6 problems

For $m,n\in\mathbb{N}$, let $d=\gcd(m,n)$, let $p_{m,n}$ be the symmetric function in the conjecture, and let $\mathsf{LRP}(m,n)$ denote the relevant labeled rectangular paths, wit…

For $n,k\in\mathbb{N}$, let $\Theta_{e_k}$, $\nabla$, and $\omega$ be the indicated symmetric-function operators, let $p_n$ and $p_{n,n(k+1)}$ be the relevant symmetric functions,…

For $m,n,k,d\in\mathbb{N}$ with $m>0$ and $d\leq\min\{m+k,n+k\}$, let $e_{m,n}$, $\Theta_{e_k}$, $s_{(m-1)^k}^{\perp}$, $h_d$, $e_{n+k-d}$ and $\langle\,\ \rangle$ have their usual…

For $m,n\in\mathbb{N}$, let $e_{m,n}$ be the rectangular symmetric function, let $\Theta_{e_k}$ be the theta operator indexed by $e_k$, and let $\mathsf{LRD}(m+k,n+k)^{\ast k}$ den…

Let $e_n$ be the $n$th elementary symmetric function, let $\nabla$ denote the Bergeron–Garsia nabla operator, and let labeled Dyck paths be Dyck paths equipped with the labeling da…

Let $m,n>0$, let $\mu=\mu_1\cdots\mu_k\vdash n$ be a composition as used in the paper, and define … Let $\mathrm{Q}_{m,n}$ be the rational shuffle operator, $1$ the unit symmetric…