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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26 views

Integer partitions with distinct parts

Let $~~p(n)~~$ denote the number of all partitions of positive integer $~~n~~$ with distinct parts. I would like to find some effective algorithm for calculating $~~p(n)~~$. It seems that dynamic ...
4
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1answer
108 views

How to partition $nk$ objects $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size $k$, so that no combination of $k$ is repeated.

What is an algorithm to partition $nk$ objects a total of $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size exactly $k$, so that no subset of size $k$ is ever repeated? For example, ...
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4answers
134 views

# of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers?

How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?
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0answers
26 views

Verification of Rogers-Ramanujan identities

In Hardy's book 'Ramanujan', section 6.8 on the Rogers-Ramanujan identities, it states: None of these proofs can be called both "simple" and "straightforward", since the simplest are essentially ...
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1answer
49 views

Show that the equivalence classes give a partition of $X$. [duplicate]

"... the following subset of $X$ denoted by $[x]$ $(x\in X)$ is called the equivalence class generated by $x$: $$[x]:= \{y \in X; x\sim y \}$$ Show that the equivalence classes give a partition of ...
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0answers
14 views

Interpreting the table of classification of the partitions of $n$

I am going through A NON-RECURSIVE EXPRESSION FOR THE NUMBER OF IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUP $S_n$ by AMUNATEGUI. In table I, the classification of the partitions of n according ...
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1answer
68 views

Isn't x/E = y/ E ⇔ x E y deduced, not just x/E = y/ E ⇒ x E y from (a) x/ E≠ Ø, (b) x/E ∩ y/ E ≠ Ø ⇔xEy?

"Theorem 3. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. Then (a) Each $x/\mathscr E$ is a nonempty subset of $X$. (b) $x/\mathscr E \cap y/\mathscr E \neq \emptyset$ if ...
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1answer
27 views

The lattice of partial partitions

In perusing A kind of Eulerian numbers connected to Whitney numbers of Dowling lattices I have gotten confused over what seems a very elementary point. The beginning of section 2 of the paper is ...
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0answers
38 views

How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
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2answers
67 views

How many distinct, non-negative integer solutions are there for $2x_0+\sum_{i=1}^{m}{x_i}=n$?

We are given constants $m$ and $n$. How many non-negative integer solutions are there for $2x_0+\sum_{i=1}^{m}{x_i}=n$ satisfying the condition that$x_i\neq x_j$ if $i\neq j$? I thought a good first ...
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0answers
24 views

Interesting orderings open sets of a topology

Let X be a set of points an $\mathcal{Q}$ be a partition on X. The intuition I want to model is that X is a set of worlds considered possible by an agent and $\mathcal{Q}$ is a question whose ...
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1answer
38 views

Use generating function to check how many solutions are to get balls from boxes and roll dices

In the box are 4 red balls, 5 blue balls and 2 yellow balls. How many possible solutions there are to get 7 balls from box to have at least 1 red ball and exactly 2 blue balls? I'm not sure I am ...
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0answers
29 views

Proof that two generating function are equals for the sequence which $n$-th number is:

I am not sure I am doing this exercise good 1) $p_n $ | all parts are pairs different and 2) $p_n $| all parts are not higher than $m$ I found these functions in book, first is: $$ ...
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0answers
24 views

Partition of numbers

Find all $\alpha_i, 1\le i \le n$ and $\beta_j, 1 \le j \le n$, $\alpha_i, \beta_j \in \mathbb{N} $ satisfying the following: (1) $\alpha_i \ge \alpha_{i+1}$ for $1 \le i \le n-1$ and $\beta_j ...
0
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1answer
31 views

Proof that $ \sum_{i=1}^k p_i = (n-k) $ where $p_i (n)$ is the number of partitions of n into exactly i parts.

I have to proof that $p_i(n) = p_i(n − i) + p_{i−1}(n − i) + . . . + p_1(n − i).$ for every $ 1 \le i \le n $, where n is number of n partitions has exactly i parts. Then I have to calculate ...
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1answer
27 views

Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$

I just computed the Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$. It would be nice if anyone could confirm it's correctness. Thanks.
7
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1answer
65 views

What is the significance of this identity relating to partitions?

I was watching a talk given by Prof. Richard Kenyon of Brown University, and I was confused by an equation briefly displayed at the bottom of one slide at 15:05 in the video. $$1 + x + x^3 + x^6 + ...
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1answer
309 views

Equivalence relation and partitions [closed]

Define an equivalence relation on the set R that partitions the real line into subsets of length 1.
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0answers
23 views

Time-scale law of Bernoulli-stopped process

This post is quite long, but the problem stated carries no computational burden. Consider the equally spaced partition $t_{i}^n=\frac{i}{n}$ with $i=0,...,n$ of the interval $[0,1]$ into $n$ ...
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1answer
17 views

Set partition, partition into X set with Y elements.

How do you partition a set into X number of new sets all that have Y elements? Example: How to partition 18 unique cards are divided to six persons, and each person gets 3 cards each. How many ...
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5answers
101 views

Does {$\Bbb Z_0$,$\Bbb Z_1$, $\Bbb Z_2 ,\cdots$, $\Bbb Z_{m-1}$} form a partition of $\Bbb Z$?

"Definition 5. Let X be a nonempty set. By a partition P of X we mean a set of nonempty subsets of X such that (a) If $A, B \in \mathscr P$ and $A \neq B$, then $A \cap B = \emptyset$, (b) ...
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1answer
87 views

Set within a partition

Say I have a partition of the set $\{1,2,3,4,5\}$. The partition is $\{\{1,3\},\{2\},\{4\},\{5\}\}$ Is there a word for set within a partition e.g. I want to say, 'one of the sets of the partition ...
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0answers
99 views

Number of ways to write a tuple of positive integers as a sum of tuples with certain constraints

There are $N$ boxes into which we put $mn$ balls in $m$ steps, where in each step we insert $n$ balls, each of which goes into a different box. In how many ways can we do this so that box $B_i$, $1 ...
1
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1answer
33 views

partitioning of a set with kn members into k subsets such that each subset has n members

we know that $S(n,k)$ is the number of ways we can partition a set with $n$ members to $k$ subsets ( each subset has at least one member). imagine we have a set with $k*n$ members. we want to ...
4
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3answers
53 views

Partitioning of sets

Question: Consider set $A= \{ 1, 2, 3, ..., n\}$. For what values of $n$ can $A$ be partitioned into 3 subsets $A_1, A_2, A_3$, such that sum of the elements of each of them are equal? My Attempt: ...
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1answer
28 views

Problem 4 from Section E of Chapter 12 of Pinter's Book of Abstract Algebra

The question: Let $f: A\rightarrow B$ be a function, and let $\{ B_i : i \in I\}$ be a partition of $B$. Prove that $\{f^{-1}(B_i):i \in I\}$ is a partition of $A$. My work so far: For any given ...
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1answer
66 views

Partition (number theory)

Please someone explain the reasoning behind the recurrence relation p[k][n] = p[k][n − k] + p[k − 1][n − 1], where p[k][n] denotes the number of partitions of n into exactly k parts. For details, ...
7
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1answer
98 views

Special Vertex Partitioning

When can we partition the vertices of a graph $G$ into $n$ subsets such that every vertex is adjacent to vertex from every subset? For example, in the following graph, we have partitioned the vertices ...
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1answer
60 views

Digital line topology on $\mathbb{Z}$ and partitions

Consider $\mathbb{Z}$ with the digital line topology, which has as a basis the sets $\{n\}$ for $n$ odd, and $\{n-1,n,n+1\}$ for $n$ even, and consider the partition of $\mathbb{Z}$ created using ...
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0answers
31 views

Quadratic variation along a sequence of subpartitions

Given a sequence $\Pi=\{\pi_n\}$ of partitions of an interval $[0,T]$ with $|\pi_n|=\max\limits_{t^n_i,t^n_{i+1}\in \pi^n}|t^n_{i+1}-t^n_i|\to_{n\to +\infty} 0$ the quadratic variation of a path ...
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1answer
30 views

Partitions of $n=a+b+c$ with restriction $b+2c=k$

Given a natural number $n$. How many partitions into three parts $a+b+c=n$ are there with the additional restriction that $b+2c=k$. E.g. for $n=12,k=18$ I get four partitions $(a,b,c)$: ...
7
votes
1answer
104 views

find the least natural number n such that if the set $\{1,2,…,n\}$ is arbitrarily divided into two nonintersecting subsets

Find the least natural number $n$ such that if the set $\{1,2,\dots,n\}$ is arbitrarily divided into two non intersecting subsets then one of the subsets contains three distinct numbers such that ...
2
votes
2answers
36 views

Partition into “fibers” $f^{-1}(y) \in Y$

Consider any surjective map f from a set X onto another set Y. We can define an equivelance relation on X by $x_1Rx_2$ if $ f(x_1)=f(x_2)$. Check that this is an equivelance relation. Show that the ...
4
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0answers
59 views

Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$

I am currently taking discrete math and have been given the following question to answer. Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$ where ...
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1answer
16 views

Trouble understanding noncrossing partitions

I am trying to understand what a non-crossing partition means. I was reading a paper and it states A partition is noncrossing if there do not exist four distinct elements $$a < b < c < d$$ ...
4
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4answers
526 views

Computing partition numbers

Today a friend and myself came up with the question of computing partitions of numbers, i.e.: given a number $n$, what is the number $p(n)$ of was of different ways writing $n$ as a sum of non-zero ...
3
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1answer
31 views

asymptotic approximation for number of partitions of integer that do contain 1 nor 2

Hardy and Ramanujan provided a famous asymptotic approximation to $P(n)$ the number of partitions of an integer $n$ when $n$ gets large. I wonder if there is an asymptotic approximation to ...
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2answers
30 views

Counting the max integer in each partition of $p(n)$

Assume $n=5$. We have $p(n)=$ 5 4 + 1 3 + 2 3 + 1 + 1 2 + 2 + 1 2 + 1 + 1 + 1 1 + 1 + 1 + 1 + 1 What I want to get is the number of partitions in which the maximum integer is $m$, for each ...
7
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0answers
360 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 ...
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3answers
2k views

Partitioning a natural number $n$ in order to get the maximum product sequence of its addends

Sorry if i ask this question. probably it's already answered somewhere else but i didn't find it. Suppose to have a natural number $n \ge 0$. Given natural numbers $\alpha_1,\ldots,\alpha_k$ such ...
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0answers
73 views

Proof of Riemann integral as limit of Riemann integral sum

I want to Prove the following statement, I will be appreciate if some one help me to do that. Let $f:[a,b]\to R$ and $f$ is bounded, show that if $f \in R$ ( Riemann integrable) and $\int_a^b ...
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1answer
48 views

Number of partitions of $n$ formed by combinations of $2$ and $4$

I'm trying to find the number of partitions of a natural number that are a combination of $2$ and $4$. For example: $$6 = 2+2+2 = 2+4 \Rightarrow p_6 = 2$$ So I start by defining $p_n$ as the ...
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0answers
11 views

Use of the “Optimality Principle” in a problem regarding maximization of a product indexed by a partition of a natural number.

I am currently reading a paper (Iterated Binomial Coefficients by S.W. Golomb, The American Mathematical Monthly, 1980, 719-727) that makes use of the "optimality principle" in a couple of proofs. One ...
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2answers
102 views

There are n objects and n boxes, how many ways can we place the objects so exactly one box remains empty

A) if both objects and boxes and indistinguishable B) if objects are indistinguishable and boxes are distinguishable My attempt: I know there are n! ways to but n objects into n boxes (both ...
0
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1answer
46 views

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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32 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
62 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
52 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
31 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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0answers
30 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 ...