30 problems

Consider the seven Weyl models in Table … . Weyl-model algebraicity conjecture. For every one of these seven models, $A(x,y)$ is algebraic; consequently, $C(x,y)$ is D-finite. In a…

For a small-step quarter-plane model with stepset $S$, let $Q(x,y;t)$ be its generating function, and let $Q(0,1;t)$ and $Q(1,0;t)$ denote its two boundary specializations. The mod…

A small-step quarter-plane lattice model is determined by a stepset $S\subseteq\{-1,0,1\}^2\setminus\{(0,0)\}$, and its counting generating function is … where the sum ranges over…

Let $fin \mathbb{Q}[[z]]$ be a globally bounded and D-finite power series, and let $L^{\mathrm{min}}_f$ denote its minimal annihilating differential operator. Christol–André conjec…

The paper considers D-finite power series and the classes of broad and strict G-functions. Siegel's conjecture. The classes of broad G-functions and strict G-functions should coinc…

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 $y(x)=\sum_{k\geq 0}a_kx^k\in\mathbb{Q}[[x]]$ be D-finite. Assume that the sequence $(a_k)_{k\geq 0}$ has at most geometric growth, that there exists $N\in\mathbb{N}$ such that…

Let $C(1,1;\lambda)$ be the lambda-extension of the square Ising-model two-point correlation function, and let $G_n(t)$ denote the form-factors arising from a deformation around it…

Consider the $317$ symmetry classes of permutations avoiding three patterns of length $4$. For each class, the paper reports finding a specification; the number of catalytic variab…

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…

Non-D-finiteness conjecture. The generating function for $\mathcal{A}v(A_{5,1})$ is not D-finite.

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…

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…

Let $u_n$ denote the popularity of the pattern $11$ in the set of $n$-letter, $n$-ary faro words, equivalently the diagonal sequence extracted from $G_{11}(x,y)$. Let…

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…

Let $\operatorname{Av}(4123,4231)$, $\operatorname{Av}(4123,4312)$, and $\operatorname{Av}(4231,4321)$ be the three permutation classes whose generating functions are not known. A…

A weighted lattice path model with short steps in the quarter plane is said to have a weighted D-finite generating function when its generating function is D-finite in the relevant…

Let the singular models be the $43$ models for which the algorithmic irrationality proof does not apply, and let the generating function of a model count walks confined to the non-…

Let $P_{1423}(0,z)$ be the generating function for permutations avoiding the consecutive pattern $1423$, and write … A formal power series is D-finite if it satisfies a linear diff…

For a consecutive pattern $\tau$, let $P_{\tau}(0,z)$ be its avoidance generating function and let $\omega_{\tau}(0,z)=1/P_{\tau}(0,z)$ be its reciprocal. A formal power series is…

For a classical pattern $ath]σ$ of permutations, let $a_n(σ)$ denote the number of permutations of length $n$ avoiding $ath]σ$, and let … Here a classical pattern is one whose occu…

Let $S(x)$ be the generating function defined by … A generating function is D-finite if it satisfies a linear differential equation with polynomial coefficients. The non-D-finitene…

Let $\mathrm{BS}(N,M)$ be a Baumslag–Solitar group, and let $G(z;q)$ be the generating function for trivial-word enumeration considered in the paper. Write $[q^0]G(z;q)$ for its co…

Let $\mathrm{BS}(1,M)$ be the Baumslag–Solitar group with $M>1$. Let $G(z;q)$ be the bivariate generating function for the trivial-word enumeration considered in the paper, and let…