All Questions
Tagged with bell-numbers combinatorics
51
questions
5
votes
3
answers
158
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 $1, 2, 5, 15$ ...
- 817
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:
$$\...
- 119
0
votes
0
answers
57
views
number of possible arrangements of n coins
Had this question in a programming class and was meant to be solved using a recursive algorithm. But I was wondering if there was a combinatorics solution.
I tried counting the number of possible ...
- 342
0
votes
0
answers
84
views
number of "equivalence relations" on a set with "n-elements"
I am trying to find a formula for number of equivalence relations on a set with n-elements however I am confused.
I have already encountered the idea of "bell's number" and "Stirling ...
1
vote
0
answers
34
views
On deriving a 'simple' formula for the taylor series of $\exp^{f(x_1,x_2)}$
It is written explicitly in wikipedia, https://en.wikipedia.org/wiki/Bell_polynomials#Generating_function, how one obtains a simple analytic expression for the Taylor series of the exponential of a ...
- 31
2
votes
1
answer
313
views
Extension of the Multivariate Faa di bruno's formula with more than two composite functions
The Faa di bruno formula for one variable (Wikipedia) is
The combinatorial forms in terms of bell polynomials are also included
Similarly, the multivariate formula (Wikipedia) is expressed ...
- 89
1
vote
1
answer
186
views
Complete bell polynomial coefficients
I would like to know if it is possible to calculate the coefficient of a given Complete Bell Polynomial 's monomial by its indexes and powers:
$B_{n}(x_1,x_2,...,x_n)= c_n(1,n) x_1^n + c_n((1,n-2),(2,...
8
votes
1
answer
311
views
Strange polynomial analog of the Bell numbers
Let $\vec{x} = (x_0, x_1, x_2, \dots)$ and $\vec{y}=(y_1,y_2,y_3, \dots)$ be two systems of parameters/variables. The Motzkin
polynomials $P_n(\vec{x},\vec{y})$ for $n \geq 0$ are defined by the ...
- 567
3
votes
1
answer
54
views
Sum of Two Arguments in Bell Polynomials of Second Kind [closed]
I understand the complete Bell polynomial $B_n$ satisfies the identity:
$$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,...,...
- 107
5
votes
0
answers
165
views
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 ...
- 270
4
votes
1
answer
308
views
Bell Polynomials
The complete Bell polynomials $B_n(x_1, x_2, \ldots, x_n)$ are defined through the relation
$$\sum_{n=0}^{\infty} B_n(x_1, x_2, \ldots, x_n) \frac{t^n}{n!} =\exp\Big( \sum_{n=1}^{\infty} x_n \frac{t^n}...
- 551
4
votes
1
answer
128
views
Group Action and the Bell Number
I am struggling on solving the inequality related to the group action and Bell numbers.
Let $G$ be a finite group acting on a set $X$ with $m$ elements. Prove that for each $1 \leq r \leq m$, $$\frac{...
- 497
3
votes
1
answer
161
views
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\; $ ($...
- 3,679
1
vote
0
answers
85
views
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 $...
- 590
0
votes
1
answer
105
views
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