20 problems

Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(q,t)$, let $\{\widetilde H_\mu\}$ be the modified Macdonald basis, and define $\Delta'_f$ by … where…

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 $n,k\in\mathbb{N}$ with $n>0$, let $\Theta_{e_k}$, $\nabla$, and $\omega$ be the indicated symmetric-function operators, let $p_n$ be the power-sum symmetric function, and let…

For $n,k\in\mathbb{N}$, let $\Theta_{e_k}$, $\nabla$, and $\omega$ be the indicated symmetric-function operators, let $p_n$ be the power-sum symmetric function, and let…

Let $n$ and $k$ be the parameters in the Delta Conjecture, let $s_\lambda^\perp$ denote the operator adjoint to multiplication by $s_\lambda$, let $H_\lambda(x;q)$ be the Hall–Litt…

Let $m,n\in\mathbb{N}$, let $d=\gcd(m,n)$, and let $p_{m,n}$ be the symmetric function associated with rectangular paths. Let $\Theta_{e_k}$ be the Theta operator indexed by $e_k$.…

Let $m,n\in\mathbb{N}$, let $e_{m,n}$ be the rectangular Dyck-path symmetric function, and let $\Theta_{e_k}$ be the Theta operator indexed by $e_k$. For…

Let $n,k\in\mathbb{N}$ with $k<n$, and let $\mathsf{LSQ}'(n)^{\bullet k}$ denote the partially labelled square paths with $k$ decorated valleys. For such a path $P$, let…

Let $n,k,r\in\mathbb{N}$ with $k<n$, and let $\mathsf{LD}(n)^{\bullet k}$ be the partially labelled Dyck paths with $k$ decorated valleys. For $P$ in this set, let…

Let $m,n,k\in\mathbb{N}$ with $k<n$, and let $\mathsf{LD}(m,n)^{\bullet k}$ denote the partially labelled rectangular Dyck paths with $k$ decorated valleys. For such a path $P$, le…

Let $n\geq1$, let $\mathsf{PSQ^E}(n)$ be the relevant set of square paths, and let $\delta(P)$ be the dinv reading word of $P$. Write $\mathsf{Des}(\delta(P))$ for its descent set…

Let $m,n\in\mathbb{N}$, let $\mathsf{LPP}(m+1,n+1)$ be the set of labelled parallelogram polyominoes of the indicated dimensions, and let $\mathsf{area}(P)$, $\mathsf{pmaj}(P)$ and…

Let $m,n,k$ be in the range for which $\mathsf{PLD}(m,n)^{\ast k}$ is defined, and let $\mathsf{pmaj}(D)$ denote the proposed partial-major-index statistic on a partially labelled…

Let $m,n,k$ be in the range for which the partially labelled Dyck-path set $\mathsf{PLD}(m,n)^{\ast k}$ is defined. Let $\Delta_{h_m}$ and $\Delta'_{e_{n-k-1}}$ be Delta operators,…

Let $\mathcal{E}_n=\sum_{\mu\vdash n}\mathcal{C}_\mu\otimes s_\mu(\bm{z})$ be the pure bosonic multivariate coinvariant characteristic, where…

Haglund–Remmel–Wilson's conjecture. For $m\geq 0$, $n\geq 0$, and $k\geq 0$,

The pmaj formula. For $m\geq 1$ and $n\geq 1$,

Partially labelled Dyck path Delta conjecture. One has

New square conjecture. For every $n\geq 1$,

Let $\mathcal{LD}_n$ be the set of labeled Dyck paths of size $n$. For $P\in\mathcal{LD}_n$, let $a_i(P)$, $\ell_i(P)$, $\operatorname{area}(P)$, $\operatorname{dinv}(P)$, $d_i(P)$…