85 problems

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 $m,n\in\mathbb{N}$, let $d=\gcd(m,n)$, let $p_{m,n}$ be the symmetric function in the conjecture, and let $\mathsf{LRP}(m,n)$ denote the relevant labeled rectangular paths, wit…

Let ${\mathcal S}\subset\mathbb{Z}^2$ be a non-singular step set, let $Q(1,1;t)$ be the corresponding generating function, and let … Let $(x^*,y^*)$ be the unique point minimizing…

Consider lattice paths in $\mathbb{Z}^2$ with unit north ($N$) and east ($E$) steps. Fix positive integers $r,s$ and identify points $v,w\in\mathbb{Z}^2$ when $v-w=\ell(r,s)$ for s…

Let $Y$ be a one-sided ladder-shaped region of the plane, with the partial order defined by $(i,j)\leq(i',j')$ if $i\geq i'$ and $j\leq j'$. For points $P_i=(i,n)$ and $Q_i=(m,i)$…

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ be a fixed Ferrers diagram of a partition, placed in the northwest corner of a grid whose vertices are $(i,j)$ for $i,j\geq 0$.…

Let $A$ be the parameter in the model and let $\log Z(\alpha)$ denote the stated formula for the logarithm of the partition function as a function of $\alpha$. Limiting-value conje…

Let $Z_-$ and $Z_+$ be the two partition-function expressions in the critical non-intersecting lattice-path model, and let $\sigma^2(N_c)$ denote the variance of the number of edge…

Let $a=1$, let $b=c=1/2+\Theta(1/n)$, and suppose that the aspect ratio is nearly a simple rational $p/q$. For sufficiently large $\alpha$, follow a loop for $mp+nq$ steps; because…

Let $a=1$, let $b=c=1/2+\Theta(1/n)$, and suppose that the aspect ratio is nearly a simple rational $p/q$. At the critical point, let $\alpha$ denote the parameter governing the pa…

Let $\frak S$ be a set of steps defining a slit-plane model, and let its generating function be the generating function that counts walks avoiding the forbidden half-line. A walk e…

Let $\frak S$ be a set of steps defining walks on the slit plane, and let $S(x,y;t)$ be the corresponding generating function. Say that the forbidden half-line is crossable without…

Catalan classification conjecture. There are exactly $C_n$ families of size $2^n$ in which every two distinct paths have an even number of common edges.

Let $Q(x,y)$ be the generating function associated with the walk model under consideration, and let the parameters defining the model determine its phase diagram. The curve contain…

Qu and Zhang's area-depth symmetry conjecture. This variant of $\widetilde{C}_M(q,t)$ remains symmetric in $q$ and $t$.

Let $b(k)$ denote the number of points in the longest lattice path with steps in $\{(1,0,0),(0,1,0),(0,0,1)\}$ avoiding $k$ collinear points. Gerver and Ramsey constructed an infin…

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…

For $m,n,k,d\in\mathbb{N}$ with $m>0$ and $d\leq\min\{m+k,n+k\}$, let $e_{m,n}$, $\Theta_{e_k}$, $s_{(m-1)^k}^{\perp}$, $h_d$, $e_{n+k-d}$ and $\langle\,\ \rangle$ have their usual…

For $m,n\in\mathbb{N}$, let $e_{m,n}$ be the rectangular symmetric function, let $\Theta_{e_k}$ be the theta operator indexed by $e_k$, and let $\mathsf{LRD}(m+k,n+k)^{\ast k}$ den…

Let $G$ be the formal power series defined by $G=(S-1)/S_1$, where $S$ and $S_1$ are the power series used in the Geode construction. For parameters $u_1,cots,u_m$, write…

Let $P1$ denote the property described in the paper as “constant on blocks.” The P1 conjecture. Property $P1$ (constant on blocks) holds. This conjecture arose from preliminary inv…

Let a grand zigzag knight's path be a path whose number of steps has an expected value determined by the corresponding family of paths ending on the $x$-axis at size $n$. Asymptoti…

Let an N-excursion be a nondeterministic walk of excursion type, and let its generating function count N-excursions by the relevant parameters, such as length, minimal reachable po…

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…