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
70 views

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

Is it possible to define a partition of a closed set, such that the union of the partitions will give $[2,3]$ and their intersection to be empty?
0
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1answer
37 views

Why are is partitions counting technique wrong?

I recently heard about partitions. I tried to count them using the following technique: 1) Ways to write $5$ as a sum of five positive integers: $$1+1+1+1+1$$ 2) Number of ways to write $5$ a sum of ...
0
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1answer
35 views

List the partitions of the set $S = \{1, 2, 3\}$.

I write the partition sets of set $S=\{1,2,3\}$ as follows: $\{\{1, 2, 3\}, \{1\}, \{2\}, \{3\}\}$ can someone show me why and how to complete the list?
0
votes
1answer
27 views

Problem on partioning

On reading the book 'Aha! Solutions' by Martin Erickson I came to know that the number of partitions of $n$ ($n \in \mathbb{N}$) into three parts is $\left\{ {\frac{{{n^2}}}{{12}}} \right\}$ where ...
1
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1answer
10 views

Number of Blocks of Size $i$ in Set Partitions

Given a set of $N$ labeled elements $\{1, 2, ..., n\}$, we know that there are $S(N, k)$ ways to partition the elements into $k$ non-empty subsets (where $S(N, k)$ is the Stirling number of the second ...
-1
votes
1answer
45 views

Conjecture: Partitioning $\Bbb N$ into parts that sum to $13^i$ [on hold]

Recently I was thinking and came up with a conjecture that goes as follows: Conjecture: There exists a $\Omega$ such that $$\Omega = \Bigg\{A_i \ \Bigg| \ \forall i,j:i\not=j, \ A_i\cap ...
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0answers
19 views

Partitioning the set of mappings.

The following is first two steps of an algorithm given from a research paper. I understood the first step. But please explain the second step: what does mean " Rearrange the partition according to ...
0
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0answers
26 views

How can I correctly catalog this partition problem?

Studying the partition problems, I tried to do an special version to apply it to a kind of model of "orbits and energy levels" (explained below), but I am having problems to properly catalog this. ...
0
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1answer
43 views

Partitions of a number with greatest product

For $n\in\mathbb{N}$ choose $k_1,\dots,k_l\in\mathbb{N}$ so that $\sum_{i=1}^{l}k_i = n$. Set $k = \prod_{i=1}^{l}k_i$. What is the largest $k$ that one can get? Is there an explicit formula? What ...
1
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0answers
25 views

Generating functions, Schur's identity

Let $S=\{n\in \mathbb{Z}_+ \mid n \equiv 1, 5 \,\,(\text{mod 6})\}.$ Let $a(n)$ be the number of partitions of $n$ into parts belonging to $S,$ and $b(n)$ be the number of partitions of $n$ into ...
1
vote
1answer
20 views

Index set of dyadic partition

Imagine if you have a line from 0 to 1, and you begin partitioning it dyadically. The first point will be at 0, the second at 1 and the third at 0.5, the fourth at 0.25, the fifth at 0.75 etc. Let's ...
0
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0answers
70 views

Partitions into non-negative powers of $2$

Let $c(n)$ denote the number of partitions of $n$ into non-negative powers of $2.$ (Thus $c(5)=4$ since $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1).$ (a). Prove that $1+\sum\limits_{n=1}^{\infty} c(n) ...
1
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0answers
53 views

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

Given $g \ge 2$, $k \ge 1$ 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; ...
1
vote
1answer
16 views

Describing a partition as a function

Let P be a partition such that P $=\{t_0,...,t_n\}$ over the interval $[a, b]$, then to refer to a point $i$ in the partition $P$, some would say $t_i$. So my question would be then, in this case ...
1
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0answers
22 views

Partitions of $n$ vs $(n-k_0)! $

Let $p(n)$ denote the partitions of $n$. It's easy to prove that $p(n) < n!$ (for n>2). I want to prove that if $n \ge 6$ then $(n-2)! > p(n)$. Or more generally let $k_0\in \mathbb{N}$ be a ...
0
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0answers
15 views

Inclusion/exclusion argument for partitions

My question regards Frobenius partitions, or $F$-partitions for short, of a number $n$. A short explanation of the concept is linked below. Specifically, my question is as follows. $F$-partitions of ...
-2
votes
1answer
36 views

Number of partitions into parts not greater than 9 [closed]

I'm looking for a closed-form formula for the number of partitions of integer $n$ into integer parts less than or equal to 9. Thanks.
-2
votes
1answer
50 views

Number of different groups given a list of repeating digits

Suppose that you are given the list[1,1,2,2] . The different groups that can be formed with this list are - ...
1
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0answers
23 views

Understanding the expansion of product notation.

I have a question regarding the expansion of product notation in the picture below. Equation 3.1 in the attached picture is ...
0
votes
1answer
13 views

Determine if given relation is an equivalence relation, and describe the partition?

Can someone walk me through how to do these types of problems in my Discreet Mathematics II Textbook? (ex.): Determine whether the given relation is an equivalence relation on the set. Describe the ...
3
votes
4answers
50 views

Equivalence relation on $\mathbb{N}$ with infinitely many equivalence classes

Does there exist an equivalence relation, defined on $\mathbb{N}$, with infinitely many equivalence classes, all of which contain infinitely many elements? I see no reason for such a relation not to ...
1
vote
1answer
37 views

Decomposition of $\mathbb N$ into mutually disjoint infinite subsets

$$\mathbb N =\bigcup_{j\in \mathbb N}\Delta_j $$ where each $\Delta_j$ is an infinite subset of $\mathbb N$ and $\Delta_j\cap \Delta_i=\Phi \ for\ i\neq j.$ Now what I need is a few examples of such ...
1
vote
2answers
41 views

Formula for Number of possible N element sequences such that their Sum is S?

How many ways I can choose a $N$ element sequence such their cumulative is $S$? Is there any formula for it? Values of $N$ will be greater Than $0$. Here are few examples Let $ N=4$ and $S=5$. Their ...
3
votes
0answers
24 views

How to call a partition of $X$ which consists of all singleton subsets of $X$? [duplicate]

In other words, if $X$ is a set, then how do we call $Y=\{\{x\}:x\in X\}$? $\{X\}$ is already named the trivial partition, so that cannot be it. Complete partition and total partition did not yield ...
4
votes
2answers
159 views

A formal name for “smallest” and “largest” partition

Consider a set $A=\{1,2,3,4,5\}$, is there any terminology for the following partitions of $A$ ? (1) $A=\{ \{1\},\{2\},\{3\},\{4\},\{5\} \}$ (2) $A=\{\{1,2,3,4,5\}\}$.
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0answers
22 views

Finding asymptotycs of partition function

I have been stuck in this problem and have no idea of how to solve it. I have a hint from the book but don't really see how to use it. Any suggestion or hint would be really appreciated. Thanks! ...
1
vote
1answer
36 views

Generating function for the partition function [duplicate]

Could someone explain what is the reasoning behind the following equality? Or maybe direct me to a proof of the following equality? $$\sum_{n=0}^{\infty}p(n)x^n = \prod_{k=1}^{\infty}(1-x^k)^{-1}$$ ...
2
votes
2answers
82 views

How many combination of $3$ integers reach given number?

I have 3 numbers $M=10$ $N=5$ $I=2$ Suppose I have been given number $R$ as input that is equal to $40$ so in how many ways these $3$ numbers arrange them selves to reach $40$ e.g. ...
0
votes
2answers
32 views

Formula for the number of partitions of 2N elements [duplicate]

We have a set $S$ of $2N$ distinct elements. I want to partition it into $N$ parts each containing 2 elements. My motivation is partitioning a group of people into pairs. What is the formula that ...
15
votes
1answer
201 views

Partition of ${1, 2, … , n}$ into subsets with equal sums.

The following is one of the old contest problems (22nd All Soviet Union Math Contest, 1988). Let $m, n, k$ be positive integers such that $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers ...
1
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1answer
35 views

How many partitions of $n$ are there?

Considering a partition to be an ordered $n$-tuple $(m_1, m_2, m_3, ..., m_n)$ with all the numbers $m_i$ natural, $m_1 \le m_2 \le m_3 \le ... \le m_n$, and $m_1+m_2+...+m_n=n$: how many of those ...
2
votes
1answer
24 views

Partition $\lambda/\mu$ notation?

When talking about two partitions $\lambda$ and $\mu$, what does the operation $$\lambda/\mu$$ mean? When Macdonald introduces partitions in the first chapter of "Symmetric functions and Hall ...
2
votes
0answers
30 views

$ \forall n \in N$ edges of $K_{2n+1}$ can be partitioned to hamiltonian cycles.

A hamiltonian cycle is a cycle which visits each vertex of the graph exactly once. A hamiltonian path is a path which visits each vertex of the graph exactly once. We need to prove that: ...
0
votes
2answers
35 views

If $n$ is a natural number $\geq 2$, how many ways are there to partition $n$ with natural numbers $\geq 2$?

The partitions are "ordered", meaning that for 5, from (2,3), (3,2) and (5) all are valid. So for the first few numbers one gets: ...
1
vote
4answers
3k views

Partition of a set, definition not clear

From wikipedia: Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and: The union of the elements of P is equal to X. (The elements of P are said ...
0
votes
1answer
60 views

Partition function and Fibonacci n-th number upperbound

i need to proof this upperbound: "Call $p(n)$ the number of partitions of n (integer) and $F_n$ the $n-th$ Fibonacci number. Show that $p(n)\leq p(n-1) + p(n-2)$ and that $p(n)\leq F_n$ for $n\geq ...
5
votes
2answers
182 views

How many ways to choose $a<b<c<d<e$ from the set $\{1,2,3,\cdots,100\}$ such that $100<a+b+c+d+e<145$?

I would appreciate if somebody could help me with the following problem In how many ways can I choose a five number $a,b,c,d,e(a<b<c<d<e)$ from the set $\{1,2,3,\cdots,100\}$ such that ...
6
votes
3answers
198 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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0answers
22 views

Using Young Diagram to show the number of partitions of n is equal to number of partitions of 2n into n parts

Show that the number of partitions of the integer $n$ is equal to the number of partitions of $2n$ into $n$ parts using a Young Diagram. I can't seem to figure out any way to create a bijection ...
0
votes
0answers
44 views

Pairs of Numbers such that the sum of their digits is Equal

How many pairs of numbers $(n,m)$ whose digits add up to the same sum, where $n\ne m$ and $(n,m)=(m,n)$ such that $m,n\le k$ , are there for a given $k$? Observing this in base 10 we are looking at ...
2
votes
2answers
44 views

Find the exponential generating function for the number of ways to distribute $r$ distinct objects into five different boxes

Find the exponential generating function for the number of ways to distribute $r$ distinct objects into five different boxes when $b_1<b_2\le 4$, where $b_1,b_2$ are the numbers of objects in boxes ...
3
votes
2answers
33 views

Non-Theoretical Applications of Partitions of Unity

I am studying partitions of unity in Munkres' $\textit{Analysis on Manifolds}$ book. Are partitions of unity just theoretical tools, i.e. used to prove theorems, or do people actually apply them ...
1
vote
1answer
29 views

Riemann Sum to show convergence help?

I have the function $f(x)=\sin\left(\dfrac{\pi x}{2}\right)$ on the partition of $[0,1]$ given by $$P_{n}: 0 < \frac{1}{n} < \frac{2}{n} < ... < \frac{n-1}{n} < 1$$ I have shown that ...
2
votes
2answers
449 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 ...
2
votes
0answers
20 views

show $\sum_{j=0}^n (-1)^j {n \brack j}_q =0$ for n odd

I would like to show $\sum_{j=0}^n (-1)^j {n \brack j}_q =0$ for n odd, or preferably even more generally that $\sum_{j=0}^n (-1)^j {n \brack j }_q =\frac{1}{2}((-1)^n+1)(q;q)_{\frac{n}{2}}$. Using ...
1
vote
1answer
33 views

Schur Decomposition Upper Triangular Matrix Partition

On page 21 of Matrix Differential Calculus by Magnus and Neudecker (3rd ed, ISBN:0-471-98632-1), the book states, without any apparent justification, that to prove the statement: If $A$ has $r$ ...
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0answers
19 views

How to derive the proof for the “intermediate series” in the Euler Transform?

Consider the description in Wolfram Alpha of a "third kind" of Euler Transform (http://mathworld.wolfram.com/EulerTransform.html), expressions (5), (7), and (8): (5) $1+\sum_{n=1}^\infty ...
0
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0answers
19 views

Is there a standard representation for the composition of an integer?

In the same way as "$\vdash$" represents integer partition, what would be the symbol for composition (ordered partition)?
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0answers
28 views

How to find last digit of number of partitions?

Is there simpler way to find last digit or last k digits (in any base) of p(n) than calculating full number?
10
votes
2answers
183 views

Identity involving pentagonal numbers

Let $G_n = \tfrac{1}{2}n(3n-1)$ be the pentagonal number for all $n\in \mathbb{Z}$ and $p(n)$ be the partition function. I was trying to prove one of the Ramanujan's congruences: $$p(5n-1) = 0 \pmod ...