15 problems

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…

Christol's conjecture. Every D-finite globally bounded function is the diagonal of a rational function.

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…

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…

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…

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

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 $P_{1423}(0,z)$ 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…

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…

Let $C_n(r)$ denote the number of colored set partitions of $[n]$ with colors from a set of size $r$. A formal power series is D-finite if it satisfies a linear differential equati…

Let $\mathcal{L}$ be a regular language, and let its shuffle closure be the union of all finite shuffles of $\mathcal{L}$: … If $L(z)$ is the ordinary generating function of this l…