37 problems

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…

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.

Peter Borwein's conjecture. The polynomials $A_n(q)$, $B_n(q)$ and $C_n(q)$ have nonnegative coefficients.

Bergeron's conjecture. If $1\leq a<b<c<d$ are integers with $ad=bc$, then

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…

Guo's strengthened positivity conjecture. For all $1\leq b\leq r$, one has

Guo's positivity conjecture. $C_{m,n}(q)$ is positive, that is, $C_{m,n}(q)\in\mathbb{Z}_{\geq 0}[q]$.

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 $n\geqslant k\geqslant0$, 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 $n$ be a nonnegative integer, and let $k$ and $l$ be integers such that the two $q$-binomial coefficients with top argument $n$ are defined. Consider their quotient, assuming t…

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…

Consider symmetric differences of two $q$-binomial coefficients having the same degree, so that no shift by a power of $q$ is required. Zagier's conjecture. All such differences ar…

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…

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…