All Questions
Tagged with bell-numbers number-theory
7
questions
0
votes
0
answers
64
views
$Li(x)$ function and Bell polynomials
I found one formula connecting the logarithmic integral function, $li(x)$, to polynomials as following:
$$li(x) = \frac{x}{\ln(x)} + \sum_{k=1}^{+\infty} \frac{P_k(\ln(x))}{x^k}$$
where $P_k(x)$ is ...
- 87
1
vote
1
answer
235
views
Number of unordered factorizations of a non-square-free positive integer
I recently discovered that the number of multiplicative partitions of some integer $n$ with $i$ prime factors is given by the Bell number $B_i$, provided that $n$ is a square-free integer. So, is ...
- 1,611
3
votes
1
answer
78
views
representation through special numbers
Let $n,r\in N$ and let $S(n,m)$ represent Stirling's number of the second kind. It is known that $\sum_{m=0}^n S(n,m)m!=F_n$ is a Fubini number. Is it possible to represent (or estimate from above) ...
- 311
0
votes
1
answer
85
views
Literature on bounds of Fubini's numbers
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
- 301
1
vote
0
answers
78
views
Is the number of sub-boolean algebra of a set with size n , Bell(n)?
In boolean algebra (P(S),+,.,’) we must have S as 1 and {} as 0 in every possible sub-boolean algebra to hold id elements.
We must have S-x for every subset x⊆S to hold complements.
It seems like ...
- 313
3
votes
2
answers
417
views
Calculating large Bell number modulo a composite number
I have been trying to solve http://www.javaist.com/rosecode/problem-511-Bell-Numbers-Modulo-Factorial-askyear-2018
It is not an ongoing contest problem.
We can calculate $n$th Bell number modulo a ...
- 547
2
votes
1
answer
532
views
Exponential generating function for the Bell numbers
I've recently come across the Bell numbers, defined as:
\begin{equation*}
B_{n+1} = \sum_{k=0}^{n}\binom{n}{k}B_{k}.
\end{equation*}
The exponential generating function of the Bell numbers is known to ...
- 143