# 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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### On deriving a 'simple' formula for the taylor series of $\exp^{f(x_1,x_2)}$

It is written explicitly in wikipedia, https://en.wikipedia.org/wiki/Bell_polynomials#Generating_function, how one obtains a simple analytic expression for the Taylor series of the exponential of a ...
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### Given Bell numbers as moments, derive the Poisson distribution

The Poisson distribution (with $\lambda=1$) has probability mass function $\frac{e^{-1}}{k!}$ where $k\in\{0,1,2,\cdots\}$. Its moments are the Bell numbers $B_n$, which count the possible partitions ...
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### Periodicity of Bell numbers modulo $n$

After doing some numerical simulations, I rediscovered that the Bell numbers are periodic modulo $n$, that is to say we have the following identities : \begin{align} B_{n+3} &= B_n\mod{2} \\\\ ...
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### Stirling Numbers Exponential Generating Function Induction

I was reading the solution to a question written here, and it uses a fact which can be proved by induction. The question is to show that an EGF for Stirling Numbers of the second kind with fixed $k$, ...
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### Closed Form for Geometric-like Finite sum of Bell Polynomials

I'm trying to see if there's a nice closed form expression for the following sum: $\sum_{k=0}^{M} \cos(\pi k t) B_k(x)$ where $M \in \mathbb{N}$, $t \in (0,1)$, and $x \in \mathbb{R}^+$. Notation: ...
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### Calculating factorization for large numbers

My mission is to calculate the factorization of large numbers, for example, from $start=1e11$ to $end=1e12$. To do that, one approach that I was thinking of is to calculate for each number his ...
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On this wikipedia page there is the following recurrence relation for the complete Bell polynomials $B_n$: $$B_{n+1}(x_1,...,x_{n+1})=\sum_{i=0}^n\binom{n}{i}B_{n-i}(x_1,...,x_{n-i})x_{i+1}$$ with $... • 570 1 vote 1 answer 82 views ### Bell number, equivalent. The$n$-th Bell number$B_n$can be defined by$\displaystyle e^{e^x-1}=\sum_{n=0}^{+\infty}\frac{B_n}{n!}x^n$or$\displaystyle B_n=\frac 1e\sum_{k=0}^{+\infty}\frac{k^n}{k!}$or$\displaystyle B_{n+...
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The number $f(n)=B(n)-\sum_{k=1}^n B(n-k) {n \choose k}$ counts certain partitions of $[n]$.Which partitions? From Wiki, The Bell numbers satisfy a recurrence relation involving binomial coefficients:...
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### Bell numbers and a card shuffle

A deck of n cards may be 'shuffled' by moving the top card to any (random) position in the deck, and performing this operation n times. Martin Gardner asserts (Scientific American, May 1978) that the ...
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### How many different equivalence relations with exactly two different equivalence classes are there on a set with $n$ elements

I came across with this topic. It looks straight forward for $5$ elements, but what if I want to find how many different equivalence relations with exactly two different equivalence classes are there ...
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### $F(n)$ is number of ways to partition set of $n$ without singleton blocks. Prove that $B(n) = F(n) + F(n+1)$

In this case $B(n)$ is $n$-th Bell number. To be honest, I would really love to know if there is a combinatorial proof for that. If there is not, other proofs are appreciated too.
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### How to prove that $B(n) < n!$ for all $n \geq 3$ where $B(n)$ is $n$-th Bell number

When I approached this problem I thought that it can be easily solved by applying induction. However something went completely wrong and I haven’t managed to prove it by induction. Maybe there is some ...
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### Constructing a bijection to show that the number of equivalence relations on a finite set is equal to the bell numbers.

It is said that the Bell numbers count the number of partitions of a finite set. How can we prove that what they count is actually the number of partitions? I don't want to take it as a definition; I ...
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### 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) ...
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### 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
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### 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 ...
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### Proof Bell-Number $B(n)=\sum_{k=0}^{n-1}\binom{n-1}{k}B(k)$

Let's say that $B(n)$ (the Bell number) is the number of ways to split $\{1, \ldots ,n\}$ in non-empty blocks. Prove that, for $n \geq 1$: $$B(n)=\sum_{k=0}^{n-1}\binom{n-1}{k}B(k)$$ I don't really ...
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### An approximation of the ordered Bell numbers

So my problem is the following: I have $n$ ice-cream flavors and I must rank them, allowing that I can place more than one flavor in some ranks. So for example if I have 4 flavors, I can put in the ...
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### Interesting ways to show that there are infinitely many equivalence relations on an infinite set (including Bell numbers).

I am trying to answer the question "Is there infinitely many equivalence relations on any infinite set?" My intuition says yes, and when I try to prove this, I feel like my reasoning is not ...
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I am trying to prove that an upper bound for the nth Bell number is n factorial. I am trying to do this by induction. Firstly, the nth Bell number is given by: $B_{n}=\sum\limits^{n-1}_{k=0} B_{k}{n-... • 131 1 vote 1 answer 161 views ### Prove$B_n\le n! $for Bell numbers How using induction it can be shown that: $$B_n\le n! \;\;\;\;\;\;\;\;\;\left( n\in \mathbb N \right)$$ Where$B_n$is the nth Bell number. The base case is true, since $$1=B_0\le 0!=1 \;\;\;\;\;... 4 votes 1 answer 83 views ### On the ratio \frac{F_n}{B_n} One of the interesting limits that I came up with is:$$\lim_{n\to\infty} \frac{F_{n}}{B_{n}}\;\;\;\;\;\;\;\;\;\; \left( n \in \mathbb N^+\right)$$Where F_n is the nth Fibonacci number and B_n... 4 votes 1 answer 108 views ### Dividing 12 people into any number of groups, such that person A and B are not in the same group? In how many ways can you divide 12 people into any number of groups, such that person A and B are not in the same group? I am trying to solve this question and so far I am thinking of this in terms ... • 365 3 votes 1 answer 274 views ### Combinatorial proof for Touchard's congruence Bell number denoted B_n is the number of ways to partition a set with cardinality n into k indistinguishable sets , where 0\le k\le n It's known that Bell numbers obey Touchard's Congruence ... 2 votes 0 answers 107 views ### Number of preorder relations on a set related to the open problem about preorder relations Consider a set A=\left\{1,2,3\right\},I want to count the number of preorder relations on this set, so there is two cases two consider,either the relation is symmetric or it is not, if the relation ... 2 votes 1 answer 106 views ### New wrong recurrence formula for Bell numbers Bell numbers are the numbers counting the total partitions on a set with n distinct elements. Explanation: Consider a set like A:=\left\{x_{1},x_{2},...,x_{n}\right\} A partial equivalence ... 1 vote 1 answer 363 views ### generating function for Bell polynomial How it can be proved that :$$\sum_{n=0}^{ ∞}B_{n}\left(x\right)\frac{t^{n}}{n!}=e^{x\left(e^{t}-1\right)}$$Where B_n is the n^{th} complete Bell polynomial. I know that$$\sum_{n=k}^{∞ }S\left(... • 369 3 votes 1 answer 541 views ### Formula for computing the coefficients of Bell polynomial I'm working on Bell polynomials and have learned some of its properties, but I've never seen any formula for calculating the coefficient in Bell polynomials. My trying to find these coefficients was ... • 369 2 votes 0 answers 66 views ###$\left( k, n \right)$symmetry? As background: there are$k!$strict orders on the set$K = \left\{ 1, \ldots k \right\}$. Under symmetry, each order occurs with probability$\tfrac{1}{k!}$. Now consider the set of weak orders on$...
I know the generating function for the total number of partitions of [n] is given by $$B(x)=e^{e^x-1}$$ I am struggling to find $V(X)$, the exponential generating function for the number of ...