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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8 views

partitioning a set into subsets while considering preferences

i am looking for an algorithm to partition a set of P (p=~70) people into minimum G (G=~3) subsets/groups so that no group would have more than M (M=~30) maximum people/elements. Each person ...
10
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2answers
161 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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2answers
44 views

Distribution of the sum of $N$ loaded dice rolls

I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$. I have found answers for the ...
2
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2answers
110 views

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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2answers
282 views

Coin Change Problem - Find the number of ways to make change

I read the solution here http://www.algorithmist.com/index.php/Coin_Change basically to find the number of ways to make change: We are trying to count the number of distinct sets. it says " Since ...
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2answers
30 views

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
20 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
27 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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1answer
20 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 ...
5
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2answers
139 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
19 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
100 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, ...
6
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4answers
105 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
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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1answer
29 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
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
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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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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0answers
37 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
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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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
34 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 ...
0
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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.
7
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1answer
62 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
302 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
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) ...
1
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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: ...
1
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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
45 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 ...
1
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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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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
29 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
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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 ...
4
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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 ...
0
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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 $$a < b < c < d$$ ...
4
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4answers
489 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 ...
2
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1answer
26 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
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 ...
7
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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 ...
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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 ...