53 problems

Let $G(x,y,t)$ be the full generating function for Gessel walks with step set $\\{\nearrow,\swarrow,\leftarrow,\rightarrow\\}$, and let $G(0,0,t)$ be the generating function of Ges…

Let $Q(x,y,t)$ be the generating series of a lattice walk in the quarter plane with small step set, and associate to the walk the group of birational automorphisms of…

Let a finite group model be a lattice-walk model with finite orbit group, and let $b$ be an admissible integer starting-coordinate parameter. Consider three-quadrant walks starting…

Non-algebraicity conjecture. This latter generating series is not algebraic.

The parametric algebraic-form conjecture.

Let $a$ and $b$ be positive integers with $\gcd(a,b)=1$, and let … Let $f$ denote the generating function for excursions, meanders, or bridges with step set $\mathcal{S}$. Minimal-…

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 $V(y)$ be a Laurent polynomial with positive coefficients, highest exponent $\operatorname{max_v}$, and lowest exponent $-\operatorname{min_v}$. Let $\tau$ be the unique positi…

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…

Roques's conjecture. For walks of length $2n+1$, the number of walks is

Roques's conjecture. The number of such walks of length $2n+1$ is

Consider a quarter-plane lattice-walk problem and its orbit sum. Zero-orbit-sum algebraicity conjecture. If the orbit sum is $0$, then the solution is algebraic. Zero orbit sum is…

For $n\ge 2$, define the large-step model $\mathcal{H}_n$ by the Laurent polynomial … For $a,b\ge 1$, consider the orbit of $\mathcal{H}_n$ and whether the monomial fraction…

For a weighted or unweighted quadrant walk model, let its generating function be algebraic in $X$ and $Y$, let the model admit non-trivial Galois invariants, and let $XY$ have a Ga…

Melczer–Wilson conjecture. Zero-drift mostly symmetric models have asymptotic growth of the form $O(S({\mathbf{1}})^n n^{-(d+1)/2})$. This conjecture was stated incorrectly: the pa…

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…

Consider walks in the three-quadrant cone starting at $(p,0)$ with $p>0$ (equivalently, at $(0,q)$), and restrict to models with a finite group. Trotignon–Raschel conjecture. The g…

For a non-singular step set $S$, let ${\sf C}(x,y;t)$ be the generating function for walks in the three-quadrant cone and let ${\sf Q}(x,y;t)$ be the corresponding quarter-plane ge…

Let $S$ be one of the $74$ non-singular step sets for walks in the three-quadrant cone, and let ${\sf C}(x,y;t)$ be its generating function. Let ${\sf Q}(x,y;t)$ denote the corresp…

The associated excursion sequence of a quadrant walk model may have an asymptotic expansion with some initial exponents rational and a later exponent irrational. Non-occurrence con…

Let $\mathcal Q$ be the class of 6909 non-trivial quarter-plane rules. Let $\mathcal E_{\mathrm{alg}}$, $\mathcal E_{\mathrm{Df}}$ and $\mathcal E_{\mathrm{nonDf}}$ be the sets of…

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…

A small-step model is a step set $\mathcal S\subset\{-1,0,1\}^2\setminus\{(0,0)\}$, considered up to isomorphism, with walks restricted to the quarter plane. Bousquet-Mélou and Mis…