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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Elementary Set Theory ~ Partitions

I tried searching for a related thread to this, so please don't roast me too hard if one already exists. Anyways, if I have a set $A = \{a, b, c\}$ then $\{a, b, c\}$ would not be considered a ...
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0answers
17 views

Conjugate classes of Symmetric group, $S_n$ and partition number of $n$.

In the book "Topics in Algebra", 2nd edition, By I.N. Herstein, the following lemma is given on page 89, Lemma 2.11.3 :The number of conjugate classes in $S_n$(the symmetric group of order $n$) is ...
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0answers
26 views

Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$

Let $f$ be a generalized function on $\Omega$. Let $G$ be an open subset of $\Omega$. Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$ Let $\phi ...
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2answers
137 views

Number of ways to partition $40$ balls with $4$ colors into $4$ baskets

Suppose there are $40$ balls with $10$ red, $10$ blue, $10$ green, and $10$ yellow. All balls with the same color are deemed identical. Now all balls are supposed to be put into $4$ identical baskets, ...
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1answer
18 views

From sets of subsets to partitions

Let S be a non-empty set, and Q be a set of non-empty subsets of S such that $\bigcup Q=S$. Let $P'$ be the set of all non-empty subsets x of S such that: $\forall q\in Q. x\subseteq q \lor x\cap ...
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0answers
18 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 ...
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2answers
130 views

How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...
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0answers
15 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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2answers
88 views
+50

Splitting a set into two disjoint sets five times, minimizing pairs in the same set

Suppose you have a class of 11 students . I want to split the class into two groups five different ways, minimizing the number of times that any two students are in the same group. In more ...
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1answer
28 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 ...
4
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1answer
98 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
99 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
13 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
23 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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1answer
67 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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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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2answers
62 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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1answer
33 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
28 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 ...
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1answer
30 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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0answers
22 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
95 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
81 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 ...
1
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0answers
88 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
29 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
45 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: ...
7
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1answer
61 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
27 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
44 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
93 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
41 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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1answer
23 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.
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0answers
29 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
28 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
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1answer
103 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
34 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 ...
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0answers
42 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
12 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 $$a < b < c < d$$ ...
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2answers
29 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 ...
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0answers
14 views

Charmed Bracelets and their Equivalence relations

The company Charmed, I’m Sure makes bracelets. Each bracelet has four charms, Apple, Banana, Cherry, and Fig (or $\{A,B,C,F\}$ for short). The way these bracelets are made is by sending a line of ...
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0answers
50 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
47 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
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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
71 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 ...
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1answer
35 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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0answers
28 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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4answers
50 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 ...
0
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1answer
30 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 ...