24 problems

Let $(m,n)$ be a coprime pair, let $Q_{m,n}$ be the operator recursively defined from the splitting of $(m,n)$, and let $\operatorname{PF}_{m,n}$ denote the corresponding set of…

Let $\rho=(a\,|\,b)$ be a hook, let $\Gamma_a$ be the Dyck path defined in the source, and let $\mathbb{L}_\beta(t;\bm{z})$ be the LLT polynomial associated with a Dyck path…

Let $p=(p_1,p_2,\ldots,p_k)$ be a composition, and let $\mathcal{PF}(p)$ denote the parking functions whose Dyck path hits the $0$-diagonal according to $p$. For a parking function…

Loehr–Warrington's Square Paths Conjecture.

The original Shuffle Conjecture.

Let $\alpha$ be a composition of $n$, let $C_\alpha[X;q]$ denote the corresponding compositional symmetric function, and let $\pi$ range over Dyck paths of size $n$. Write…

The rational Shuffle Conjecture. The following equation holds:

Let $(m,n)$ be coprime and let $k>1$. Define operators $Q_{0,n}$ and coefficients $c_\lambda(q,t)$ by … and set … Let $\operatorname{PF}_{km,kn}$ be the corresponding set of parkin…

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 $(m,n)$,

Haglund–Morse–Zabrocki B-operator conjecture. For any composition $\alpha$ of $n$,

HHLRU-2005 conjecture. For all $n\geq 1$,

Let $\mathsf{DR}_n$ be the doubly graded diagonal coinvariant ring, and let $\mathsf{PF}_n$ be the labeled classical parking functions. For each $P$, let $\mathsf{area}(P)$,…

Let $n$ be a positive integer, let $\mu=(\mu_1,\mu_2,\ldots,\mu_k)\vdash n$, and let $E_1,E_2,\ldots,E_k$ be successive segments of the word $123\cdots n$ of respective lengths…

Let $n$ be a positive integer, let $\mu=(\mu_1,\mu_2,\ldots,\mu_k)\vdash n$, and let $E_1,E_2,\ldots,E_k$ be successive segments of the word $123\cdots n$ of respective lengths…

Let $\operatorname{Fr}_{n,m}$ be the symmetric function obtained by summing over $m/n$ parking functions with the statistics and Gessel quasisymmetric functions defined in the sour…

Refined big-car conjecture. For $(p_1,\ldots,p_k)\models n+1-j$,

Diagonal-composition refinement. For all compositions $p\models n$ and partitions $\mu\vdash n$,

Let $\mathscr{H}_n^{(r)}(\mathbf{w};q,t,1)$ denote the graded Frobenius characteristic of the trivariate diagonal higher-harmonic space. Let $\Dyck{r}{n}$ be the set of $r$-Dyck pa…

Let $\mathscr{H}_n^{(r)}(\mathbf{w};q,t)$ denote the graded Frobenius characteristic of the bivariate diagonal higher-harmonic space. Let $\delta_n^{(r)}$ be the relevant $r$-stair…

Let $\alpha\models n$ be a composition, let $C_\alpha[X;q]$ be the generalized Hall–Littlewood symmetric function indexed by $\alpha$, let $\mathcal{W}P_D$ denote the set of words…