All Questions
Tagged with bell-numbers stirling-numbers
20
questions
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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
1
answer
108
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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 $k$, ...
- 5
8
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1
answer
311
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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 ...
- 567
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105
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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:...
- 695
6
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3
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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
$$\frac{d^n}{dx^n}\Big[f\left(\sqrt{x+1}\right)\Big]=\sum_{k=...
- 6,692
1
vote
3
answers
503
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How to prove that $B(n) < n!$ for all $n \geq 3$ where $B(n)$ is $n$-th Bell number
When I approached this problem I thought that it can be easily solved by applying induction. However something went completely wrong and I haven’t managed to prove it by induction.
Maybe there is some ...
- 983
1
vote
1
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202
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Constructing a bijection to show that the number of equivalence relations on a finite set is equal to the bell numbers.
It is said that the Bell numbers count the number of partitions of a finite set. How can we prove that what they count is actually the number of partitions? I don't want to take it as a definition; I ...
- 121
2
votes
1
answer
232
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Exponential generating function with Stirling numbers
I want to prove in particular this result-
$$
\newcommand{\gkpSII}[2]{{\genfrac{\lbrace}{\rbrace}{0pt}{}{#1}{#2}}}
\sum_{k \geq 0} \gkpSII{2k}{j} \frac{\log(q)^k}{k!} =
\frac{1}{\sqrt{2\pi}} \...
- 63
1
vote
2
answers
185
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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 ...
4
votes
1
answer
111
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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
292
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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 ...
user715522
1
vote
1
answer
422
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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 $$\sum_{n=k}^{∞ }S\left(...
- 379
3
votes
1
answer
577
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Formula for computing the coefficients of Bell polynomial
I'm working on Bell polynomials and have learned some of its properties, but I've never seen any formula for calculating the coefficient in Bell polynomials. My trying to find these coefficients was ...
- 379
1
vote
1
answer
55
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How can I find $f(a,b,c)=e^{-c^a/a}\sum\limits_{n=0}^{\infty}\left(\frac{c^a}{a}\right)^{n}\frac{(an)^{b}}{n!}$?
Inspired by Dobinski formula, by lucky guess I find, that (for natural $a,b$)
$$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 ...
- 1,475
4
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1
answer
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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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