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

partitions of finite set in same-size parts having at most one element in common

Given g ≥ 2, k ≥ 1 and a population of p = kg workers, I'm trying to figure out the longest series of work shifts such that: during each shift, all workers work in k teams of g people; any two ...
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0answers
71 views

Expansion for r-associated Stirling numbers of the second kind

I am looking for a paper or guidance for expanding the r-associated Stirling numbers of the second kind $S_r(n,k)$. $S_r(n,k)$ is the number of ways to partition a set of n objects into k subsets, ...
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0answers
44 views

Expected Max Pseudotree Size

I'm working on a problem where I need to calculate the expected maximum pseudotree size in a randomly-generated pseudoforest with $n$ nodes. Expected maximum value is of course: $$ E(x) = ...
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2answers
105 views

Counting Set Partitions with Constraints

I'm trying to count set partitions under the following constraints: The number of partitions on $n$ elements where the largest cell(s) have exactly $k$ elements All cells have at least two elements ...
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0answers
51 views

Truncation of partitions generating function question

$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $ $$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
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1answer
33 views

Union of each family is not the whole set

Let $n\geq k>0$, and consider all $\binom{n}{k}$ subsets of $A=\{1,2,\ldots,n\}$ of size $k$. We want to partition it into families so that the union of each family is not equal to $A$. At least ...
6
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1answer
210 views

Number of Sets of Partitions

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
31 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
114 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
62 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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1answer
36 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
35 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?
1
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1answer
651 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) ...
0
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1answer
45 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 ...
0
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0answers
53 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 ...
3
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2answers
32 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 ...
0
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0answers
80 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
42 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 ...
6
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1answer
90 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 ...
0
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0answers
40 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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1answer
54 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
37 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
80 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
36 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
52 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 ...
3
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0answers
64 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
42 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
111 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 ...
1
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1answer
140 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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3answers
25 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
434 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
172 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
160 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
37 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
163 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
25 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
37 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
22 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 ...
0
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0answers
89 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
91 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
47 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 $y_1=x_1-1$ ...
0
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0answers
42 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
29 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
45 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
178 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
40 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
96 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
48 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 ...
6
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3answers
162 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
53 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 ...