Questions tagged [bell-numbers]
For questions related to the Bell numbers, a sequence of natural numbers that occur in partitioning a finite set.
86
questions
6
votes
3
answers
148
views
The number of partitions of $\{1, \ldots, n+1\}$ into subsets of nonconsecutive integers
For each integer $n$, let $a_n$ be the number of partitions of $\{1, \ldots, n+1\}$ into subsets of nonconsecutive integers.
I found (by listing) that $ a_1, a_2, a_3, a_4$ are $1, 2, 5, 15$ ...
- 721
2
votes
0
answers
24
views
Bell numbers - Cardinality of odd number of parts in partitions of the finite set $[n]$.
As it well known, Bell numbers denoted $B_{n}$ counts distinct partitions of the finite set $[n]$. So for example if $n=3$ there are 5 ways to the set $\left\{ a,b,c\right\}$ can be partitioned:
$$\...
- 119
2
votes
1
answer
68
views
Closed-form expression for the infinite sum in Dobiński's formula
In combinatorial mathematics, I learned about Dobiński's formula for the $n$-th Bell number $B_n$, which states that:
Dobiński's formula gives the $n$-th Bell number $B_n$ (i.e., the number of ...
- 2,288
0
votes
0
answers
57
views
number of possible arrangements of n coins
Had this question in a programming class and was meant to be solved using a recursive algorithm. But I was wondering if there was a combinatorics solution.
I tried counting the number of possible ...
- 342
0
votes
0
answers
77
views
number of "equivalence relations" on a set with "n-elements"
I am trying to find a formula for number of equivalence relations on a set with n-elements however I am confused.
I have already encountered the idea of "bell's number" and "Stirling ...
1
vote
0
answers
34
views
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 ...
- 31
3
votes
2
answers
83
views
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 ...
1
vote
0
answers
47
views
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} \\\\
...
- 888
0
votes
1
answer
105
views
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$, ...
- 5
1
vote
1
answer
84
views
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: ...
- 317
0
votes
1
answer
70
views
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 ...
- 119
2
votes
1
answer
175
views
Integral representation of Bell Polynomial?
From Wolfram Alpha: https://functions.wolfram.com/IntegerFunctions/BellB/07/01/0001/,
we have an integral representation for Bell numbers as:
$B_n = \frac{2n!}{\pi e} \displaystyle{\int_{0}^{\pi}} e^{...
- 317
2
votes
1
answer
306
views
Extension of the Multivariate Faa di bruno's formula with more than two composite functions
The Faa di bruno formula for one variable (Wikipedia) is
The combinatorial forms in terms of bell polynomials are also included
Similarly, the multivariate formula (Wikipedia) is expressed ...
- 89
2
votes
1
answer
98
views
Closed form for $H(n,k)=\frac{1}{k!}\sum_{j=0}^\infty H(n-k, j)j^k$ where $H(1,k)=\frac{1}{k!}$
Let $H(n,k)$ be defined such that
$$H(1,k)=\frac{1}{k!}\text{, and }H(n,k)=\frac{1}{k!}\sum_{j=0}^\infty H(n-1,j)j^k$$
As pointed out in the comments, I should mention that we must define $0^0=1$ as ...
- 4,472
0
votes
0
answers
64
views
$Li(x)$ function and Bell polynomials
I found one formula connecting the logarithmic integral function, $li(x)$, to polynomials as following:
$$li(x) = \frac{x}{\ln(x)} + \sum_{k=1}^{+\infty} \frac{P_k(\ln(x))}{x^k}$$
where $P_k(x)$ is ...
- 87