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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2answers
784 views

Partitions and Bell numbers

Let $F(n)$ be the number of all partitions of $[n]$ with no singleton blocks. Find the recursive formula for the numbers $F(n)$ in terms of the numbers $F(i)$, with $i ≤ n − 1$ Find a formula for ...
0
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1answer
137 views

Partitions of $n$ into $k$ blocks, without single blocks.

So I'm trying to come up with a recursive formula $f(n)$ which counts the number of all partitions of $[n]$ into $k$ identical blocks, where the number of elements in each box is more than 1. What ...
2
votes
0answers
169 views

Partition minimizing maximum of Euler's totient function across terms

Given natural numbers $M$ and $N$, I'd like to find a partition of $2^N$ with $M$ or fewer terms, $t_1 + t_2 + ... + t_M$, such that $\max(\phi(t_1), \phi(t_2), ..., \phi(t_M))$ is minimized, where ...
4
votes
7answers
530 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 ...
4
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0answers
137 views

A general Combinatorics problem (Coefficients of the q factorial)

I was solving a combinatorics problem when I encountered difficulties. The problem was: $x_1 \in \{0,1\}$ $x_2 \in \{0,1,2\}$ . . $x_{n-1}\in\{0,1,2..,n-1\}$ We have to find the number of ways ...
4
votes
0answers
115 views

Why limit Euler's Partition function P to $k\leq\sqrt n$ instead of $k\leq n$?

I solved a Project Euler problem (I won't say which one) involving the Partition Function P. I used equation #11 from the above link: $$P(n) = \sum_{k=1}^n (-1)^{k+1}\bigg(P\Big(n-{1\over ...
3
votes
2answers
693 views

Equivalence relations on natural numbers

How many equivalence relations are there on $\mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers? How can one compute it?
2
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0answers
41 views

How to prove some properties of partitions of finite subsets of N? [duplicate]

Possible Duplicate: Any partition of {1,2,..,9} must contain a 3-Term Arithmetic Progression The problem is as such: Prove that there is not a partition of $N_9 = \{1, 2, 3, 4, 5, 6, 7, ...
5
votes
1answer
105 views

Elementary proof of a bound on the order of the partition function

I am interested in the asymptotic order of the partition function $p(n)$. The paper Asymptotic Formulae in Combinatory Analysis proves there are constants $A$,$B$ such that $e^{A\sqrt{n}} < p(n) ...
2
votes
1answer
408 views

How do I implement an induction argument to show that we may assume that a partition $Q$ of $[a,b]\in{R}$ which refines $P$ has only one more point?

There is a lemma (32.2) in Kenneth A. Ross's Elementary Analysis text, that states: Let $f$ be a bounded function on $[a,b]$. If $P$ and $Q$ are partitions of $[a,b]$ and $P\subseteq{Q}$, then ...
7
votes
0answers
354 views

Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$. Is there any ...
1
vote
1answer
1k views

Ways to partition an n-element set

I've done a couple of searches and haven't found a solution to this here, but if I've missed it please feel free to close the question! I was wondering how many different equivalence relations I ...
1
vote
0answers
42 views

What is the big-$\mathcal{O}$ bound for the sum of function applied to the partitions of a set?

Consider a set $A$ that is partitioned into $n$ subsets $A_1 | A_2 | ... | A_n$ and a function $f \in \mathcal{O}(g)$. Question: what is the tightest bound I can establish for $\sum_{i=1}^n ...
3
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0answers
133 views

If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$

Question: If $f$ is quasi and left continuous on $[a,b]$ and $\alpha$ is increasing and right continuous then $f$ is Riemann-Stieltjes integrable over $[a,b]$ with integrator $\alpha$. ...
2
votes
1answer
1k views

Partitions of an integer into k parts.

I am interested in knowing whether an exact formula (analogous to the Hardy-Ramanujan-Rademacher formula for $p(n)$) for the number of partitions of a positive integer into k parts is known. I tried ...
10
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4answers
4k views

Understanding equivalence class, equivalence relation, partition

Im having difficulty grasping a couple of set theory concepts, specifically concepts dealing with relations. Here are the ones I'm having trouble with and their definitions. 1) The collection of ...
4
votes
1answer
182 views

Showing two generating functions to be equal

Let $\mathcal{A}$ be the set of partitions in which each part may occur 0, 1, 4, or 5 times and let $\mathcal{B}$ be the set of partitions which have no parts congruent to 2mod4, and in which parts ...
6
votes
1answer
303 views

Creating generating functions for integer partitions

Say I have a generating function $\Phi_\mathcal{A}$ for the set of partitions $\mathcal{A}$ which have no parts congruent to 2 mod 4, and I have the generating function for $\Phi_\mathcal{B}$ for the ...
1
vote
1answer
471 views

Integer Partitions Formulas [duplicate]

Possible Duplicate: Identity involving partitions of even and odd parts. How would I go about to show the following: Let $pe(n)$ be the number of partitions of size n with an even number of ...
2
votes
3answers
112 views

Number decomposition

Recently I encountered a problem I was not familiar with. So hope someone can help me for this. Here is the problem. Given any odd integer, how many different ways of decomposition into sum of three ...
0
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3answers
126 views

Combination Question

I currently have an open question about counting the possible ways of summing numbers. I am still exploring all the ideas provided - those within my level of understanding. This is a question ...
4
votes
1answer
2k views

Median of medians algorithm

I am referring to the algorithm presented here used to find a good pivot: http://en.wikipedia.org/wiki/Selection_algorithm#Linear_general_selection_algorithm_-_Median_of_Medians_algorithm My ...
0
votes
1answer
121 views

What is minimum Eulerian partition?

I have a homework in my graph course. It asks something about minimum Eulerian partition but it doesn't give any information about it. I googled it but couldn't come up anything useful and clear. Is ...
1
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1answer
120 views

Nonnegative integers with unique restricted partitions

I have come over the following problem in elementary number theory: suppose that a positive integer $k$ is given. Is it possible to find, for every such $k$, a nonnegative integer $N(k)$ and a set of ...
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2answers
2k views

Distribution of the sum of a multinomial distribution

I have distilled an error analysis problem into the following: I have a multinomial distribution, $X$, consisting of $n$ independent trials where each trial takes on the values $\{0,1,\ldots,k-1\}$ ...
4
votes
2answers
806 views

Enumeration of partitions

The Stirling number of the second kind $S(n,k)$, where $S(n,k) = \frac{1}{k!}\sum\limits_{j=0}^k(-1)^{k-j}\left(\begin{array}{l}k\\j\end{array}\right)j^n$ Gives the number of unique unlabeled, ...
2
votes
3answers
999 views

Partition of a set of size n into subsets of size 1 and 2.

Before I ask the question, I must admit that combinatorics has never been my forte. I am given a set X of size $n$, we may assume assume $X=\{1,2,...,n\}$. I want to count the partition of this set ...
1
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1answer
125 views

Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ [closed]

Count the number of integer solution to $\sum_{i=1}^ {2}{a_i\times b_i} \geq 6 $ such that condition 1: $1 \leq a_i \leq 7$ condition 2: $1 \leq b_i \leq 4$ condition 3: $\sum_{i=1}^{2} {a_i} = 8$ ...
1
vote
4answers
3k 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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2answers
2k views

Equivalence Relations and Partitions

My university "textbook" for discrete math is Schaum's Outline. In this outline he goes over Equivalence Relations and Partitions, and I got confused at a particular theorem. From the book: Theorem ...
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3answers
3k views

How many ways to merge N companies into one big company: Bell or Catalan?

There's a famous interview question variously credited to Microsoft, Google and Yahoo: Suppose you have given N companies, and we want to eventually merge them into one big company. How many ...
1
vote
1answer
189 views

Combinatorics problem based on Ferrers graph

Need help with this proof using Ferrers' graph or otherwise. Show that the number of partitions of $r+k$ into $k$ parts is equal to The number of partitions of $r + {k+1 \choose 2}$ into $ k $ ...
3
votes
1answer
47 views

cosets aren't forming a partition?

If you have a group $G$ and $H \leq G$, the cosets of $H$ should partition $G$. Suppose $G=\mathbb{Z}_2\times\mathbb{Z}_4$ and $H=\langle (0,1)\rangle = \{(0,0), (0,1),(0,2),(0,3)\}$. Then both ...
1
vote
2answers
4k views

How many ways you can make change for an amount [duplicate]

I am looking for a formula or at least something to use when trying to compute how many ways I can make change for an amount. Example: there are $3$ ways to give change for $4$ if you have coins ...
1
vote
3answers
506 views

Finite or infinite set?

Due to my not-so-advanced math skills, this question may take a few attempts to state clearly: Consider the unordered pair (2-tuple) partitions of n (e.g. with n=4, we have {{4,0},{3,1},{2,2}}). ...
3
votes
0answers
63 views

Ordered integer partition [duplicate]

Possible Duplicate: Proving an equality involving compositions of an integer A sequence of natural numbers $\langle a_1,\ldots,a_k \rangle$ is an ordered partition of $n$ if ...
2
votes
1answer
119 views

Partitions of binary numbers into binary numbers with fixed digits?

If we are to have (two, for example) binary numbers, such that their sum is $100111010_2$, and given that the first number has 5 ones, and the second number has 3 ones, can I find the numbers that ...
0
votes
1answer
33 views

Partitioning a sequence of numbers based on a percentage

I have a sequence $s$ of rational numbers where $|s|=n$. I am given a percentage $p$ to partition the sequence $s$ into $\frac{1}{p}$ parts, such that each part contains $p\%$ number of elements of ...
1
vote
2answers
408 views

Integer partitions with restrictions for elements [duplicate]

Possible Duplicate: Same number of partitions of a certain type? Prove that the number of partitions of $n$ without elements divisible by $4$, is equal to the number of partitions of $n$ ...
3
votes
1answer
520 views

Number of partitions of a set with size constraints

One of my students made some experiments on partitions of sets. He found some results and I asked him, if he can prove some statements. After two weeks he had no result, so maybe one of you can help. ...
5
votes
2answers
583 views

Find all ways to factor a number

An example of what I'm looking for will probably explain the question best. 24 can be written as: 12 · 2 6 · 2 · 2 3 · 2 · 2 · 2 8 · 3 4 · 2 · 3 6 · 4 I'm familiar with finding all the prime ...
9
votes
1answer
408 views

Average Number of Blocks in a Partition Containing a Large Block

Notation Let $[n]$ denote the set of integers from 1 up to $n$, inclusive. Let $A_n$ denote the average number of blocks over all partitions of $[n]$. Let $B_n$ denote the number of partitions of ...
4
votes
2answers
506 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 ...
1
vote
3answers
182 views

Probability - multinomial partitioning

Question: Joe is in his hunting blind when he locates $20$ geese, $25$ ducks, $40$ eagles, $10$ cranes, and $5$ flamingos. Joe randomly selects six birds to target, what is the probability that at ...
2
votes
5answers
979 views

General questions about equivalence classes and partitions

1) If a set is partitioned into non-overlapping, non-empty subsets, then those subsets are equivalence classes. If each element in a set is unique, how can a set be partitioned into subsets with ...
1
vote
0answers
145 views

Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

I would like to find articles or documentation regarding this process: Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook ...
1
vote
1answer
204 views

Integer Partition by Counting Repetition : Conjecture ??

I would like to find informations regarding this way of doing Integer Partitions or this conjecture, Suppose you have all the ordered partitions of 5: 5 4 1 3 2 2 2 1 3 1 1 2 1 1 1 1 1 1 1 1 Then ...
3
votes
4answers
539 views

Two hard number partition problems

For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$. Also define $p(0)=1.$ Problem 1. Prove that ...
1
vote
1answer
315 views

Two number partition problems

Let $p_k(n)$ be a number of ways to express $n$ as a sum of $k$ positive integers. For example $p_2(3)=1$. Problem 1. Prove that following recurrences are correct: ...
2
votes
2answers
446 views

Proof of a Proposition on Partitions and Equivalence Classes

I stumbled upon a seemingly rudimentary proposition that I am having trouble writing out a proof for. The proposition goes something like, Proposition: If $\{A_i|i\in I\}$ is a partition of ...