6 problems

Let $N=(n_1,\dots,n_n)$ be a multinomial vector with parameter $n$ and probabilities $(n^{-1},\dots,n^{-1})$. For each $C>0$, let $c>0$ be a constant independent of $n$ such that t…

Let $(X_1,X_2,\ldots)$ be an infinite exchangeable sequence, and let $K_n$ be the number of distinct values among its first $n$ terms. For $m\in\{1,2,\ldots,\infty\}$, let…

Let $(w_j)_{j\geq 1}$ be the decreasingly ordered frequencies of the extended geometric stick-breaking process with integer scale parameter $s\geq 2$, and let $K_n$ denote the numb…

Average-bin monotonicity conjecture. For fixed $k$, the average number of bins increases as the number of balls $n$ increases.

Maximum and minimum occupancy conjecture. The actual numbers of balls in the maximum-loaded and minimum-loaded urns can be approximately found using the upper and lower bounds, res…

Sharpness conjecture. Theorem is likely to be sharp when the first terms of the sequence $(\mathbb{E}K_r(n))_{r\geq 2}$ grow at the same rate as $\mathbb{E}K(n)$, or at least as…