29 problems

Let $n$) and $k$ satisfy $0\le k<n$, and let $\Delta'_{e_{n-k-1}}$ be the Delta operator on modified Macdonald polynomials. Let a module of super-diagonal coinvariants denote the m…

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…

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 $n,k\in\mathbb{N}$ with $n>k\geq 0$. For a labelled Dyck path $P$ of size $n$ with positive labels and $k$ decorated rises, let $\mathsf{area}(P)$, $\mathsf{dinv}(P)$, and…

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

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 $n,k\in\mathbb{N}$, 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 $e_r$ denote the elementary symmetric function, let $e_{n-k}^{\perp}$ be the operator adjoint to multiplication by $e_{n-k}$ under the Hall inner product, and let…

Let $\Theta_{e_k}$ be the Theta operator, let $\nabla$ be the nabla operator, and let $\mathsf{LD}(n)^{\ast k}$ and $\mathsf{LD}(n)^{\bullet k}$ denote the rise- and valley-decorat…

Let $TDR_n$ be the quotient of the polynomial-superpolynomial ring in two sets of commuting variables and two sets of anti-commuting variables by the ideal generated by the invaria…

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 $n,k\in\mathbb{N}$ with $n>k\geq 0$, and let $\alpha\vDash n-k$ be a composition of $n-k$. For a partially labelled Dyck path $P\in\mathsf{LD}(n)^{\ast k}$, let…

Let $n>k\geq0$, let $\mathsf{LDD}^{\ast k}(n)$ be the set of decorated labelled Dyck paths in the rise model, and let $\mathsf{dinv}(D)$, $\mathsf{area}(D)$ and $x^D$ denote the co…

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…

Let $D_k$ be the plethystic operators defined by … where $\operatorname{Exp}[X] = \sum_{n \geq 0} h_n[X]$ and let $\Theta_k=\Pi\,\underline{e}_k^*\,\Pi^{-1}$, with…

Let $\mathbb{V}_n({\mathbf {a}})$ be the extension of the superspace module $V_n({\mathbf {a}})$ obtained by adjoining commuting variables $y_1,\dots,y_n$ and closing under polariz…