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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How to show that $\sum_{l=0}^{\infty} \dfrac{(l+1)^{n-1}}{l!}=\sum_{l=0}^{\infty} \dfrac{l^{n}}{l!}$ (proof of Dobiński's formula)?
I am reading a proof of Dobiński's formula in Béla Bollabás book "The Art of Mathematics" (p. 144). There he uses
$$\frac{1}{e}\sum_{l=0}^{\infty} \dfrac{(l+1)^{n-1}}{l!}=\frac{1}{e}\sum_{l=...
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Complete bell polynomial coefficients
I would like to know if it is possible to calculate the coefficient of a given Complete Bell Polynomial 's monomial by its indexes and powers:
$B_{n}(x_1,x_2,...,x_n)= c_n(1,n) x_1^n + c_n((1,n-2),(2,...
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Strange polynomial analog of the Bell numbers
Let $\vec{x} = (x_0, x_1, x_2, \dots)$ and $\vec{y}=(y_1,y_2,y_3, \dots)$ be two systems of parameters/variables. The Motzkin
polynomials $P_n(\vec{x},\vec{y})$ for $n \geq 0$ are defined by the ...
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Sum of Two Arguments in Bell Polynomials of Second Kind [closed]
I understand the complete Bell polynomial $B_n$ satisfies the identity:
$$B_n(x_1+y_1,x_2+y_2,...,x_n+y_n) = \sum_{k=0}^n \left(\matrix{ n \\ k }\right) B_{n-k}(x_1,x_2,..,x_{n-k})\, B_k(y_1,y_2,...,...
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Is there a simple lower bound or approximation for the Bell numbers?
I'm not quite certain what descriptors to use to describe the solution I'm looking for, but is there an approximation or useful lower bound for the Bell numbers, for which the amount of terms used in ...
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Bell Polynomials
The complete Bell polynomials $B_n(x_1, x_2, \ldots, x_n)$ are defined through the relation
$$\sum_{n=0}^{\infty} B_n(x_1, x_2, \ldots, x_n) \frac{t^n}{n!} =\exp\Big( \sum_{n=1}^{\infty} x_n \frac{t^n}...
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Group Action and the Bell Number
I am struggling on solving the inequality related to the group action and Bell numbers.
Let $G$ be a finite group acting on a set $X$ with $m$ elements. Prove that for each $1 \leq r \leq m$, $$\frac{...
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Confusion about a factor in a composition of series/Faa di Bruno formula
In the wikipedia article on the Faà di Bruno formula, one considers the composition of two "exponential/Taylor-like" series $\displaystyle f(x)=\sum_{n=1}^\infty {\frac{a_n}{n!}} x^n\; $ ($...
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Reference request for recurrence relation of the complete Bell polynomials $B_n$
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 $...
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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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Problem about counting partitions
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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Show that $\left| A_n \right| = \sum_{k=1}^{n} (-1)^k \binom{n}{k} B(n - k)$ where $B(n)$ is the nth Bell number.
I am having some trouble solving the following problem:
Let $A_n$ be the set of set partitions of $\{1, . . . , n\}$ without any singleton blocks. Show that $$\left| A_n \right| = \sum_{k=0}^{n} (-1)^...
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prove for bell number using induction on n [duplicate]
hi guys I have to prove this equality $$B_n=e^{-1}\sum_{k=0}^{\infty}\frac{k^n}{k!},$$ that is called bell equality only using induction on $n$ . How can i do this? I have tried by substituting the ...
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Compact formula for the $n$th derivative of $f\left(\sqrt{x+1}\right)$?
I'm interested in a general formula for
$$\frac{d^n}{dx^n}\Big[f\left(\sqrt{x+1}\right)\Big].$$
In particular, Fàa di Bruno's formula gives
$$\frac{d^n}{dx^n}\Big[f\left(\sqrt{x+1}\right)\Big]=\sum_{k=...
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