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.

learn more… | top users | synonyms

3
votes
1answer
168 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 $f(n)=\sum_{i=1}^{\Omega(n)}{g(n,...
0
votes
0answers
26 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, … D....
0
votes
0answers
63 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 ...
2
votes
1answer
2k 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 ...
0
votes
1answer
23 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 $5....
0
votes
1answer
52 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 ...
1
vote
1answer
95 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 ...
1
vote
1answer
17 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 ...
0
votes
0answers
31 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 ...
0
votes
1answer
25 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 ...
2
votes
1answer
69 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: $$B(n)=\left\{(b_1,\ldots,b_n)\in\mathbb{N}^n \ \ ; \sum_{i=1}^{n} i\cdot b_i=n \right\}$$ ...
0
votes
0answers
32 views

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: ...
16
votes
1answer
231 views

Partition of ${1, 2, … , n}$ into subsets with equal sums.

The following is one of the old contest problems (22nd All Soviet Union Math Contest, 1988). Let $m, n, k$ be positive integers such that $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $...
1
vote
1answer
26 views

Determining the number of different ways a partition can occur?

Say for instance a food stall keeps a record of how many complaints it receives during its 4 day work week. Complaints are classified as Mon, Tues, Wed or Thurs, depending on the day during the week ...
1
vote
0answers
16 views

How to construct a partition of X given a sigma-algebra on X (when X countably infinite) [duplicate]

I am attempting to construct a partition of a countable set X, given a sigma-algebra on that set. Eventually I will want partitions of X to be in 1-1 correspondence with sigma-algebras on X. See my ...
9
votes
0answers
89 views

Closed form for $\sum_{n=1}^\infty \frac{1}{P(n)}$, where $P(n)$ is the partition function.

Is there a closed form for the following infinite series? $$\sum_{n=1}^\infty \frac{1}{P(n)}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$$ where $P(n)$ is the partition function.
4
votes
1answer
42 views

Is this integer partition studied?

First, this is my firt post here; I'm not a rigorously trained mathematician so apologies for the abuse of language, or if the problem is too trivial :) In a finite-system problem in statistical ...
0
votes
1answer
67 views

Maximizing the sum of exponentials whose exponents sum to $N$

Let $N \geq 1$ be a sufficiently large integer, let $a > 1$ be a real number, and let $n_1, \dots, n_t$ be integers between $0$ and $K$, where $K$ divides $N$. I want to determine the following: $$...
0
votes
1answer
31 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 ...
1
vote
0answers
42 views

Find a polynomial with evaluation equal to # of partitions of n into at most k parts

Fix a positive integer $k$. For positive integer $n$ let $p(n;\le k)$ denote the number of partitions of $n$ into at most $k$ parts, and let $p(0;\le k)=1$. (1) Show that there is a polynomial $P(x)$ ...
4
votes
3answers
81 views

Help with partitions, equivalence classes, equivalence relations.

The following definitions and results are from my textbook. A partition $\mathcal{P}$ of a set $X$ is a collection of nonempty sets $X_1, X_2, \dots$ such that $X_1 \cap X_j = \emptyset$ for $i \...
3
votes
4answers
52 views

Into how many equivalences classes does $R$ partition $\mathbb{Z}$?

Let $R= \{ (a,b) \in\mathbb{Z}\times\mathbb{Z} \mid a^2\equiv b^2 \bmod 7\}$. Into how many equivalences classes does $R$ partition $\mathbb{Z}$? My best guess is that there are $7$ equivalence ...
1
vote
1answer
38 views

Tool for the partition problem with planar rectangles

The classical "partition problem" asks how many ways one can write a given natural number as a sum of smaller numbers. One variant of this would be to ask if a positive real number can be expressed ...
3
votes
0answers
38 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
0
votes
1answer
32 views

Proving $F . G$ is the greatest lower bound

This is one of the problem I have been solving from Velleman's How to Prove book: Suppose $A$ is a set. If $F$ and $G$ are partitions of $A$, then we'll say ...
-1
votes
1answer
95 views

partitions of the number n

I'm having difficult with the following question : show that the number of partitions of n into parts of size 3,5,7,9,... equals to the number of partitions of n into different parts which are not 1,...
1
vote
1answer
34 views

Prove that $R$ is anti-symmetric

This is one of the problem I have been solving from Velleman's How to Prove book: Suppose $A$ is a set. If $F$ and $G$ are partitions of $A$, then we'll say ...
1
vote
0answers
54 views

Finite partitions of $\mathbb{N}$ and relations betweens sets of natural numbers

Suppose that $R\subseteq \mathcal{P}(\mathbb{N})\times\mathcal{P}(\mathbb{N})$ is a relation such that $(x,y)\in R$ only if $|x|=|y|$. Say that a partition $P$ divides a set $x$ if $x$ is the union ...
0
votes
1answer
175 views

Coin Change Problem with Fixed Coins

Problem: Given $n$ coin denominations, with $c_1<c_2<c_3<\cdots<c_{n}$ being positive integer numbers, and a number $X$, we want to know whether the number $X$ can be changed by $N$ coins. ...
1
vote
0answers
43 views

Generating function for writing an even number as a sum of at most k squares

I would like to find the exact number of ways in which $n$ can be represented as a sum of at most $k$ squares such that each term is less than or equal to say, $N$. A generating function for this ...
0
votes
1answer
154 views

NP-complete proof of subset with sum zero

I'm trying to proof that a problem of subset from a group has a sum of zero. I know that i can use the partition problem that is known to be NP-complete, but i can't seems to find what i need to ...
1
vote
2answers
149 views

Partitioning $[0,1]$ into pairwise disjoint nondegenerate closed intervals

My friend threw me a challenge. He told me that he proved the follow proposition: The topological space $[0,1]$ cannot be partitioned into pairwise disjoint nondegenerate closed intervals (except ...
1
vote
1answer
32 views

What is the name of the transform which finds the number of ways to make partitions of the given sizes?

I'm looking for the name of a transform which takes a sequence giving the number of 'prime' elements of a given size to the number of ways to make a number out of a sum of 'prime' elements, up to ...
2
votes
0answers
46 views

Which are partitions are for $\mathbb{Z}$? Which are covers for $\mathbb{Z}$? Which are both or neither?

(a) {{x : x is an even integer}, {x : x is an odd integer}} (b) {{x : x is an even integer}, {x : x is divisible by 3}} Note - divisible by 3 includes negative integers, yes? -3, -6, -9, ... and ...
2
votes
1answer
48 views

Can someone present a visualization of the partitioning of a $L^p$ space into equivalent classes?

I am a bit confused by what it means for a $L^p$ space to be partitioned into equivalent classes instead of functions. I understand that give two or more functions $f$, $g$, $h,\ldots$ of which are "...
0
votes
0answers
17 views

Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? [duplicate]

Question: Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? Someone comes along and gives us the partition $P=\{2,2,3,3,4\}$ of $14$. How can we ...
0
votes
0answers
66 views

What is wrong with my proof by exhaustion?

$n$ colored balls are placed in an urn, with $c$ colors such that there are an equal number of balls of each color. What is the expected number of distinct colors in $k$ randomly picked balls, ...
0
votes
0answers
24 views

k-way graph partitioning with bounded size constraint

A typical $k$-way graph partitioning problem is to partition a weighted graph into $k$ components, with the constraint that all $k$ components have the same size. However, if we drop the same size ...
1
vote
1answer
38 views

Prove that family of sets generates partition

Let $X$ be any set. Let $\mathcal{F}$ be family of subsets of $X$ closed under aritrary intersections and complements. Let $P(x)=\bigcap \{A\in\mathcal{F}:x\in A\}$. I need to prove that family $\{P(x)...
2
votes
4answers
66 views

What is a simple example that shows equivalence classes constitute a partition?

Can someone illustrate using a simple concrete example that the equivalence classes defined by $\sim$ constitute a partition of a set $A$?
1
vote
1answer
28 views

Writing a Sum of Partition Items in Combinatorial Form

For each partition $\lambda$ we can define \begin{equation} n(\lambda) = \sum_{i \geq 1}(i-1)\lambda_i. \end{equation} According to my book this is equivalent to \begin{equation} n(\lambda)=\sum_{i ...
16
votes
1answer
419 views

On the inequality $\frac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
1
vote
0answers
40 views

Why does the number of ways that $n$ can be summed with at least one $1$ equal the partition function for $n-1$?

For some reason I was counting the number of partitions of $n$ that have at least one $1$ as an addend. The beginning sequence for these numbers, starting with $n=1$, is $\{1, 1, 2, 3, 5, 7, 11, 15, ...
1
vote
0answers
62 views

Partition Function Breakdown

Let $g(n,k)$ be the number of partitions of $n$ into exactly $k$ parts, in which no part is a $1$. Show that $$g(n,k) = g(n-2,k-1) + g(n-k,k).$$ How would I show this?
0
votes
1answer
46 views

Counting functions and stirling numbers

Let S= { f | f: A $\rightarrow$ B, |Image(f)|=k}. |A|=m, |B|=n. where k $ \le n, k \le m $ |S|=$ {n \choose k} $ S(m,k) k!. where S(m,k) are the striling numbers of the second kind. What I can't ...
3
votes
1answer
109 views

How to measure similarity of partitions / partitioning?

Suppose a set of elements of finite size. E.g.: $X = \left\lbrace a,b,c,d,e,f,g \right\rbrace$ There are several ways to partition $X$. E.g: $P_1 = \left\lbrace \left\lbrace a,b \right\rbrace, \...
3
votes
1answer
85 views

Integer Partitions and distinguishable permutations

I'm not a mathematician but I'm faced with a problem where I can't find an answer, also because I do not know what I shall ask for: I have to deal with partitions of an integer k, only small values, ...
0
votes
2answers
81 views

Partition and Equivalence Relation: In $\mathbb{Z}$, let $m\sim n$ iff $m - n$ is a multiple of 10

Prof. Pinter's "A Book of Abstract Algebra" presents this exercise: Prove that each of the following is an equivalence relation on the indicated set. Then, describe the partition associated with ...
1
vote
1answer
70 views

Facing problem to understand the equivalence class concept.

Currently I am trying to understand the equivalence class from the Herstein's book. And I am facing problem to understand the concept and also can't understand the theorem given in the page number 8. ...
0
votes
1answer
40 views

Minimum number of partitions of a set given list of numbers that can't appear in the same partition

I wish to calculate the minimum number of partitions of a set required given a list of pairs of numbers that cannot appear together in the same partition. Example 1: $$S = [1,2,3,4,5,6]\\[5pt] \text{...