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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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 ...
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28 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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1answer
49 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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4answers
50 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 ...
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1answer
30 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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28 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
41 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 ...
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1answer
24 views

Counting number of balanced two-way partition of the set

Given a set with $2n$ elements, show that the number of balanced two-way partition of the set $$P(2n)=\frac{2n!}{2\times n!\times n!}$$ I'm getting is as P(2n)=${2n}\choose{n}$. But I'm getting ...
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4answers
5k views

Algorithm for generating integer partitions

I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate all of them and then discard the ones that ...
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1answer
22 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 ...
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2answers
781 views

In how many ways can you split a string of length n such that every substring has length at least m?

Suppose you have a string of length 7 (abcdefg) and you want to split this string in substrings of length at least 2. The full enumation of the possibilities is the following: ...
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0answers
10 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 ...
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0answers
27 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
76 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 ...
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1answer
18 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 ...
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1answer
76 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 ...
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2answers
61 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 ...
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1answer
40 views

partitions are the elements of the free abelian monoid on $\aleph_0$ generators

although the following insight may not be particularly profound, it is always of interest when we see a canonical isomorphism between objects of apparently different types. it is a commonplace that ...
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32 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
42 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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0answers
48 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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2answers
15 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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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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0answers
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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 ...
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0answers
39 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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2answers
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Identity involving partitions of even and odd parts.

First, denote by $p_E(n)$ be the number of partitions of $n$ with an even number of parts, and let $p_O(n)$ be those with an odd number of parts. Moreover, let $p_{DO}(x)$ be the number of partitions ...
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1answer
300 views

Demystifying the asymptotic expression for the partition function

A partition of an integer $n$ is a way of writing $n$ as a sum of integers. The partition function $p(n)$ counts the number of distinct partitions of $n$. In 1918, Hardy and Ramanujan proved the ...
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1answer
30 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 ...
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1answer
33 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
28 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
50 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 ...
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0answers
23 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 ...
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2answers
37 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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5answers
958 views

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 ...
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1answer
36 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
28 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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2answers
779 views

Counting integer partitions of n into exactly k distinct parts size at most M

How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$? The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at ...
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1answer
111 views

How many ways to write a number $n$ as the product of natural numbers $\geq 2$?

I am looking for a closed form (or efficient algorithm) for $f(n)$, the number of ways in which $n$ can be written as a product of natural numbers $\geq 2$. Note that ...
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0answers
23 views

Deciding whether a collection of sets is a partition

Which of the following collections of sets is a partition of [0,∞)? A. $S_i$ = (i-1,i) for i =1, 2, 3, … B. $S_i$ = (i-1,i] for i =1, 2, 3, … C. $S_1$ = (0,1], $S_i$ = [i-1,i] for i = 2, 3, 4, … ...
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0answers
57 views

Bin Packing and Partition.

I am trying to do my assignment and got really confused and hard to understand with particular question. I need to show or prove that Partition ≤p Bin Packing. I read through the lecture slides and ...
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1answer
386 views

Number of possible combinations of x numbers that sum to y

I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10. An asymptotic ...
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1answer
21 views

Is $\mathbb R/R$ a partition of $\mathbb R$ given by some equivalence relation $R$?

Let $aRb $ iff $b - a$ is an integer. $5 - 0$ is an integer, so $5 \in [0].$ In fact, $[0] = \mathbb Z$. Does it mean $\mathbb Z \in \mathbb R/R$? $5.14159265359 - \pi$ is an integer, so ...
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1answer
48 views

Equating summations of an integer partition and its conjugate

I've been given an integer partition A = (A1, A2, ... ,An) and its conjugate B = (B1, B2, ... ,Bm). Using that information, I'm tasked with proving that $$\sum_{i=0}^n (i-1)A_i = \sum_{j=1}^m B_k(B_k ...
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1answer
65 views

How to prove that we can increase the precision of Riemann sum if we refine a partition

When learning Riemann Integral, I was introduced to the concept of partitions and refining them. It stated that refining a partition (e.g. controlling its norm) increases its efficiency. What I mean ...
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1answer
16 views

Show that the following Set $\Lambda$ is a Partition

Consider the two sets $$A_r = \{x \in \mathbb{Z} \enspace | \enspace x \enspace = \enspace 5q + r, \enspace 0 \enspace \leq \enspace r \enspace < \enspace 5\}$$ $$\Lambda = \{A_r\}$$ We must ...
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0answers
25 views

Lower bounds/upper bounds for Qbinomials

Is there any lower bound or upper bound known for Q-binomials? I know that number of partitions function p(n)>2^(\sqrt n). But, I don't know any lower bounds for Q-binomials which are the generating ...
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1answer
23 views

Partitioning a Set: Need help with Notation.

If I have a relation: $$R=\{(x,y)\in\mathbb R^2 : \cos(x)=\cos(y)\},$$ it is clear to me that $[x]= \{x,-x : x \in \mathbb R\}$ What I'm trying to say is that the equivalence class $x$ is partitioned ...
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1answer
61 views

Proof of an identity for $n!$ involving integer partitions of $n$

Let $B(n)$ be the set of the integer partitions of the integer $n\gt0$, with the notation: $$B(n)=\left\{(b_1,\ldots,b_n)\in\mathbb{N}^n \ \ ; \sum_{i=1}^{n} i\cdot b_i=n \right\}$$ ...
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0answers
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Partition Problem with discontiguous sets

I'm trying to solve a variant of the partition problem. I have two important twists. I need to solve for k partitions, not just 2, as in the classic partition problem. The following code does that: ...