3
votes
2answers
39 views

Question about partitions in intervals of the real numbers.

I have to prove the following: Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any kind, except intervals containing a single element. Prove $ \mathcal{D} $ is countable. ...
0
votes
2answers
94 views

Finding Distinct Elements and Permutation in Partitioned Set

I am having a hard time figuring out where to start on a homework problem. The question is: A set of $nk$ elements is partitioned into $k$ subsets in two ways, each subset having size $n$: one ...
1
vote
2answers
145 views

Partitions of a set into three parts

How many partitions of the set $\{1,2,3, \ldots , 100\}$ are there such that both a) there are exactly three parts and b) elements $1,2,3$ are in different parts. Any help on this question would ...
1
vote
0answers
60 views

Explanation of block systems and group action

According to Wikipedia: "If $B$ is a block then $gB$ is a block for any $g$ in $G$. If $G$ acts transitively on $X$, then the set $\{gB \mid g \in G\}$ is a block system on $X$." i.e., $\{gB \mid g ...
1
vote
2answers
30 views

Partioning/Enumeration

How many ways can one distribute A) 15 Balls into 3 bags. Both bag and balls are distinct (labelled) and each bag must contain at least one ball. B) 10 balls into 3 bags. again both bag and balls ...
1
vote
1answer
39 views

About the parity of the partition function.

I am reading this Kolberg's article, where he proofs that the partition function takes both even an odd values infinitely often. http://www.mscand.dk/article.php?id=1555 Although I'm sure it's ...
-1
votes
1answer
113 views

I'm taking an advanced math paper and I have no idea how to start this question!

How would I go about working this out? I honestly don't know where to start! Any help is appreciated.
0
votes
2answers
178 views

Give combinatoric argument for partition counting: $P(n, k) = P(n -1, k -1) + P(n-k,k)$

Suppose you have $n$ identical pieces. You want to split them in $k$ groups. (each group must have $> 0$ pieces) First, I was ask to answer the basic cases $1 \le k \le n \le 5$ For examble, ...
4
votes
2answers
430 views

Partitions and Bell numbers

Let $F(n)$ be the number of all partitions of $[n]$ with no singleton blocks. Find the recursive formula for the numbers $F(n)$ in terms of the numbers $F(i)$, with $i ≤ n − 1$ Find a formula for ...
4
votes
1answer
126 views

Showing two generating functions to be equal

Let $\mathcal{A}$ be the set of partitions in which each part may occur 0, 1, 4, or 5 times and let $\mathcal{B}$ be the set of partitions which have no parts congruent to 2mod4, and in which parts ...
6
votes
1answer
232 views

Creating generating functions for integer partitions

Say I have a generating function $\Phi_\mathcal{A}$ for the set of partitions $\mathcal{A}$ which have no parts congruent to 2 mod 4, and I have the generating function for $\Phi_\mathcal{B}$ for the ...
0
votes
0answers
163 views

Give an expression for lower and upper sum, concluding $f$ is integrable

$f : [a,b] \rightarrow \mathbb{R}$ is non-increasing, which means that $f(y) \le f(x)$ when $y>x$ The two questions are : Give an explicit expression (without infima and suprema) of ...
3
votes
2answers
124 views

a formal name of this partition

Consider a set $A=\{1,2,3,4,5\}$, is there any terminology for the following partitions of $A$ ? (1) $A=\{ \{1\},\{2\},\{3\},\{4\},\{5\} \}$ (2) $A=\{\{1,2,3,4,5\}\}$.
2
votes
0answers
209 views

Generating Function of Integer Partition Such that at Least One Part is Even

I've been having a few issues coming up with a generating function for an integer partition such that at least one part is even. What I have got so far is: The generating function with no ...
0
votes
1answer
115 views

Bijective Proof - partitions

what is a bijective proof of #93 in the link provided http://math.mit.edu/~rstan/bij.pdf: The number of partitions of $n$ with $k$ parts equals the number of partitions of $n + \binom{k}{2}$ with ...
28
votes
2answers
580 views

Very curious properties of ordered partitions relating to Fibonacci numbers

I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon. We call an ordered ...
2
votes
2answers
161 views

On the path diagram of a partition

A path diagram is a Ferrers diagram for a partition $\lambda$ in which the dots have been replaced with numbers computed as follows. The number in position $(i,j)$ the number of paths (going either ...
1
vote
2answers
331 views

Proving that exactly half the partitions of $n$ into powers of 2 have an even number of parts

Could you help me how to prove that exactly half the partitions of $n$ into powers of 2 have an even number of parts, please?
2
votes
2answers
211 views

Two combinatorics problems. I'm not 100% confident in my answers

These are two problems from my combinatorics assignment that I'm not quite confident in my answer. Am I thinking of these the right way? Problem 1: On rolling 16 dice. How many of the $6^{16}$ ...