If anybody can suggest where I can find a literature for a known upper and lower bounds on Fubini numbers https://en.wikipedia.org/wiki/Ordered_Bell_number
2026-03-25 11:14:46.1774437286
Literature on bounds of Fubini's numbers
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QING ZOU, "THE LOG-CONVEXITY OF THE FUBINI NUMBERS", http://toc.ui.ac.ir/article_21835_684378fec55e5c66c7fccd4321a84637.pdf
gives the bounds on $f_n$, the $n^{\text{th}}$ Fubini number:
$$ 2^n < f_n < \frac{n!}{(\ln 2)^{n+1}} < (n+1)^n \text{.} $$ The lower bound holds for $n \geq 3$ and the upper for $n \geq 1$.
(Zou cites Barthelemy, "AN ASYMPTOTIC EQUIVALENT FOR THE NUMBER OF TOTAL PREORDERS ON A FINITE SET ", from which we determine the intended base of the logarithm.)