42 problems

Let $T$ be an $m\times n$ table with $2n\geqslant m$. Let $\mathcal{C}(T)$ be the coefficient matrix whose entries are the coefficients arising when the relevant quantities…

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.

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…

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 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 $m,n\in\mathbb{N}$, 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 $m,n\in\mathbb{N}$, let $d=\gcd(m,n)$, and let $p_{m,n}$ be the symmetric function associated with rectangular paths. For a rectangular path $\pi\in\mathsf{LRP}(m,n)$, let…

$L^1$ convergence conjecture. There exists a constant $M\in(-\infty,+\infty)$ such that

Differential-equation conjecture. The generating function $D(s,z)$ is D-finite in $z$ but not algebraic. Specifically, it satisfies a fifth-order linear differential equation with…

Fix integers $n$ and $k$ with $k\le n$. A Dyck path is a lattice path of semilength $n$ from $(0,0)$ to $(n,n)$ that stays weakly on one side of its diagonal; a nested pair…

Fix integers $n$ and $k$. Consider pipe dreams $P$ with $\omega_P\in\mathfrak{S}_n$ whose factorization into atomics consists of $k$ identity permutations, and quarter-plane walks…

Let $b_n^{(p)}$ denote the coefficients appearing in the recurrence referenced in the source, with $n,p$ positive integers. The first two rows are $b_1^{(p)}=p/12$ and…

Crucial–Lundqvist–Nevo conjecture. The Hilbert series of $\Lambda(V)/(f,g)$ is

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…

Axis-minimization conjecture.

Let $\mathcal{I}_n(n)$ denote the number of the relevant perfect lattice paths in an $n\times n$ table. Povolotsky's recurrence conjecture. For positive integers $n$, the following…

Bijection conjecture. The following families are in bijection: hesitating excursions of length $2n$ in the quadrant; hesitating axis-walks of length $2n$ in the octant; and hesitat…

Riehl's conjecture. The number of basketball walks of length $n$ starting at the origin and ending at altitude $1$ that never touch or pass below the $x$-axis equals the number of…

Let ${\mathscr{S}}$ be a step multiset and let ${\textsc{H}}_{\theta}({\mathscr{S}})$ be a half-plane model, with $h_n(\theta)$ denoting the number of walks of length $n$ in this m…

Let $G_2$ be the signed peak-count generating function for the quarter-planar paths under consideration. Define…

Let $\mathcal{V}$ be a positive vertical path with $u$ up-steps and $d$ down-steps, let $Q$ be the set of quarter-planar paths with $r$ right-steps, $l$ left-steps, and vertical pr…

Let $\mathcal{V}$ be a positive vertical path with $u$ up-steps and $d$ down-steps, let $Q$ be the set of quarter-planar paths with $r$ right-steps, $l$ left-steps, and vertical pr…

Let $\mathcal{V}$ be a positive vertical path with $u$ up-steps and $d$ down-steps, let $Q$ be the set of quarter-planar paths with $r$ right-steps, $l$ left-steps, and vertical pr…