18 problems

Let ${\mathcal Y}$ be a step set for walks in the quarter plane, and let $Q_{\mathcal Y}$ denote its generating function. Assume that the step set group ${\mathcal Y}$ is not singu…

Let $\mathfrak S\subset\mathbb Z^2$ be a finite set of steps. For walks on the slit plane starting at the origin, define the generating function counted by length and final coordin…

Let $G(x,y,t)$ be the full generating function for Gessel walks with step set $\\{\nearrow,\swarrow,\leftarrow,\rightarrow\\}$. Gessel's conjecture. The generating function…

Let $\mathcal Q$ be the class of two-step rules, and let $\mathcal N_\ell$ denote the set of rules in $\mathcal Q$ whose symmetry group has order $\ell$, with $\ell=\infty$ allowed…

Let $\mathcal Q$ be the class of two-step rules, and let $G_e,G_n,G_w,G_s$ denote the corresponding groups for the four directions. Directional group-order conjecture. For every tw…

Let $P_m^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}$ be the number of quarter-plane spiral walks of length $m$. Let…

Let $G(t;a,b)$ denote the generating function for the corresponding model, with parameters $t$, $a$, and $b$. Models 19 and 20 are the Kreweras and reverse Kreweras walks, respecti…

Let $G(t;a,b)$ denote the generating function for the corresponding model, with parameters $t$, $a$, and $b$. A formal power series is D-finite if it satisfies a linear differentia…

Let $G(t;a,b)$ denote the generating function for the corresponding model, with parameters $t$, $a$, and $b$. A formal power series is D-finite if it satisfies a linear differentia…

The parametric algebraic-form conjecture.

Consider the model … Let $Q(0,0;t^{1/2})$ be its excursion generating function after extracting the even subsequence. The hypergeometric closed-form conjecture. … The formula is in…

Let $d=2$ and consider a model whose orbit is finite. A section-free equation is a linear combination of the orbit equations in which the boundary sections do not occur. If the mod…

Let ${\mathcal{S}}$ be one of the 19 small-step lattice-path models with finite non-zero orbit sum, and consider its centrally weighted walk model. Let $Q(1,1;t)$ be the correspond…

The converse characterization of central weightings. If there exist constants $\alpha_1,\dots,\alpha_d,\beta$ such that

Let $Q(a,s,u)$ be the generating function for quarter-plane loops, refined by the variable $s$, and let $Q^{\bullet}(a,s,u)$ be the corresponding generating function counting primi…

Let $Q(a,u)$ be the generating series for quarter-plane loops, and write $Q(a,\mathord\cdot)$ for the resulting univariate series in $u$. For $a\ge -1$, let $\rho_Q(a)$ denote its…

Let $Q(a,u)$ be the generating series for quarter-plane loops. $(a+1)$-positivity conjecture. The series $Q(a,u)$ can be expanded as … where $P_n(x)\in\mathbb{N}[x]$. Thus, the coe…

Square non-zero configuration conjecture. The cardinality satisfies