_{n,r,a,j}(q)$ denote the Laurent polynomial expression in t…

Let $m$, $n$, $s$ and $t$ be positive integers. Let $r$ be a non-negative integer with $r+s+t\equiv 1\pmod{2}$, and let $j$ be an integer. Let $B_{m,k}(q)$ and $B_{n,k}(q)$ denote…

Let $u$ and $i$ be nonnegative integers and let $q$ be a prime power. Write $\binom{n}{r}_{q}$ for the Gaussian binomial coefficient, namely the number of $r$-dimensional subspaces…

Fix $a,b\in\mathbb{N}$. Let $\mathrm{R}(a,b)$ be the set of partitions fitting in the $b\times a$ rectangle, let $\mathsf{ml}_{b,a}(\mu)$ be the minimum level on the frontier of…

Let $p$ be a prime, let $n$ be a power of $p$, and define the $q$-integer by … For Gaussian binomial coefficients, write ${a\brack b}_q$. Prime-power $q$-binomial congruence. Then…

Let $p$ be a prime, $a\geq1$, let $\Phi_{p^a}(q)$ be the $p^a$th cyclotomic polynomial, and let $\left(\frac{p^a}{3}\right)$ denote the Legendre symbol. Sun–Tauraso q-conjecture. O…