For $n\ge0$, let
where $S(n,k)$, which can be generated by the exponential function
and can be computed by the explicit formula
denotes the Stirling numbers of the second kind.
One calls these numbers $F_n$ the Fubini numbers in the paper  below, ordered Bell numbers in the paper , or geometric numbers in the paper  below.
A Fubini number $F_n$ has been interpreted in the papers [2, 3] combinatorially: it counts all the possible set partitions of an $n$ element set such that the order of the blocks matters. In the paper , the Fubini numbers $F_n$ were connected with preference arrangements and the recursion for $F_n$ was derived. In the papers [2, 4] below, the exponential generating function
and an asymptotic estimate for $F_n$ were established.
- M. E. Dasef and S. M. Kautz, Some sums of some importance, College Math. J. 28 (1997), 52--55.
- O. A. Gross, Preferential arrangements, Amer. Math. Monthly 69 (1962), 4--8; available online at https://doi.org/10.2307/2312725.
- S. M. Tanny, On some numbers related to the Bell numbers, Canad. Math. Bull. 17 (1974/75), no. 5, 733--738; available online at https://doi.org/10.4153/CMB-1974-132-8.
- R. D. James, The factors of a square-free integer, Canad. Math. Bull. 11 (1968), 733--735; available online at https://doi.org/10.4153/CMB-1968-089-7.
- F. Qi, Determinantal expressions and recurrence relations for Fubini and Eulerian polynomials, Journal of Interdisciplinary Mathematics 22 (2019), no. 3, 317--335; available online at https://doi.org/10.1080/09720502.2019.1624063.