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

7
votes
1answer
153 views

Partition Generating Function

a) Let $$P(x)=\sum_{n=0}^{\infty} p_nx^n=1+x+2x^2+3x^3+5x^4+7x^5+11x^6+\cdots$$ be the partition generating function, and let $Q(x)=\sum_{n=0}^{\infty} q_nx^n$, where $q_n$ is the number of ...
1
vote
0answers
39 views

Partition Generating Function (Truncation)

Let $P(x)=\sum_{n=0}^{\infty} p_nx^n$ be the partition generating function, and let $P^*(x)=\sum_{n=0}^{\infty} p^*_nx^n$, where $$p^*_n = \binom{\text{number of partitions of }n}{\text{into an even ...
2
votes
1answer
41 views

Maximal Triangle Partitioning in n lines

Recently I was given the following problem at work: Given a 5 pointed star, draw two straight lines through it so that there are 10 minimal triangles within the drawing. It took some work but I ...
2
votes
3answers
36 views

Generating Functions Partitions

Let $U(x)=\sum_{n=0}^{\infty} u_nx^n$, where $u_n$ is the number of partitions of $n$ into at most two parts. For example, $u_4=3$ because $4$ can be partitioned into at most two parts as $4$, $3+1$, ...
0
votes
2answers
32 views

Generating Functions Proof

Let $A=\{2,6,10,14,\ldots\}$ be the set of integers that are twice an odd number. Prove that, for every positive integer $n$, the number of partitions of $n$ in which no odd number appears more than ...
2
votes
2answers
44 views

Write a number N as a sum of K numbers

I need to find the no of ways of partitioning a number N as a sum of K non-negative numbers. Zeroes are also needed to be included in the sum. Ordering does matter. Example- For $N=2,K=3 $ ...
1
vote
3answers
37 views

Prove: In $\mathbb{Z}$, m ~ n iff $|m|=|n|$

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 ...
0
votes
1answer
54 views

How many possibilities of writing a natural number $M$ as a sum of $N$ natural numbers between $0$ and $M$?

How many possibilities are there of writing a natural number $M$ as a sum of $N$ natural numbers between $0$ and $M$? For example, I need to write $4$, using $4$ numbers between $0$ and $4$. The ...
1
vote
3answers
40 views

Prove: $\lbrace A_r : r \in \mathbb{R}\rbrace$ is a partition of $\mathbb{R} \times \mathbb{R}$

Given the following: For each $r \in \mathbb{R}$, let $A_r = \lbrace(x,y) \in \mathbb{R} \times\mathbb{R} : x - y = r\rbrace$ Prove: $\lbrace A_r : r \in \mathbb{R}\rbrace$ is a partition of ...
1
vote
1answer
31 views

Prove: $\langle A_0, A_1, A_2, A_3, A_4 \rangle$ is a partition of $\mathbb{Z}$

The following exercise from Dr. Pinter's "A Book of Abstract Algebra" presents this exercise: Prove that each of the following is a partition of the indicated set. Then describe the equivalence ...
0
votes
1answer
49 views

An expression to represent all possible arrangments of a set of length n into 3 (infintely sized) bins?

I have a set, A = {1,2} And I generate a set, B, of all possible arrangements of the above set across 3 "bins" (note where 1 and 2 are together, they are summed): ...
-1
votes
1answer
47 views

Prove $\lbrace A_n : n \in \mathbb{Z}\rbrace$ is a partition of $\mathbb{Q}$

Dr. Pinter's "A Book of Abstract Algebra" presents the following exercise: Prove that each of the following is a partition of the indicated set. Then describe the equivalence relation with that ...
2
votes
2answers
63 views

Integer partitioning

Suppose we have an integer $n$. I we want to partition the integer in the form of $2$ and $3$ only; i.e., $10$ can be partitioned in the form $2+2+2+2+2$ and $2+2+3+3$. So, given an integer, how to ...
3
votes
0answers
51 views

Problem for number theory

Here it is: $c , n \in \mathbb{N}$ and $x_1,x_2,\ldots,x_n \in \mathbb{N}\cup \{0\}$ $c= 1 x_1 + 2 x_2 + \ldots + n x_n$ How many solutions $\{x_1,x_2,\ldots,x_n\}$ are there? I do not know number ...
4
votes
2answers
387 views

The fibres of a map form a partition of the domain.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
0
votes
0answers
6 views

Relation between singular/eigen values of PCA partition with singular value of all the data

I'm trying to find a relationship between the singular values (or eigen values) of data with the singular value of a partition of that data obtained as follows. The data is partitioned based on the ...
1
vote
3answers
503 views

Proof of the duality of the dominance order on partitions

Could anyone provide me with a nice proof that the dominance order $\leq$ on partitions of an integer $n$ satisfies the following: if $\lambda, \tau$ are 2 partitions of $n$, then $\lambda \leq \tau ...
0
votes
2answers
39 views

If S is finite then the equivalence classes will not exceed the size of set S

If S is a finite set and $\sim$ is an equivalence relation on it, then the total number of equivalence classes can never exceed $\vert S\vert$ and it can be any integer number $1\leq k\leq\vert ...
1
vote
0answers
52 views

Partitioning real numbers with sum $1$ to sets

If the sum of a finite number of positive real numbers is $1$ and each of them is less than $x$, then those real numbers can be partitioned into $50$ sets (some of which may be empty) such that the ...
0
votes
2answers
22 views

What is an algorithm for describing the partition of this equivalence relation?

Let $\mathbb{R}$ be the set of real numbers,$ f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ ℝ $ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid ...
1
vote
1answer
24 views

Quick question for proof on unimodal sequence formula in Enumerative Combinatorics

I am looking at page 238 of Stanley's Enumerative Combinatorics where he says that $\#V_n = \#D_n - \#V_n^1$ because every element in $V_n^1$ appears twice as a value of $\Gamma_1$. Can someone ...
2
votes
0answers
38 views

What are the combinatorial numbers appearing in these repeated derivatives?

Let $f$ be a $C^\infty$-function and define $g(x) = \exp(f(x))$. I am interested in the higher derivatives $g^{(1)}, g^{(2)}, \ldots$ of $g$. Let $\lambda$ be a partition of $n$, i.e. a tuple of ...
0
votes
0answers
26 views

Edge-partition graph into maximum distinctive paths

Is anyone aware of a graph partitioning algorithm in which each partition is a path and where the criteria for partitioning is to maximise the difference between partitions where two paths are ...
11
votes
1answer
248 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $$S(\alpha)=\big\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \big\}.$$ Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers. ...
1
vote
3answers
71 views

Show that ~ creates a partition of $M_2(\mathbb{R})$

Let $M_2 (\mathbb{R})$ be the set of 2x2 matrices over $\mathbb{R}$: $$ M_2 (\mathbb{R}) = \biggl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \; \biggm| \; \text{with } a,b,c,d \in ...
11
votes
8answers
3k views

How many ways can $133$ be written as sum of only $1s$ and $2s$

Since last week I have been working on a way, how to sum $1$ and $2$ to have $133$. So for instance we can have $133$ $1s$ or $61$ $s$2 and one and so on. Looking back to the example: if we sum: $1 + ...
2
votes
1answer
68 views

Covering a rectangle of size $n\times1$ with dominos

A rectangle of size $n\times1$ is given. (a) In how many ways the rectangle can be covered with dominoes of size $1\times1$ and $2\times1$? (b) In how many ways the rectangle can be covered with ...
0
votes
0answers
29 views

Number Partitioning of summands

So, I need to partition the number 133 in 1, 2 and 3. Like $$133 = 128*1 + 1*2 + 1*3$$ $$133 = 126*1 + 2*2 + 1*3$$ $$133 = 125*1 + 1*2 + 2*3$$ Where I always must use at least one 1, 2 or 3. I ...
1
vote
2answers
45 views

Number of Partitions proof

How do I prove that the # of partitions of n into at most k parts equals the # of partitions of n+k into exactly k parts? I was trying to improve my ability of bijective-proofs, unfortunately I was ...
0
votes
0answers
13 views

Indexing for partitions

I'm using spatial hashing for broad-phase collision detection and I'm trying to squeeze some more performance out of it. Currently, it creates a new hashset for every partition which works well ...
0
votes
1answer
18 views

Combining an outcome of a score

Hey I was wondering how many are the possibilities of combining the scored points of a result such $133:75$ from a basketball game? Considering that there are fouls($1$ point), normal($2$ points) and ...
1
vote
1answer
37 views

partitions are the elements of the free abelian monoid on $\aleph_0$ generators

although the following insight may not be particularly profound, it is always of interest when we see a canonical isomorphism between objects of apparently different types. it is a commonplace that ...
0
votes
1answer
30 views

Proving that two equivalence classes are disjoint?

I am having trouble with the following proof: Define the relation $R$ on $\mathbb{Z}$ by $nRm$ if $n-m$ is divisible by $2$. Prove that the equivalence class for $0^{(\bar{0})}$ and the equivalence ...
2
votes
1answer
59 views

Circles partitioning the plane [duplicate]

What is the equation for the maximum number of regions into which N circles can partition a plane? Is there a name for this equation? A single circle partitions the plane into two regions, inside ...
0
votes
1answer
58 views

Partition on a Closed Set A= [2,3]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
1
vote
3answers
229 views

Multiplication partitioning into k distinct elements

Let's say I have a list with the prime factors of a number $n$ and their corresponding exponents. Is there a formula to calculate how many multiplications with $k$ distinct factors of $n$ are ...
0
votes
1answer
40 views

Partition and equivalence relation

Consider the equivalence relation between non-empty subsets $A , B$ of $\{ 1,2,3, 4,\dots,100\}$ defined by the condition: the greatest element of $A$ is the same as the greatest element of $B .$ ...
0
votes
1answer
26 views

Prove an identity with integer partitions

I have already proven this identity: $\prod_i (1+st^i) = 1 + \sum_r \frac{s^rt^{r(r+1)/2}}{(1-t)(1-t^2)\cdots(1-t^r)}$ I expanded the product, grouped the s terms, and then made an argument about ...
6
votes
2answers
86 views

Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.

I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$ where $B_n$ is the $n$th Bell number. I found this result by exploring the series on wolframalpha and looking up the ...
0
votes
0answers
32 views

Can you partition a rectangle into exactly 3 congruent non-rectangular parts?

Recently I came upon the following result: Theorem (*): Let $n$ be a positive integer not equal to $1,3,5,7,9$. Then it is possible to partition a rectangle into exactly $n$ congruent non-rectangular ...
0
votes
1answer
71 views

Exponential generating function of partitions of set [n]

Find the exponential generating function of the partitions of the set [n], all of whose classes have a prime number of elements. The only thing I came up with was $$\sum\limits_{\textrm{t prime}} = ...
2
votes
1answer
136 views

Jee Main 2015 Question. Probabilty

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is: (1) $22 \times(\frac{1}{3})^{11}$ (2) $\frac{55}{3} \times ...
1
vote
0answers
129 views

Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
1
vote
1answer
52 views

How many solutions of equation

How many solutions of equation $x_1+x_2+x_3+x_4=n$ in $N_0$ such that $x_1\leq x_2\leq x_3 \leq x_4$? I found solutions of $x_1+x_2+x_3=n$ in $N_0$ , $x_1\leq x_2\leq x_3 $ in the following way : ...
2
votes
0answers
33 views

Notation for the set of all integer partitions

I'm working on a project that involves that set $P = \{\{n_1, \ldots, n_k\} \mid k \in \mathbb{N}, n_i \in \mathbb{N} \text{ and } n_1 + \cdots +n_k = n\}$ of all integer partitions of a number $n$. ...
0
votes
1answer
38 views

Counting unordered partitions on nested concentric disks

The idea is to think of each layer outside of the [core] as a rotatable disk and then only count a single member from each of the resulting equivalence classes, which I think can be done by requiring ...
3
votes
2answers
377 views

For which $k$ are there most partitions of $n$ into $k$ parts?

Let $P(n,k)$ denote the number of partitions of $n$ into $k$ parts. I would like to know for given $n$, which $k$ does maximize $P(n,k)$? Additionally, information on the maximum of $P(n,k)$, for ...
3
votes
1answer
119 views

Introduction to proofs: proving a set is a partition.

I've been really trying to understand how some of these proofs work; I've spent a majority of my time studying the material for this class, but I'm still performing poorly in it. It doesn't help that ...
0
votes
1answer
19 views

Number of ways to build a collection of numbers where $sum = k$, each $0 < n_i <= d_i$ for some corresponding $d_i$, and sum of all $d_i >= k$

I apologize for any (mis|ab)use of notation since I'm not a mathematician. My background is software engineering and computer science. I ran into this problem while trying to figure out the ...
3
votes
1answer
73 views

Partition problem for consecutive $k$th powers with equal sums (another family)

This is the partition problem as applied to a special set, namely the first $n$ $k$th powers. Assume the notation, $$[a_1,a_2,\dots,a_n]^k = a_1^k+a_2^k+\dots+a_n^k$$ I. Family 1 The following ...