21 problems

For each integer $k\geqslant 2$, let $\gamma_k$ be the constant in the asymptotic growth … of the maximal ranks of at-most-$k$-furcating rooted trees with $n$ leaves. Monotonicity…

Refined recurrence conjecture. For $n\geq 3$ and $1\leq i\leq n-2$, we have

Let $n\geq 0$, let $T_n$ denote the first $n$ levels of the rooted binary tree, and let $H\leq \operatorname{Aut}(T_n)$ be a group that contains an element acting transitively on…

Let $f$ be the post-critically finite quadratic polynomial under consideration, and let $G_n(f)$ and $M_n(f)$ denote, respectively, its level-$n$ Galois group and even Markov group…

Let $f\in \mathbb{C}[x]$ be a post-critically finite rational function. Write $\mathrm{IMG}(f)$ for its iterated monodromy group, and say that it has $(\mathrm{LR})$ when it has th…

Local-to-global conjugacy conjecture. If $H\leq\operatorname{Aut}(\mathcal{T}_n)$ is locally conjugated into $M$, then $H$ is globally conjugated into $M$.

Left LBS-tree edge-activity conjecture. For $0\leq k\leq n-1$, the number of left LBS trees with $k$ edges of the form $(i+1,i)$ is equal to the number of left LBS trees with $k$ e…

Let $n$ be a positive integer, let $\mathsf{RTT}j(1^{n+1})$ denote the set of standard fully tiered rooted trees on $n+1$ vertices whose root is labelled $j$, and let…

Let $u$ be a composition, let $u( u)$ denote the partition obtained by rearranging its parts, and let $\mathsf{RTT}(nu)$ be the set of fully tiered rooted trees associated with…

Let $T$ be a rooted tree of order $n$, and let $\rho(T)$ denote its Perron value and $\mu(T)$ its moment. Logarithmic bound conjecture. There exists $\alpha_0\in\mathbb{R}$ such th…

Let $(T,R)$ be a balanced rooted tree, let $T^p$ be the graph formed from the union of $p$ distinct labelled copies of $T$ that agree on the roots $R$ and are otherwise disjoint, a…

Logarithm–rooted-tree degree conjecture. The degree of the rooted tree associated with a prime $p$ is asymptotic to $\ln(p)$. This is proposed as a natural measure on the linearize…

Let $\lambda=(\lambda_1,\ldots,\lambda_m)$ be a partition, let $T_\lambda$ be the rooted tree with $n+1$ vertices defined by … let $k\geq 3$ be an odd integer, and let $\Phi_d$ den…

Let $T$ be a rooted tree, let $\sympawn_T$ be its coefficient, and let $\square_T$ be the Newton polygon of the numerator of $\sympawn_T$. Write $\#T$ for the number of vertices of…

Let $\mathtt{Crl}_n$ denote the corolla with $n$ leaves, let $\sympawn_{\mathtt{Crl}_n}$ be its coefficient, and let $\Phi_d$ denote the $d$th cyclotomic polynomial. The denominato…

Let $\mathscr{Y}=\mathscr{E}\circ\mathscr{D}^{-1}$ be the quotient series in the group $\mathsf{G}_{\operatorname{PreLie}}$, let $\mathscr{Y}_T$ be the coefficient of a rooted tree…

Let $\mathtt{Frk}_{i,n-i}$ be the fork with $n$ vertices and stem of size $i$, and let $\#\mathbb{F}(\mathtt{Frk}_{i,n-i},k,0)$ denote the number of closed flows of size $k$ on thi…

Let $T$ be a rooted tree, let $v$ be a vertex of $T$, let $u$ be its parent, let $w$ be a rooted tree grafted at $v$, and let $S$ and $S'$ be the rooted trees defined by removing e…

A rooted tree $(T,r)$ is a tree $T$ with a distinguished vertex $r$. Two rooted trees $(T,r)$ and $(T',r')$ are twins, or mutually embeddable, when there are injective graph homomo…

Freeness conjecture. The $\fontfamily{U}\fontshape{n}\selectfont{As}^2$-subalgebra of $NCK$ generated by elements of degree $1$ is a free algebra over the operad of two compatible…

Compatibility conjecture. For any rooted tree $t$,