7 problems

Interior-coefficient unimodality conjecture. This sequence is unimodal when $n$ is sufficiently large; the source notes that computations seem to support the claim for $n\geq16$. T…

For $m\geq3$, let … and define … where $\operatorname{PT}'_q$ is the positive-part operator used in the source. Write $[q^i]G_m(x,q)$ for the coefficient of $q^i$. Almost-positivit…

Positivity conjecture. If $n=2m+1$ is odd, then

For an integer $n\geq 2$, define the MacMahon $q$-Catalan polynomial … where $[k]_q=(1-q^k)/(1-q)$ and $[k]!_q=[1]_q\cdots[k]_q$. Chen's even-derivative conjecture. All even-order…

Let $a$ and $b$ be relatively prime positive integers. For an $(a,b)$-core partition $\kappa$, let $\operatorname{sl}(\kappa)$ be its skew length and let…

Let $\mathsf{Cat}_{a,b}(q,t)$ be the rational $q,t$-Catalan number and let $[m]_q$ and $\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q$ denote the $q$-integer and $q$-binomial coefficient. Rati…

Fix $a,b\in\mathbb{N}$ with $\gcd(a,b)=1$. Let $\mathrm{D}(a,b)$ be the set of partitions whose diagrams fit in the triangle with vertices $(0,0)$, $(0,a)$, and $(b,a)$. For a cell…