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Questions tagged [bell-numbers]

For questions related to the Bell numbers, a sequence of natural numbers that occur in partitioning a finite set.

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What is the type of combinatoric function is used to define partitions of partitions of partitions etc.? [closed]

I am interested in recurring or nested bell number sequences and polynomials that changes term by term. I am sure there is an overarching pattern that can be establish this more concisely wherein bell ...
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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 ...
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Bell number modulo prime power

I'd like to ask how to fastly calculate the Bell number $B_n$ modulo a prime power, where $n$ is around one million.
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Number of cycle partition of a set with repeating elements

We have a set $S$ with $E$ elements of which only $N$ are unique. We of course know how many repetitions of each of the $N$ elements are present: element $s_i$ is repeating $t_i$ times. I would like ...
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On the sign of the complete exponential Bell polynomials

Can somebody help me to show that for each positive integer m, there exists a sufficiently large r such that: $$ Y_{2r}(-m x_1,..., -m x_{2r})=\sum_{j=1}^{2r}(-1)^jm^jB_{2r,j}(x_1,\dots, x_{2r-j+1})&...
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Exponential generating function for the Bell numbers

I've recently come across the Bell numbers, defined as: \begin{equation*} B_{n} = \sum_{k=0}^{n}\binom{n}{k}B_{k}. \end{equation*} The exponential generating function of the Bell numbers is known to ...
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An $n$ element set can be written as a union of disjoint sets, in $B_n$ different ways. But what if the sets are not disjoint?

If we let $X=\{1,2,3,\ldots n\}$ then what is the value of $\left|\{\mathcal{F}\subseteq\wp(X):\bigcup\mathcal{F}=X\}\right|?$ I know that if one requires the sets in $\mathcal{F}$ be pairwise ...
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How many partitions are there of a 5 element set into 3 parts (of a specific form)?

I am currently trying to understand the number of ways of partitioning a 5 element set into 3 parts. However, I am only interested in partitions with the form $$ \left\{ \{ a, b \}, \{c, d\}, \{e \} \...
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How can I find $f(a,b,c)=e^{-c^a/a}\sum\limits_{n=0}^{\infty}\left(\frac{c^a}{a}\right)^{n}\frac{(an)^{b}}{n!}$?

Inspired by Dobinski formula, by lucky guess I find, that (for natural $a,b$) $$f(a,b)=e^{-1/a}\sum\limits_{n=0}^{\infty}\frac{(an)^{b}}{a^{n}n!}=\sum\limits_{k=1}^{b}{b\brace k}a^{b-k}$$ but I have ...
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Complicated recursion formula, seems similar to Bell numbers?

I came up with a recursive formula for a problem I was working on. It is as follows. $$a_n = \Big(\frac{1-q^{f \cdot n}}{1-q^n}\Big)\displaystyle\Big(1+\sum_{i=0}^{n-1}\binom{n}{i}p^{n-i}q^ia_i\Big)$$...
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Taylor series of a power tower

I recently proved that the Taylor Series of $\exp(\exp(x))$ is given by $$\exp(\exp(x))=\sum_{n=0}^\infty \frac{eB_n x^n}{n!}$$ where $B_n$ are the Bell Numbers. However, I can't figure out a Taylor ...
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Probability of set partition

Let $A = \{1\dots n\}$. Do partition of set $A$ on pairwise disjoint two- and three-element subsets randomly. For each n determine probability that number of two-element subsets is equal to three-...
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Exponential Generating Function Stirling Numbers

In class we found the exponential generating function for the Bell numbers $B_n$ which are defined by the recurrence $B(0) = 1$, $B(1) = 1$ and $B(n+1) =\sum_{i=1}^n\dbinom{n}{i}B(n-i)$ for all$ n\geq ...
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Sum of product of Bell number and Stirling number of the first kind

Prove that $$\sum_{k=0}^n s(n,k)*B(k) = 1$$ where s(n,k) - Stirling number of the first kind; B(k) - Bell number. I've tried to use $$B(k)=\sum_{j=0}^kS(k,j)$$ where S(n,j) - Stirling numbers of the ...
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Counting partitions of a finite set excluding singletons

I am not a combinatorialist by training, but I need insight on the following question for a current project. I cannot find this as a duplicate here. Let $[n]$ be the set $\{1, 2, \ldots, n\}$. I ...
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Extending Bell Numbers to Fractional Values

An identity of the Bell numbers is given by $$B_n=\frac{1}{e}\sum_{x=1}^\infty \frac{x^n}{x!}$$ and I was wondering if it would be valid to define fractional Bell numbers in the same way, to preserve ...
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Compute $S_n=\sum\limits_{a_1,a_2,\cdots,a_n=1}^\infty \frac{a_1a_2\cdots a_n}{(a_1+a_2+\cdots+a_n)!}$

It is tagged as an open problem in the book Fractional parts,series and integrals. If this proof is valid , I don't have any idea how to get it published so I posted it here . $\displaystyle \sum_{...
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Catalan Numbers vs Bell Numbers

I did a lot of research on Catalan Numbers and I came across one interesting fact that the nth Catalan numbers never exceeds the nth Bell number. I know that the nth bell numbers counts the number of ...