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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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 ...
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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: $a(0)=1$, ...
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1answer
79 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 ...
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1answer
40 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 ...
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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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7answers
472 views

Book recommendation for Integer partitions and q series

I have been studying number theory for a little while now, and I would like to learn about integer partitions and q series, but I have never studied anything in the field of combinatorics, so are ...
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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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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
243 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
79 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
81 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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4answers
291 views

Algorithm to partition sum between buckets in all unique ways

The Problem I need an algorithm that does this: Find all the unique ways to partition a given sum across 'buckets' not caring about order I hope I was clear reasonably coherent in expressing ...
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1answer
96 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
170 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
183 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
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2answers
107 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
39 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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2answers
403 views

Lower bounds for the partition function

In this question we consider the partition function $p(n)$ - that is, the number of ways to express $n$ as a sum of positive integers. One easy exercise is to show that $$ p(n) \geq 2^{\lfloor ...
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1answer
37 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
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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1answer
101 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
107 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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0answers
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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0answers
32 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
49 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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0answers
32 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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0answers
44 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
214 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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3answers
214 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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2answers
496 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
52 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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23 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
120 views

Partition function with restrictions

What is the number of ways of partitioning a positive number $k\leq mn$ using non-increasing parts such that the number of parts can be at most $n$ and the value of each part can be at most $m$?
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1answer
208 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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1answer
65 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 ...
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0answers
84 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 ...
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0answers
72 views

Monoid on ordered partitions of a natural number

Fix a natural number $n$, and let $O_n$ be the set of ordered partitions of $n$. For example $O_3=\{1+1+1,1+2,2+1,3\}$ which can also be written as $\{1|2|3,1|23,12|3,123\}$. We can define two ...
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1answer
26 views

Partioning Mystery

Who has the wisdom to answer the following: 9 distinct marbles distrubted into 4 distinct bags with each bag receiving at least 1 marble,how many ways can this be done? Thankyou for contributing! ...
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1answer
40 views

Finding Partition, Riemanns Integral

Define $f:[0,2]\rightarrow\mathbb{R}$ by setting $f(x)=1$ if $x\not=1$ and $f(1)=3$. Find a partition $D$ of $[0,2]$ for which $S_D-s_D<2^{-1000}$.
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2answers
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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 ...
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1answer
34 views

Let $P_1(r;n)$ denote the number of partitions of n into parts that are either even and not congruent to 4r-2(mod 4r)

Let $P_1(r;n)$ denote the number of partitions of n into parts that are either even and not congruent to 4r-2(mod 4r) or odd and congruent to 2r-1 or 4r-1(mod 4r). Let $P_2(r;n)$ denote the number of ...
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4answers
1k views

Partition of a set, definition not clear

From wikipedia: Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and: The union of the elements of P is equal to X. (The elements of P are said ...
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0answers
47 views

Partitions and divisor functions: what is known about their relations?

If $i\geq 1$ is an integer, we have the following integer valued functions (for any integer $n\geq 0$): \begin{align} p_i(n)&=\textrm{the number of }i\textrm{-dimensional partitions of ...
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1answer
63 views

What am I missing about Schur functions?

Let's say I only know the following about Schur functions: you give me a partition $\lambda$ of $d$ such that $\lambda$ has $n$ parts $\lambda_1,\ldots,\lambda_n$, and I can compute the Schur function ...
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2answers
45 views

Partitioning $\Bbb{N}$

Can we partition $\Bbb{N}$ into a finite union $$J_1 \sqcup J_2\ldots \sqcup J_N= \Bbb{N}$$ where $\sqcup$ denotes disjoint union. I'm guessing if we can then one of the $J_i$ must be infinite and ...
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28 views

Quantiles when the population contains only one unique value

My apologies for the mixing of the terms quartile and quantile below. I am interested in the general case of quantiles, but I'm using a quartile as a specific example. Also feel free to clarify any ...
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1answer
111 views

Proof that the series for the generating function of the partition function converges?

For $|q| < 1$, the generating function of the partition function $p(n)$ is given by $$ \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1} $$ I have an intuitive ...