12 problems

There are $N$ coupon types, with draws made independently according to a probability vector $\mathbf{p}=(p_1,\ldots,p_N)$ in the open simplex … and let $T_N$ be the first time ever…

Let $P_{\mathcal{M}}$ be a stationary distribution with entropy $H$, let $H_N$ denote the LAWS hit rate after $N$ total queries, and let $N_{\min}$ be the per-node visit threshold.…

For a coupon collector model with $N$ coupon types and positive probability vector $p=(p_1,\ldots,p_N)$, let $U_j^N$ be the number of coupon types that have appeared fewer than $j$…

Grunbaum–Yaakobi conjecture. Among all fair mechanisms on fixed parameters $(n,ell)$, the full model maximizes the expected coverage time.

Finite variance extremality. The equal-probability vector $\nu$ minimizes

Let $Q_I(n,k)$ and $Q_{II}(n,k)$ denote the numbers of groups required to determine all coupon labels when the label set is known in advance and when it is not, respectively. For…

Let $n$ be the number of distinct coupons and let $s$ satisfy $2\leq s\leq n-1$. In each round, a package consisting of a set of $s$ coupons is drawn, with an arbitrary distributio…

Let the universe consist of $n$ coupons, and let $H_t$ be an $r$-subset drawn at round $t$. Starting from $H_t$, form $H_{t+1}$ by choosing a uniformly random coupon in $H_t$ and r…

Consider a sequence of coupon collections indexed by $N$, with $U_j^N$ denoting the number of unfilled coupons in a brother's album after the main collector has a complete set. A s…

Fix positive integers $m$ and $N$. Consider the coupon-collection problem in which there are $N$ coupon types, each draw yields one type according to a probability distribution, an…

Let $T_m(N)$ be the number of draws needed to collect at least $m$ copies of each of $N$ coupon types, and let $V[\,T_m(N)\,]$ denote its variance. In the equal-probability case, e…

Let $n\geq 1$, let $c\in\{1,\ldots,n\}$, and let $p=(p_1,\ldots,p_n)\in(0,1)^n$ satisfy $p_1+\cdots+p_n\leq 1$. Write … Here $T_{c,n}(p)$ denotes the time needed in the generalized…