8 problems

Permutation formulation of the rational Shuffle Conjecture. The following equation holds:

The rational Shuffle Conjecture. The following equation holds:

Rational Dyck-path constant-term conjecture. For any such $(u,v)$ and $\gamma$,

Compositional $(km,kn)$-Shuffle Conjecture. For all compositions $\alpha=(a_1,a_2,\ldots,a_r)\models k$,

Refined non-coprime rational Shuffle Conjecture. For all coprime pairs of positive integers $(m,n)$, all $k\in\mathbb{N}$, and $1\leq r\leq k$,

Non-coprime rational Shuffle Conjecture. For all coprime pairs of positive integers $(m,n)$ and any $k\in\mathbb{N}$,

Gorsky–Negut rational Shuffle Conjecture. For all coprime pairs of positive integers $(m,n)$,

Let $\mathsf{PF}_{a,b}(q,t)$ be the symmetric function defined from rational parking functions, and let ${\sf Sch}_{a,b;k}(q,t)$ denote the coefficient of $s_{(k+1,1^{a-k-1})}$ in…