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 into groups with maximal mixing

Suppose I have a class of 30 students and I want to give them 8 assignments to do in groups of 3. As far as possible I'd like the students to work with as many different students as possible. Ideally ...
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1answer
46 views

constant length of blocks in partitions

Let's assume we have partitions $P_k$ of the set $\{1,...,n\}$. If we choose two partitions it can happen, that each of them has a constant length of its blocks, but that the intersection of these two ...
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56 views

Difficulty parsing combinatorics exercise

I am working through the wonderful book Proofs and Confirmations by David Bressoud. In the section 2.2, I came across the following exercise, which has me scratching my head. (2.2.8) ~ Let ...
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33 views

Bound and parity integer-partition into fixed number of parts

Let $E(n,k)$ denote how many set of $k$ distinct non-negative integer are there such that their sum is an even number $\leq n$. Let $O(n,k)$ denote how many set of $k$ distinct non-negative integer ...
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1answer
61 views

Compactness of a set of partitions

The interval $[0,1]$ is partitioned to $n$ disjoint parts. Is the set of all possible partitions compact? There are several cases: A. All $n$ parts are connected intervals (possibly empty). In this ...
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2answers
69 views

What does George Andrews mean by i in “Theory of Partitions”?

From the first page of chapter 1 of George Andrews "Theory of Partitions" (Rather ominous place to get stuck): What do these last two sentences mean? I don't get "where exactly $f_l$ of the ...
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28 views

An identity relating sum of number of partitions to sum of number of parts

I encountered this identity while studying about the Kac determinants in CFT. $$\sum_{\{n_1 + \cdot \cdot \cdot + n_k \}} k = \sum_{pq \le N} P(N-pq)$$ Here $P(N-pq)$ is the number of partitions of ...
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39 views

How to calculate least moves of fruit.

I have a question, I'll try to abstract from the real problem to not lose people. What I'm really looking for is the name of the algorithm or class of problem to find my solution. I feel that this is ...
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2answers
78 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. ...
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1answer
57 views

How to find Partition of unity in $ \mathbb{S}^n$ with only $2$ functions

How to find Partition of unity in $\mathbb{S}^n$ with only $2$ functions?
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3answers
179 views

Ways of Distributing $n$ balls among $k$ boxes, each box containing $L \leq x_i \leq M$ or $0$ Balls

I need to calculate the number of ways of distributing $n$ balls among $k$ boxes, each box may contain no ball, but if it contains any, then it must contain $\geq L$ & $\leq M$ balls. This ...
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46 views

intersection of stabilizers of block systems

Lets assume the following situation: $G$ acts regularly on a set $M$. Then there is a bijection between the set of subgroups and the set of blocks containing a fixed element $m \in M$. The blocks ...
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1answer
36 views

Example where a cover is fewer than a partition?

Given a Cartesian product of sets $X \times Y$, A (combinatorial) rectangle is a set $A \times B$ where $A \subseteq X$ and $B \subseteq Y$. Given a function $f : X \times Y \rightarrow \{ 0, 1\}$ one ...
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0answers
28 views

generalization of bijection proof

$p(n|\text{odd parts}) = p(n|\text{distinct parts})$ I need bijection proof and generalization of this proof. for $k = 7$ odd parts are : ...
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68 views

OEIS sequence A086449

OEIS sequence A086449 http://oeis.org/A086449 is defined by: $a(0)=1$, $a(2n+1)=a(n)$, $a(2n) = a(n)+a(n-1)+\ldots+a(n-2^m)+\ldots$ $= a(n)+\sum_{i=0}^{\lfloor\lg n\rfloor}a(n-2^i)$ One can show ...
2
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1answer
73 views

Integer Partition into Powers

Is there any way to count the number of integer partitions of a number N into powers of two such that each size is repeated a power of two times? Ok so the recurrence can be expressed by: $a(0)=1$, ...
3
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1answer
77 views

Existence of a prime partition

I'm interested in finding out whether there exists a prime partition of a given positive integer $N>1$ such that the partition has specific number of parts. For instance, as given in another ...
2
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1answer
39 views

A bound on the balanced equipartition of a multi-set of integers

A balanced equipartition of a multi-set of $2n$ integers is a partition into two multi-sets $S_1$ and $S_2$ of size $n$ such that the sum of the integers in $S_1$ is as close as possible as the sum of ...
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0answers
60 views

Generation of n terms that sums to a specific value?

I'm about to conduct an integer program where I am to transport some products from different cities to docks and refinement stations. I am however required to work with simulated data where I only ...
2
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1answer
34 views

partitioning a set using a non-equivalence relation

I understand that a set can be partitioned into equivalence classes by an equivalence relation (wikipedia article). However can we partition a set into a forest of trees by a relation that's simply ...
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0answers
25 views

What are all the possible sums (and how often do they occur) of a k-subsequence of an n-sequence of integers?

Let $A_n = \{a_1,\dots,a_n\}$ be a sequence of non-decreasing non-negative integers. Let $P(A_n,k)$ be the set of all subsequences of $A_n$ of length $k$. Given $n,k\in\mathbb Z_{\geqslant0}$ with ...
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2answers
235 views

Partitions of Natural Numbers [duplicate]

This is a question from Complex analysis by Stein. The question is Prove that it is not possible to partition $\mathbb N$ into finitely many infinite AP's with distinct common differences.(other ...
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1answer
77 views

Ramsey Theory: Showing the existence of a special set of natural numbers.

Show that there is an infinite set of natural numbers such that the sum of any two elements has an even number of prime factors. My attempt: Define a coloring on the doubletons of $\mathbb{N}$, that ...
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1answer
76 views

How to prove this identity?(perhaps related to partition)

How to prove this identity? $$ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)} = \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$$ Maybe the method using generating functions is good.
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1answer
84 views

Partition induced by a Relation

Here's the problem: Let $A=\{1,2,3,4,5,6,7,8,9\}$. Define a relation $R$ on set $A$ by $xRy$ if and only if $2\mid(x+y)$ Assuming that $R$ is an equivalence relation, determine the partition of set ...
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1answer
122 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
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113 views

Cauchy's Coefficient Formula

How can I recover the coefficients of a polynomial power expansion of the type (number of partitions): $ \prod_i (1+x^{\alpha_i}z) = A_1 z + A_2 z^2 + A_3 z^3 + A_4 z^4 \ldots $ using Cauchy's ...
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1answer
164 views

Finding the amount partitions of a multiset

A multiset $A$ contains $n$ positive integers. The multiplicity of every integer is less or equal to $m$. $A$ is partitioned into $m$ subsequences in such a way that the multiplicity of all elements ...
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1answer
176 views

Number of ways to add up to a number without repetition (order does not matter)?

I have a number x and want to find how many ways there are to add up to that number using the y numbers from numbers 1-z. for example, for x=15 y=3, z=9, there are 8 ways to add up to 15 using 3 ...
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1answer
39 views

show that there is some element x∈X whose stabilizer Gx is all of G where G is a group of order p^k, where p is prime and k is a positive integer

I'm having trouble with this problem: Suppose that G is a group of order p^k, where p is prime and k is a positive integer.
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105 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 ...
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1answer
38 views

Construct matrix of ones and zeros based on sequences

We are given ($a_1,a_2,....,a_m$) and ($b_1, b_2,....,b_n$) sequences with non-negative integers. Decide whether it's possible and if it is construct a matrix $\Re^{m x n}$ of ones("1") and ...
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1answer
36 views

List 4 partitions of the alphabet containing 3 sets. How many different partitions can be made?

The first part of the question is easy; there are many right answers but I put down: $$(a), (b), (c..z)$$ $$(a,b), (c), (d..z)$$ $$(a,b,c), (d), (e..z)$$ $$(a,b,c,d), (e), (f..z)$$ The part I am not ...
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1answer
94 views

Real Analysis Riemann integrals with piece wise function

this is part of my homework assignment and I have been stuck on it for a few days. Let f be the function on [0,1] given by f(x)= { 1 if x does not = 1/2 and 2 if x=1/2 Prove f is Riemann integrable ...
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0answers
73 views

Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the ...
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21 views

Limits and integration

I have the following quick question: Consider bounded open domain $O \subset \mathbb{R}^{n}$ assume that we partition $O$ into $O_{1}^{m}$ and $O_{2}^{m}$ such that $O_{1}^{m},O_{2}^{m} \subset O$, ...
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1answer
52 views

Scalar products and partitions of Hypercubes

My questions relate to scalar products defined in $\mathbb{R}^{n}$ and partitions of hypercubes. Take $s \in \mathbb{R}$, $\xi, \eta \in \mathbb{R}^{n}$. My first question is why is it possible to ...
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31 views

Mapping a set of sets to a partitioning.

I've been experimenting with the following idea, and I wondered if there's a name for it: Suppose $S_0, S_1, ... S_{n-1}$ is an array of $n$ sets of elements in $U$. Now for any element $e \in U$ we ...
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2answers
48 views

Alternative reference for number of restricted partitions

I am looking for the number of partitions of some number $n$ into $k$ parts. Following the Wikipedia article on partitions, I ended up with Andrew's book [1]. Judging by Google's preview Chapter 3 ...
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29 views

Partition of the node set of a graph into connected subsets

What word is most commonly used in graph theory for a partition of the node set of an undirected graph into connected subsets? More rigorously: Given an undirected graph $(V,E)$, a partition $S ...
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42 views

Partition of the 3d space with circles?

Does it exist a partition of the 3d space with circles of positive radium? I know the answer is no for a plane, but I can not transpose my demonstration to the space and I have no clue on how to do ...
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3answers
199 views

integral $ \int e^{x} \sqrt{e^{x} - 1} dx $

i want to determinate the following integral \int e^{x} \sqrt{e^{x} - 1} dx my try and steps were as follow $$ \int e^{x} \sqrt{e^{x} - 1} dx $$ let $ u = \sqrt{e^{x} - 1} $ and $ v' = e^{x} $ ...
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183 views

Integral $\int \sqrt{x^{2}+a^{2}}$

i want to determinate the following integral $$\int \sqrt{x^{2}+a^{2}}dx \space | \space a> 0$$ i used integration by partition and u-substitution but i came to no result
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483 views

Integrate $ \int \sinh 2x \cosh x dx$

I want to integrate $$\int \sinh 2x \cosh x dx$$ my steps was as follow: $$ \int \sinh 2x \cosh x dx = \\ \sinh 2x . \sinh x - 2 \int \cosh 2x \sinh x dx = \\ \sinh 2x . \sinh x - 2 (\cosh 2x . ...
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1answer
48 views

How to talk about overlapping partitions of a set?

If I have a set $A$ and a number of sets $A_\sigma$ for each $\sigma \in \Sigma$ such that $$A = \bigcup_{\sigma \in \Sigma}A_\sigma$$ Is there a concise and eloquent way of saying this in plain ...
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20 views

How to prove this identity about Sylvestered partitions of n into m parts such that …

Let us say a partition $\lambda$ is $Sylvestered$ if the smallest part to appear an even number of times ($0$ is an even number) is even. For each set of positive integers $T$, define $S(T\,;m,\,n)$ ...
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1answer
19 views

Regular expression for a particular language

Several years ago I came across a paper that defined a regular expression (or collection of regular expressions?) for a specific language. The language is the language of set partitions enumerated by ...
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1answer
206 views

Two different coins on a chessboard

Two different coins are placed on squares of a standard 8x8 chessboard; they may both be placed on the same square. Let us call two arrangements of these coins on the chess board equivalent if we can ...
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2answers
348 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 ...
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1answer
64 views

A conjecture on partitions

While trying to prove a result in group theory I came up with the following conjecture on partitions: Let $b(i,j)$ be the number of partitions of $i$ with greatest part exactly equal to $j$ , for ...