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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1answer
20 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 ...
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1answer
44 views

Circles partitioning the plane

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 ...
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1answer
47 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?
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1answer
104 views
+100

Minimizing over partitions $f(\lambda) = \sum \limits_{i = 1}^N |\lambda_i|^4/(\sum \limits_{i = 1}^N |\lambda_i|^2)^2$

I'm trying to characterize the behavior of the the quantity: $$A = \frac{\sum \limits_{i = 1}^N x_i^4}{(\sum \limits_{i = 1}^N x_i^2)^2},$$ subject to the constraints that $$ \sum \limits_{i = 1}^N ...
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3answers
41 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 ...
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1answer
33 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 $B .$ ...
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1answer
19 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 ...
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2answers
76 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 ...
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0answers
26 views

Determine the number of partition of 20 into at most 5 parts.

Im stuck on these questions, I can kind of compute them on maxima, but I have to figure out why and how to get to a particular generating function or method. Determine the number of partition of 20 ...
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0answers
19 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 ...
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1answer
56 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}} = ...
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1answer
68 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 ...
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0answers
47 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 ...
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1answer
48 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 : ...
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0answers
20 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
32 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
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2answers
365 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
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1answer
65 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
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1answer
17 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
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1answer
59 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 ...
5
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0answers
83 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
7
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1answer
109 views

Characters of the symmetric group corresponding to partitions into two parts

Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
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2answers
56 views

Sums of Consecutive Cubes (Trouble Interpreting Question)

Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums. Any suggestions on what techniques should be used to ...
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0answers
12 views

On t-core partitions

How exactly can one define what is known as a t-core partition? I know (vaguely) that it involves the definition of what is known as "Hook numbers". Anyone cares to provide a link or explain it?
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4answers
26 views

Number of sequences formed of $k$ pairwise disjoint subsets of a set of $n$ elements is $(k+1)^n$.

Let $S=\{1,2,\dots,n\}$ and $P(S)$ the family of the $2^n$ subsets of $S$. Prove that the number of sequences $(S_1,S_2, \dots, S_k )$ formed by the subsets of $S$ that verify that $S_i \cap S_j = ...
3
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1answer
53 views

Does and where to does $\lim_{n\to\infty}\sum_{m} \prod_k \frac{1}{\lambda_{k,m}!}$ converge?

Given $n$ you get a number of partitions of $n$ and let's denote $\lambda_{k,m}$ to be the $k$th part of the $m$th partition. Now I built the following sum, that stimulated the following question: $$ ...
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0answers
28 views

Terminology: Opposite of “refinement”

Let A be a partition of a set, and B a refinement of A. Fill in the blanks: A is a __________ of B. I know that A is coarser than B, but how does one turn that into a noun?
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2answers
22 views

Proving a family of sets is a partition of the set of integers.

I am trying to show that for $E_n = \{10n, 10n + 1, 10n + 2 . . . , 10n + 9\}$, $\{E_n\}$, $n ∈Z$ is a partition of $ Z$. Should this be broken down into cases or is there a more general way to ...
2
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0answers
48 views

Recurrence relation / recursive formula / closed formula for partition numbers (1,2,3,5,7,11,15,22,30,42,…)

I have already read this: Number partition - prove recursive formula But the formula from the above link requires a parameter k which is the required number of ...
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0answers
28 views

dot diagram prove

I have to prove theorems using dot diagrams. a.- $P_n(k) = p_n(k-n)$ b.- $p_n(k) = p_{n-1}(k) + p_n(k-n)$ c.- The number $P_n(k)$ of partitions of $k$ into exactly $n$ parts is equal to the number ...
3
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0answers
70 views

Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity. (for example, H. H. Chan’s proof, M. Hirschhorn's ...
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0answers
18 views

Find largest regions bounded by a set of planes

Suppose we are given a set of planes that partition the unit cube into a large number of regions. Is there a computationally efficient way to find the region with the largest volume?
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0answers
43 views

Proof of Ramanujan's Identities of Euler's Function

Consider Euler's Function defined as (and not to be confused with the totient!) $$ \phi(x) = (1-x)(1 - x^2) (1 - x^3 ) .... = \prod_{i=1}^{\infty} \left[(1 - x)^i \right] = (1;x)_{\infty}$$ ...
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1answer
56 views

Sample, randomly & uniformly the partitioning of $n$ objects into $K$ groups

I wish to sample randomly and uniformly from the set of partitions of $n$ objects into $K$ groups where the number to be assigned to each group, $n_k$ ($k = 1, 2, \dots, K$) is known. I know that the ...
3
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0answers
46 views

Counting the number of partitions that are a distance d away from a fixed partition.

Given a positive integer $N \in \mathbb{Z}^{\geq 0}$ let $Partitions(N)$ denote the set of all partitions of $N$, where a tuple $\left(f_1,\ldots,f_N \right)$ is a partition of $N$ if $\sum_{i=1}^N ...
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1answer
46 views

Partitioning a set to get a sum

I have a set of numbers: 2,2,4,4,4,4,4,4,6,6,6,6,6,6 I want to enumerate the possible ways to partition this set into 4 groups, each of which sum to 16. How can I approach this short of brute force? ...
2
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0answers
23 views

Link between partition function and ordered partition function

The partition function $p(n)$ measures the number of partitions of $n$, or the number of ways in which natural numbers can be summed to produce $n$, without regard to order. For example, the ...
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1answer
29 views

Generating function for set partitions

Let $k$ be fixed. For every $n$ denote by $p_{\leq k}(n)$ the number of partitions of the integer $n$, for which each part is at most $k$. a). Compute $p_{\leq 3}(5)$ b). Compute the generating ...
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2answers
351 views

Partitions of a set into three parts

How many partitions of the set $\{1,2,3, \ldots , 100\}$ are there such that both a) there are exactly three parts and b) elements $1,2,3$ are in different parts. Any help on this question would ...
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0answers
17 views

Can I partition a non-separable set?

This question flashed in my mind, as I was studying quotient partition of a set. Is there any non-separability condition for which a topological set cannot be partitioned? Particularly, I tried ...
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0answers
48 views

Prove; regarding partitions and refinements

If $ f $ is a bounded function, then for every $ \epsilon >0 $ there exists a partition of a real interval $ X $ such that if $ Y $ is a refinement of $ X $. Why is this true?
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1answer
47 views

Prove that A001399 == A069905

While solving PE and reading SICP I found, that there are two problems, that produce the same OEIS sequence: http://oeis.org/A001399 is a number of partitions of n into at most 3 parts ...
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2answers
58 views

Prove: Partitions and refinements

Problem: Let $ R $ be the set of partitions of a real interval. Then for all elements in $ R $, every pair of elements has an upper bound. I am having trouble structuring the proof; and intuitively ...
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1answer
31 views

A fast algorithm for a simple multi-objective minimization?

I have a set $S$ consists of n (arbitrary) integer numbers which I want to partition into $k$ subsets $S_i$ each of size $\frac{n}{k}$ (assume $k$ divides $n$). Let $A$ be the arithmetic mean of ...
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0answers
35 views

Generalization of Catalan numbers

I am looking for some kind of function describing the number of non-crossing partitions similar to those described by the Catalan numbers. Let's say $C_3$ would be the third Catalan number. $C_3=5$ ...
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1answer
16 views

Counting resticted partitions of a multiset with additional restrictions

Say I have some multiset of integers, for example $M1=\{6,6,4,4,4,2,2\}$. I have a second multiset that consists of some set of valid sums derived from picking without replacement from $M1$, say for ...
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0answers
30 views

Does Euler's recurrence relation for partitions imply that the partition function grows exponentially

Can one, just by manipulating the series, demonstrate that the partition function must be growing exponentially or at least that it is unbounded by any polynomial? If so, then how would it be done. ...
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0answers
17 views

What is wrong with this “inference” about partitions?

Given that: $p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+⋯$ Why can one not state: $p(n)≥p(n-1)+p(n-2)-p(n-5)-p(n-7)$ Here is the logic: the subsequent 2 terms of the relation are additive and the 2 ...
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1answer
598 views

Divide a set of $n$ elements into $k$ subsets having equal sum

Given $n$ ( $n$ <= 20) non-negative numbers. Is there / Can there be an algorithm with acceptable time complexity that determines whether the n numbers can be divided into $k$ ( $k$ <= 10) ...
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0answers
23 views

Famous Arithmetic property of lotteries - restricted partition of integer into exactly k distinct parts between a given set

I would like to find a complete explanation regarding a famous arithmetic property of lotteries : Let´s say a friend of us is a regular player of a lottery where 5 numbers are taken from a box ...