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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49 views

Number of Sets of Partions

I looked at the partitions of numbers, like let's say $n=5$. You get $$ \begin{eqnarray} 5&=&5\\ \hline &=&4+1\\ &=&3+2\\ \hline &=&3+1+1\\ &=&1+2+2\\ \hline ...
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1answer
20 views

What are the number of possible partitions of a set containing n elements?

This question rises immediately if we try to enumerate the number of possible equivalence relations on a set with n elements.
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2answers
41 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
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2answers
42 views

If L(P, f) = U(P, f), prove that f is a constant function on [a, b]

That is what I have for the beginning of my proof, but I'm not sure how to conclude that the function is constant.
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0answers
44 views

Riemann integral by Riemann sums

If you are confused as to what I want it follows the method listed in this question, Riemann integrals real analysis Help? What am I doing wrong Trying to apply what you did above to the integral ...
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1answer
29 views

Counting the number of possible matchups for teams

A tournament has 16 teams. How many ways are there to match up the teams in 8 pairs? Is it (16 choose 2)(14 choose 2)(12 choose 2)(10 choose 2)(8 choose 2)(6 choose 2)(4 choose 2)(2 choose 2)?
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1answer
14 views

Different flag signal questions

How many different signals can be created by lining up 9 flags in a vertical column in 3 flags are white, 2 are red, and 4 are blue? Is it 9 choose 3 * 6 choose 2 * 4 choose 4?
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1answer
376 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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1answer
34 views

Prove that the number of partitions of $n$ into $3$ parts is equal to the number of partitions of $2n$ into $3$ parts, each of size less than $n$.

Prove that the number of partitions of $n$ into $3$ parts is equal to the number of partitions of $2n$ into $3$ parts, each of size less than $n$. I do not have any idea to prove this ...
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0answers
33 views

Minimize sum of squared error

I have an array of real numbers, I want to partition them into k sets. In each set, I calculate the sum of squared error. Then, I add up all the sum of squared error for all the set. I want to ...
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2answers
28 views

Prove that the number of partitions of $2010$ into $10$ parts is equal to the number of partitions of $2055$ into $10$ distinct parts.

Prove that the number of partitions of $2010$ into $10$ parts is equal to the number of partitions of $2055$ into $10$ distinct parts. How we can prove this?My idea is to construct a bijection but I ...
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0answers
30 views

Partition problem of equal size

I have an array S of size 2n, each element in the array is an integer I want to split it into two arrays of size n, and under this condition, minimize the difference of the sum of integers in the two ...
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0answers
29 views

Partition partition with constraint of equal size

I see the problem here Polynomial complexity algorithm of partition problem with sets of equal size This is the well know partition problem but with constraint that the size of both sets must be ...
4
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1answer
60 views

Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
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0answers
34 views

number of pairs that are member of a class of one of the given partitions

Suppose I am given a set of S partition of universal set A, define pair as (a,b) iff elements a,b are members of one of set in at least one partition. I want to know how many pairs in S. Example: $$A ...
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0answers
20 views

What are the number of “minimal 2-complete” partitions for the first 100 natural numbers?

Let $λ = (λ_0, λ_1, . . , λ_n)$ be a partition of the natural number $m$ into $n+1$ parts $λ_i$ such that: 1) $m = λ_0+ λ_1+ . . . . + λ_n$ 2) $λ_0 ≤ λ_1 ≤ . . . . ≤ λ_n$ The partition $λ = ...
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1answer
32 views

Find all Combinations of 1 and 2 which sums up to k.

I have two numbers $1$ and $2$. I have to print all ordered combinations which sums up to $k$. For example: $k=1$ Its only $1$. $k=2$ It's ${1,1},{2}$. $k=3$ Its ${1,1,1},{1,2},{2,1}$ What ...
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1answer
27 views

Proof of Partitions

Let $|n,k|$ denote the number of partitions of $n$ into $k$ distinct parts. Prove $$|n,k| = |n-k,k-1| + |n-k,k|$$ Workings: LHS counts the number of partitions of $n$ into $k$ distinct parts. RHS: ...
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2answers
52 views

Combinatorics] Partition Numbers

Let $R(n,k)$ denote the number of partitions of $n$ into $k$ (non-empty) parts. That is for example $R(7,2) = 3$ because it can be expressed as $1+6, 2+5$ and $3+4$. Prove that: $R(n,1) +R(n,2) + ...
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1answer
32 views

Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...
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0answers
46 views

Degree distribution of a graph of integer partitions

Consider a graph whose nodes are the integer partitions of $n$, connected by an edge if you can get from one partition to the other by ‘sliding one square of its Young diagram’ – that is, if one ...
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0answers
55 views

The number of partitions by distinct positive numbers

Let $N>0$ be a natural number and let $P(N)$ denote the number of ways to write $N$ as a finite sum of $a_i$ such that the $a_i$ are strictly decreasing positive natural numbers. There is a paper ...
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1answer
37 views

Partitioning a finite set which sums to $n$

Given $n > 1$, we consider the finite sets of positive integers which sum to $n$, and out of these sets we want to maximize the product. For example, given $n = 6$, the set $\{1, 5\}$ does not ...
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0answers
38 views

Proving Identities using Partition and Generating Function

I have a problem with these two questions: Let $P_E(n)$ be the number of partitions of $n$ with an even number of parts, $P_O(n)$ the number of partitions of n with an odd number of parts, and ...
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1answer
39 views

Given the partition list the ordered pairs in the corresponding equivalence relation.

Given the partition $\{a,b,c\}$ and $\{d,e\}$ of the set $S=\{a,b,c,d,e\}$, list the ordered pairs in the corresponding equivalence relation. I'm not really sure how to get started on this and would ...
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4answers
23 views

Choosing three pairs out of eight items

3 students each choose two problems from a list of eight problems. How many ways can this be done? The answer in the text book gives 8!/(2! 2! 2! 2!), but don't we also have to multiply by the ...
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5answers
360 views

How many sets of distinct non-negative solutions are there to $k_1+\cdots+k_n=k$?

How many distinct $n$-tuples with distinct non-negative integer elements are there that add to $k$. For example there are $6$ triples that add to $4$. Namely $(0, 1, 3)$ and its $6$ permutations. ...
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0answers
160 views

Does $A193201$ count the partitions of $n$ of arbitrary dimension?

By my count, these sequences match for $n=0\ldots6$, where partitions that are the same after relabeling dimensions are considered equivalent (i.e., the dimensions are unordered). For example, for ...
3
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1answer
54 views

Demystifying the asymptotic expression for the partition function

A partition of an integer $n$ is a way of writing $n$ as a sum of integers. The partition function $p(n)$ counts the number of distinct partitions of $n$. In 1918, Hardy and Ramanujan proved the ...
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1answer
31 views

Equivalence classes, relation, partitions

Let $\sim$ be an equivalence relation on M, and $M/{\sim}$ to be the partition of M by $\sim$, and $\sim_{M/\sim}$ to be the equivalence relation by the partition. How do I show that the two ...
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1answer
88 views

Generating functions for partitions of n with an even number of parts and odd number of parts, and their difference.

I've been trying to figure this out for more than 10 hours. So far I have, for even number of partitions, $$P_e(x)=\sum_{k\ge1}(x^{2k}\prod_{i=1}^{2k}\frac{1}{1-x^i})$$ and for odd numbers ...
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0answers
33 views

Partitions of numbers

It needs to prove that the partitions of n, in which all the terms do not exceed k, so same number of partitions of n into at most k non-zero terms. I tried to prove it by using Stirling numbers of ...
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0answers
19 views

Existence of finite Darboux sum with infinite partition

I would like to describe the class of all functions $a\in L^1(\mathbb{R},dx)$, such that there exists $\tilde{a}=a$ a.s. and a size $h$ of an infinite partition of $\mathbb{R}$, such that ...
3
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2answers
35 views

Partitions without 2

How do I find the generating function for partitions of $n$ that have no part with size $2$? In general, how would I find this for partitions that have no part of size $k$?
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0answers
17 views

K-way Undirected Weighted Graph Partition with K Vertices Pre-Assigned

I have an undirected weighted graph to be partitioned into k subgraphs with minimal edge weight between the partitions and k of the vertices are constrained to lie in separate partitions. I am ...
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0answers
81 views

Real Analysis: Show that g is integrable on [a,b] and that $\int_a^b$ $g(x)dx=$ $\int_a^b$ $f(x)dx$

Suppose f is integrable and g is bounded on [a,b], and g differs from f only at points in a set H with the following property: For each $\epsilon>0$, H can be covered by a finite number of closed ...
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1answer
55 views

solving combinatorial problem using partition functions

How many natural numbers less than 99000 have the sum of the digits equal to 8. This is what I tried to do.Let $x_i$ be the ith digit for any $i \in \{1,2,3,4,5\}$. Ways of creating numbers less ...
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1answer
38 views

number of integer solutions combinatorial problem

Find the number of integer solutions to $x_1+x_2+x_3+...+x_7=23$ subject to $x_1\gt0,x_2\ge3$ and $x_i\gt0$ for all $i\ge3$. This is the given answer in the book: Using the substitutions $y_1=x_1-1$ ...
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0answers
35 views

Special partition of a number $n$

Given any integer $n$, how many ways can it be partitioned in which the number $1$ is not allowed? For instance, if $n=6$, then the partitions obeying the aforementioned rule are $6+0$, $4+2$, $3+3$, ...
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1answer
25 views

equivalence Relation problem with some conditions

If A be a set with $|A|=n$. if R be a equivalence Relation on A and $|R|=r$, why $r-n$ always be even ?
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1answer
41 views

Sum of positive integers estimating sum of fractions

Given $m$ fractions adding up to an positive integer $n$ For example: $m=3\\n=10=\frac{30}{6}+\frac{20}{6}+\frac{10}{6}$ How can we find $m$ positive integers that sum to $n$ (a partition of $n$), ...
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7answers
170 views

Infinite partition of $\mathbb N$ by infinite subsets

I am looking for an explicit partition of $\mathbb N$ with the following condition: $$\mathbb N=\bigsqcup_{i\in\mathbb N}A_i$$ where all the $A_i$'s are infinite. What I mean by explicit is a formula ...
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0answers
36 views

Standard form for partitions of $Z_n$

Let $A$ and $B$ be partitions of $Z_n$. Let's say that $A$ and $B$ are equivalent if $A=Bx+y$ for some $x\in{1,−1}$ and $y\in Z_n$. In other words, two partitions are equivalent if one can be obtained ...
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1answer
67 views

Friendly graph partitioning

The question is from the "Introduction to Algorithm" 3rd edition: B-2 Friendly graphs: Reword the following statements as a theorem about undirected graphs, and then prove it. Assume that friendship ...
2
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1answer
42 views

Intuition for Euler's Partition Theorem

Euler's Partition Theorem states the following: Every number has as many integer partitions into odd parts as into distinct parts. I played around with small examples (I wrote out the partitions ...
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3answers
127 views

Counting problem: generating function using partitions of odd numbers and permuting them

We have building blocks of the following lengths: $3, 5, 7, 9, 11$ and so on, including all other odd numbers other than $1$. Each length is available in two colors, red and blue. For a given number ...
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1answer
31 views

Partitions without twice odd numbers and where every odd number appears at most once

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 ...
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0answers
27 views

Ferrers Graph question

Use Ferrers diagram to show that the number of partitions of an integer into parts of even size is equal to the number of partitions into parts such that each part occurs an even number of times ...
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1answer
75 views

General term of $(1+x)(1+x^2)(1+x^3)…$?

Is there a closed for the coefficient of $x^n$ in $(1+x)(1+x^2)(1+x^3)\cdots$? If not, then what is the closest to a closed form that anyone has found? (An infinite series that approximates it ...
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2answers
58 views

How to calculate the number of ways of partitioning n identical objects into r different groups such that each group has same number of objects?

I was solving the following problem, "Given a collection of 10 identical objects calculate the number of ways in which these objects can be partitioned into 2 groups of 6 and 4 objects each" - for ...