11 problems

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…

Loehr–Warrington's Square Paths Conjecture.

The rational Shuffle Conjecture. The following equation holds:

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

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…