Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb E[f(x)]$. Define $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write $w(P)$ for the largest size of a pairwise incomparable subset. We prove three results. The first proves an old conjectural relation between width and expected-rank gaps that has appeared repeatedly, in increasingly general forms, in work of Brightwell and Trotter (2002), Biró and Trotter (2011), and Aires and Kahn (2025): $\operatorname{gap}(P)\le 2w(P)-1$. Second, for every $L>0$ we construct a width-two poset such that every maximal chain has an expected-rank gap of at least $L$, where the two endpoint spacings are included when computing this gap. Finally, for every $r\in\mathbb N$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.

翻译：暂无翻译