6 problems

Let $k\geq 3$, let $c=(c_1,c_2,c_3)$ be a cylindric partition profile with $|c|+3=m=3k+\{-1,0,1\}$, and, using cyclic symmetries, assume $c_1\geq c_2,c_3$. Let…

Cylindric-partition hierarchy conjecture.

For integers $d,r\geqslant1$, let $c=(c_0,\dots,c_{r-1})$ be a profile of level $d$, meaning … Let $Q_{n,c}(q)$ be the formal power series defined from the cylindric-partition gene…

Let $\operatorname{GK}_{c}(z,q)$ denote the two-variable generating function for cylindric partitions, and let $(q)_n$ and $\genfrac{[}{]}{0pt}{}{a}{b}$ denote the $q$-Pochhammer s…

For a nonnegative integer $n$ and a profile $c=(c_0,c_1,c_2)$, define $d:=c_0+c_1+c_2$, $\ell:=\gcd(d,3)$, and … Assume $d\not\equiv0\pmod 3$. Corteel--Dousse--Uncu's conjecture. T…

Non-negative coefficient conjecture. For any profile $c=(c_1,c_2,c_3)$, the polynomial $P_{n,c}(q)$ has non-negative coefficients. Moreover, if $\ell\not\equiv 0\pmod{3}$, then