10 problems

Let ${\mathrm{w}}=(w_k,k \ge 0)$ be a weight sequence with $w_0>0$ and with $w_k>0$ for some $k \ge 2$. For $n \ge 0$ with $Z_n({\mathrm{w}})>0$, let ${\mathcal{T}}_n$ be a simply…

Unlabelled simply generated tree conjecture. Results similar to those established in the paper for rooted and unrooted simply generated trees should also hold for unlabelled simply…

Small-component conjecture.

Simply generated forest conjecture. Lemma should also hold when the generating function has radius of convergence $0$.

Let $\rho_\Phi$ denote the radius of convergence of the generating function associated with the simply generated tree model. The trees and quantities $\mathcal{T}_n^{(1)}$,…

Let ${\mathcal T}_n$ be the conditioned simply generated tree, and let $H({\mathcal T}_n)$ denote its height. Assume that the condensation parameter satisfies $\nu=1$ and that the…

Let ${\mathcal T}_n$ be the conditioned simply generated tree, let $\widehat{{\mathcal T}}$ be its limiting size-biased tree, and let $l_k(T)$ denote the number of vertices at dist…

Let $Y_{(1)}$ be the largest box size in the random allocation, and suppose $0<\nu<\lambda$. Nonvanishing maximum-size subsequence conjecture. There exists $\varepsilon>0$ and a su…

Let $Y_{(j)}$ be the $j$th largest box size in the conditioned allocation, and let $\xi_{(j)}$ be the $j$th largest observation in the corresponding independent reference sample. W…

Let $\mathbf{w}=(w_k)_{k\geqslant0}$ be a weight sequence with radius parameter $\rho>0$. Let $Y_{(1)}$ denote the largest box size in the associated random allocation, with…