All Questions
Tagged with bell-numbers set-partition
13
questions
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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 ...
3
votes
1
answer
161
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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\; $ ($...
- 3,679
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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
1
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1
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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 $$\left| A_n \right| = \sum_{k=0}^{n} (-1)^...
- 376
2
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2
answers
313
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How many different equivalence relations with exactly two different equivalence classes are there on a set with $n$ elements
I came across with this topic. It looks straight forward for $5$ elements, but what if I want to find how many different equivalence relations with exactly two different equivalence classes are there ...
- 1,979
4
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2
answers
1k
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$F(n)$ is number of ways to partition set of $n$ without singleton blocks. Prove that $B(n) = F(n) + F(n+1)$
In this case $B(n)$ is $n$-th Bell number.
To be honest, I would really love to know if there is a combinatorial proof for that. If there is not, other proofs are appreciated too.
- 983
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Relation between Bell number and $F(n)$ the number of partitions of $[n]$ without singeton blocks
Let $F(n)$ be the number of all partitions of $[n]$ with no singleton blocks.
Prove that
$$\lim_{n\to\infty}\frac{F(n)}{B(n)}=0$$
According to this question, we can know
$$F(n+1)=\sum_{i=0}^{n-1}(-1)^...
- 153
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
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
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2
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1k
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Partitions of a set with n elements (proof)
I was reading a textbook about combinatorial mathematic which claimed that we can calculate the exact possible partitions of a set with n elements .
I searched it on wikipedia and I read about bell ...
user668140
1
vote
1
answer
338
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Number of cycle partition of a set with repeating elements
We have a set $S$ with $E$ elements of which only $N$ are unique. We of course know how many repetitions of each of the $N$ elements are present: element $s_i$ is repeating $t_i$ times.
I would like ...
2
votes
1
answer
3k
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How many partitions are there of a 5 element set into 3 parts (of a specific form)?
I am currently trying to understand the number of ways of partitioning a 5 element set into 3 parts. However, I am only interested in partitions with the form
$$
\left\{ \{ a, b \}, \{c, d\}, \{e \} \...
- 2,737
0
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1
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154
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Probability of set partition
Let $A = \{1\dots n\}$. Do partition of set $A$ on pairwise disjoint two- and three-element subsets randomly. For each n determine probability that number of two-element subsets is equal to three-...
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