Questions related to the different ways of expressing an integer as a sum of integers; or, questions related to the subdivision of a set into smaller disjoint sets; questions related to the subdivision of an interval into smaller intervals that intersect only at the endpoints.

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Partitioning of sets

Question: Consider set $A= \{ 1, 2, 3, ..., n\}$. For what values of $n$ can $A$ be partitioned into 3 subsets $A_1, A_2, A_3$, such that sum of the elements of each of them are equal? My Attempt: ...
7
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1answer
67 views

What is the significance of this identity relating to partitions?

I was watching a talk given by Prof. Richard Kenyon of Brown University, and I was confused by an equation briefly displayed at the bottom of one slide at 15:05 in the video. $$1 + x + x^3 + x^6 + \...
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1answer
28 views

Problem 4 from Section E of Chapter 12 of Pinter's Book of Abstract Algebra

The question: Let $f: A\rightarrow B$ be a function, and let $\{ B_i : i \in I\}$ be a partition of $B$. Prove that $\{f^{-1}(B_i):i \in I\}$ is a partition of $A$. My work so far: For any given $...
1
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1answer
83 views

Partition (number theory)

Please someone explain the reasoning behind the recurrence relation p[k][n] = p[k][n − k] + p[k − 1][n − 1], where p[k][n] denotes the number of partitions of n into exactly k parts. For details, ...
7
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1answer
102 views

Special Vertex Partitioning

When can we partition the vertices of a graph $G$ into $n$ subsets such that every vertex is adjacent to vertex from every subset? For example, in the following graph, we have partitioned the vertices ...
2
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1answer
80 views

Digital line topology on $\mathbb{Z}$ and partitions

Consider $\mathbb{Z}$ with the digital line topology, which has as a basis the sets $\{n\}$ for $n$ odd, and $\{n-1,n,n+1\}$ for $n$ even, and consider the partition of $\mathbb{Z}$ created using ...
0
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1answer
28 views

Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$

I just computed the Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$. It would be nice if anyone could confirm it's correctness. Thanks.
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0answers
40 views

Quadratic variation along a sequence of subpartitions

Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ with $|\pi_n|=\max\limits_{t^n_i,t^n_{i+1}\in \pi^n}|t^n_{i+1}-t^n_i|\to_{n\to +\infty} 0$ the quadratic variation of a path $x\...
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1answer
30 views

Partitions of $n=a+b+c$ with restriction $b+2c=k$

Given a natural number $n$. How many partitions into three parts $a+b+c=n$ are there with the additional restriction that $b+2c=k$. E.g. for $n=12,k=18$ I get four partitions $(a,b,c)$: $(3,0,9),(2,2,...
7
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1answer
104 views

find the least natural number n such that if the set $\{1,2,…,n\}$ is arbitrarily divided into two nonintersecting subsets

Find the least natural number $n$ such that if the set $\{1,2,\dots,n\}$ is arbitrarily divided into two non intersecting subsets then one of the subsets contains three distinct numbers such that the ...
2
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2answers
40 views

Partition into “fibers” $f^{-1}(y) \in Y$

Consider any surjective map f from a set X onto another set Y. We can define an equivelance relation on X by $x_1Rx_2$ if $ f(x_1)=f(x_2)$. Check that this is an equivelance relation. Show that the ...
4
votes
0answers
69 views

Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$

I am currently taking discrete math and have been given the following question to answer. Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$ where $R=\{(0,0),(0,4),(...
0
votes
1answer
16 views

Trouble understanding noncrossing partitions

I am trying to understand what a non-crossing partition means. I was reading a paper and it states A partition is noncrossing if there do not exist four distinct elements $$a < b < c < d$$ ...
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2answers
32 views

Counting the max integer in each partition of $p(n)$

Assume $n=5$. We have $p(n)=$ 5 4 + 1 3 + 2 3 + 1 + 1 2 + 2 + 1 2 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1 What I want to get is the number of partitions in which the maximum integer is $m$, for each $...
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0answers
90 views

Proof of Riemann integral as limit of Riemann integral sum

I want to Prove the following statement, I will be appreciate if some one help me to do that. Let $f:[a,b]\to R$ and $f$ is bounded, show that if $f \in R$ ( Riemann integrable) and $\int_a^b ...
0
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1answer
50 views

Number of partitions of $n$ formed by combinations of $2$ and $4$

I'm trying to find the number of partitions of a natural number that are a combination of $2$ and $4$. For example: $$6 = 2+2+2 = 2+4 \Rightarrow p_6 = 2$$ So I start by defining $p_n$ as the ...
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0answers
12 views

Use of the “Optimality Principle” in a problem regarding maximization of a product indexed by a partition of a natural number.

I am currently reading a paper (Iterated Binomial Coefficients by S.W. Golomb, The American Mathematical Monthly, 1980, 719-727) that makes use of the "optimality principle" in a couple of proofs. One ...
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2answers
131 views

There are n objects and n boxes, how many ways can we place the objects so exactly one box remains empty

A) if both objects and boxes and indistinguishable B) if objects are indistinguishable and boxes are distinguishable My attempt: I know there are n! ways to but n objects into n boxes (both ...
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3answers
67 views

Number of non-negative distinct integer solutions of $x+y+z+w=10$

I understand that there are already many questions relating to this, but my question is regarding some concept of mine that should be working but doesn't produce the right result. So, I follow an ...
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1answer
49 views

Proving Riemann Integrability when $x$ is a function of $n$?

(Context: I have been using this resource from UC Davis to help me out with Riemann integrability.) So, I more or less get how Riemann integrability works when we specify $x$ over an interval, say $x&...
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0answers
32 views

The number of good partitions

Consider a set $S$ of $n$ red balls and $m$ blue balls. It is well known that the number of partitions of this set is the Bell number $B_{n+m}$. We say that a partition $P \subset \mathcal{P}(S)$ of $...
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4answers
53 views

Prove that $\inf{U(f,P)}\geq 1/2$

Define $f(x)=\begin{cases} x\;\;\;\text{if the point $x\in[0,1]$ is rational}\\ 0\;\;\;\text{if the point $x\in[0,1]$ is irrational} \end{cases}$ Prove that $\inf{U(f,P)}\geq 1/2$. Let $P=\{...
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1answer
33 views

Partitioning a Queue

Imaging a situation that we have n people in a queue and each people represent with number 1 and I want to partition the queue in smaller part, there are several ways to partition the queue. For ...
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0answers
30 views

Generating functions and integer partitioning [duplicate]

Show that the number of partitions of a positive integer n where no summand appears more than twice is equal to the number of partitions of n where no summand is divisible by 3 So I begin by ...
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1answer
70 views

name or characterization for the “partition” lattice of integer partitions of some n? (Young lattice row partial order)

Is there a name or characterization for the "partition" lattice of integer partitions of some n? Young's Lattice depicts the integer partitions of numbers. Often Young diagrams are used in displaying ...
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1answer
29 views

Identity from integer partitions and Ferrers diagrams

So the problem I'm working on is as follows: Let $\lambda$ and $\mu$ be integer partitions, and let $\lambda^*$ and $\mu^*$ be their conjugates. By counting a set in two ways, prove $\sum_{i,j}\min\{\...
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0answers
11 views

Partition Theorem to show $P(W \gt Z)$

I am confused as to how to use the partition theorem on the following example? Any help is appreciated! Suppose that W has a U(0,1) distribution and suppose that W is independent of the random ...
0
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0answers
28 views

Random variables, fair die tossed twice

A fair die is tossed twice. Let $X$ = the sume of the faces, $Y$ = the maximum of the two faces, and $Z$ = $|face 1 - face2|$. (a) Write down the value of $X, Y$ and $W = XZ$ for each outcome $\...
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1answer
20 views

Recurrence Relations; partitions of a set

Need help trying to solve this problem. Let $S$ be a set of $2n$ elements and let $P_n$ represent the number of Partitions of $S$ into $n$ parts, with two elements in each part. Explain why $P_n=(2n-...
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1answer
48 views

Closed form Formula for Partitions of an Integer

I happened across the following question: Determine the number of ways of putting $m$ indistinguishable balls into $n$ indistinguishable boxes with the restriction that no box is empty. The ...
4
votes
1answer
83 views

Partition of $S = \{1,2,\dots, 3n\}$ in to three subsets $A, B, C$ such that $|A| = |B| = |C| = n$

Let $n$ be a positive integer and consider the set $S = \{1,2,\dots ,3n\}$. Show that, for every partition of $S$ into 3 subsets $A, B, C$ such that: i) $|A| = |B| = |C| = n$ (here $|X|$ denotes the ...
2
votes
1answer
88 views

What does this product converges to?

Let $p\in[0,1]$. I'm interested in computing $$\lim_{n\to\infty}\prod_{i=1}^n(1-p^i)$$ Any thoughts? EDIT: As Kibble mentioned, this is the Euler function. Also from Kibble: a simple upper ...
2
votes
2answers
67 views

Show that $S(n ,k)$…

Show that $$S(n,k) = \sum_{m = k-1}^{n-1} {n-1 \choose m} S(m,k-1) $$ -I was having trouble with this proof in class and my professor suggested to look at it as another proof of the following ...
0
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1answer
54 views

Calculate the area under $f(x) = \sqrt x$ on $[0,4]$ by computing the lower Riemann sum for $f$ with the given partition [duplicate]

Where $x_i = \dfrac{4i^2}{n^2}$ and letting $n \rightarrow \infty$ I don't know how and where to begin.
2
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0answers
33 views

Known classic problem or not?

There is a set of positive whole numbers without null. I have to find the minimal number of subsets of the original set so, that the the sum of two numbers in a subset can't be the value of a number ...
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1answer
77 views

intersection of partitions

I am trying to figure out a good way of finding the intersection of two partitioned subsets of a set (or what to call what I'm trying to do so I can read something about it). Let's say I have two ...
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2answers
22 views

Number of “Overlapping” Cells Within a Hypercube

I have a hypercube in $k$-space which is divided along each dimension into $n$ cells. Each cell in the hypercube is assigned a unique ordered set of coordinates as follows: $(a_{1}, a_{2}, a_{3}, ... ...
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0answers
53 views

Interval partition with constraints

Problem: I have a set of intervals of $\mathbb{R}$ or $\mathbb{Z}$ (large integers): $[d_1^-,d_1^+],[d_2^-,d_2^+],...,[d_n^-,d_n^+]$. My goal is to find one value $x_i$ in each interval so that the ...
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0answers
68 views

How to make a canonical coin system so that greedy solution is the only optimal solution for change-making problem

Related to the paper: http://arxiv.org/PS_cache/arxiv/pdf/0809/0809.0400v1.pdf and coin-change problem in general. We say that a coin system of coins canonical if the greedy algorithm to the coin ...
3
votes
0answers
48 views

Amount of combinations of sets summing to number

(Apologies for the confused arbitrariness here; I don't have experience in formal maths to make abstract my lay-person thoughts, but I've tried my best.) I have $x$ identical but order-important sets ...
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1answer
20 views

Counting partitions of a finite set in $\lambda_j$ $j$-element sets

Suppose we have an $n$-element set $A$ and $\lambda_1,\dots,\lambda_n \in \mathbb{N}_0$ with $\sum_{i=1}^n \lambda_i\cdot i = n$. How many partitions $P$ of $A$ are there, s.t. $\# \binom{A}{j} = \...
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1answer
41 views

Number of partitions of a set, where the partitions have specific sizes

I stumbled upon the following question: given a set of size $k$, how many partitions of sizes $(n_1, ..., n_m)$ exist, for $n_1 + ... + n_m = k$. I am not sure I can explain it exactly like this, so I'...
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1answer
34 views

Number of pairwise non-isomorphic spanning trees of the wheel $W_n$, with restrictions

I recently encountered this problem. Frankly I'm stuck; would be nice for some help. Here it is: Let $N,k$ be positive integers. By $p_k(N)$ we denote the number of integer partitions of $N$ with ...
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2answers
32 views

Lexicographic order is a partial order on the the set of all partitions of the positive integer n.

I think the above statement is false, if not then please give a hint to prove.I know majorization is a partially order.
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1answer
75 views

Formula for how many combinations of powers of 2 sum to $2^n$

Given a number $2^n, n\in\mathbb{Z}\gt 0$, I would like to find a formula for how many unique sets of powers of $2$ sum to that number. This is related to the triangular numbers but excludes non-...
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0answers
26 views

How to prove $\prod_{\lambda\vdash n}\prod_im_i(\lambda)!=\prod_{\lambda\vdash n}1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots$

Let $\lambda$ be a partition of an positive integer $n$, it can be presented as $\lambda=(\lambda_{1},\lambda_2,\cdots,\lambda_l)$ such that $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_l>0$, or $\...
0
votes
2answers
46 views

On partitions of integers

In an example in my textbook, I came across a question where it was asked to find the generating function for the number of partitions of ${n \in N}$ into summands that (a) cannot occur more than 5 ...
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1answer
38 views

On partition of integers

I came across an example in my textbook where it was asked to find the generating function for the number of integer solutions of: ${2w+3x+5y+7z=n}$ where ${0\le w, 4\le x,y, 5\le z}$ The ...
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1answer
40 views

Is there a formula for the number of equipartitions of $[n]$ into $k$ parts of size $s = n/k$?

Let $k$ divide $n$ and $Q(n,k)$ be the number of partitions of $n$ into $k$ parts, each of which has size $s = n/k$. Is there a formula for $Q(n,k)$? What is the asymptotic behavior of $Q(n,k(n))$ ...
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Prove that a set of nine consecutive integers cannot be partitioned into two subsets with same product

Problem: Prove that a set of nine consecutive integers cannot be partitioned into two subsets with same product My attempt: This problem would be solved if it could be proven that 9 consecutive ...