4 problems

Given coprime positive integers $m$ and $n$, let $\mathcal D_{m,n}$ be the set of rational $(m,n)$-Dyck paths, and let $\Phi$ be the sweep map that sorts the steps of a path accord…

Let $N$ and $M$ be positive integers, write $d=\frac{\gcd(N,M)}{}$, $N=dn$, and $M=dm$ with $n$ and $m$ coprime, and define … Let $P_{n,m}$ be the operator from the coprime case, a…

Let $n$ and $m$ be coprime positive integers, let $C_{n,m}(q,t)$ be the generating polynomial defined by … where $P_{n,m}$ is the operator acting on symmetric functions, and let th…

Let $a$ and $b$ be relatively prime positive integers. Let $\mathfrak{D}_{a,b}$ be the set of $(a,b)$-Dyck paths, and let … be the image of the pair $(\zeta,\eta)$. Uniqueness conj…