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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124 views

Partition of unity. Functional analysis

Need to find partition of unity in case of oparator $A_{f}(x)=(|x-1|+x)f(x)$. Operator $A \in L_{2}[0,2]$ Partition of Unity is set of operators $E_{\lambda}=E((- \infty,\lambda]) $, where ...
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2answers
221 views

Count the number of integer solution to $\sum_{i=i}^{n}{f_ig_i} \geq 5 $

How to count the number of integer solutions to $$\sum_{i=i}^{n}{f_ig_i} \geq 5$$ such that $\displaystyle \sum_{i=1}^{n}{f_i}=6$ , $\displaystyle \sum_{i=1}^{n}{g_i}=5$ , $\displaystyle 0 \leq f_i ...
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1answer
652 views

partition a number N into K tuples

Given an integer $N ≥ 0$ and an integer $K ≥ 0$, how many tuples $(n_1,\ldots,n_k)$ are there such that $n_i ≥0$ and $\Sigma n_i = N$? In other words, how many way can you "partition" $N$ into $K$ ...
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37 views

order of elements in a partition using Maple

I determined this whole partition but I just want to have the finer the partition for example: I have this ...
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1answer
205 views

When is the sum of first $n$ numbers equal to the sum of the next $k$ numbers?

When is the sum $1+2+\cdots + n = (n+1) + (n+2) + \cdots +(n+k)$? The easiest solution $(n,k)$ is $(2,1)$. For example, $1+2 = 3$. Do any others exist? Roots of $(n+k)^2 + (n+k) = 2n^2 +2n$ give ...
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1answer
121 views

What is the size of partitions versus subsets?

I know this might be a newbie question, but it's confusing to me. What is a partition, does it only apply to integers? And how does the size of it compare to the size of subsets? Also, suppose we ...
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1answer
204 views

Total number of solutions of an equation

What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 < x_2 < \cdots < x_r < N$ where $N$ is some natural number and $x_1, ...
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1answer
132 views

Bijective Proof - partitions

what is a bijective proof of #93 in the link provided http://math.mit.edu/~rstan/bij.pdf: The number of partitions of $n$ with $k$ parts equals the number of partitions of $n + \binom{k}{2}$ with ...
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1answer
747 views

Generating Function for Even Number of Odd Parts

How I would write the generating function for a partition of a positive integer n with an even number of odd parts? Any hints or suggestions will be greatly appreciated.
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2answers
145 views

Unique sum of $k$ triangular numbers into $k$ triangular numbers?

First things first. I've made this a seperate thread off my previous topic so that these topics don't conflict if I'd posted this to my previous question. I believe any $k$ triangular numbers will ...
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2answers
78 views

Partitions of sums of $k$ random odd/even $r$th powers from array of consecutive $r$th powers

Like my previous question, I'll pose this one too with an array. $1^r, 3^r, 5^r, 7^r, 9^r$ (all odd $r$th powers) That's array 1. And array 2; $2^r, 4^r, 6^r, 8^r, 10^r$ (all even $r$th powers) ...
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1answer
457 views

Finding a partition $P'$ over an interval for which $U(f, P')-L(f,P') \lt 2 $

If $f:x \to 2x+1$ over the interval $[1,3]$ and $\mathcal P$ be a partition consisting of the points $\{1,\frac32,2,3\} $, how do I find a partition $\mathcal P'$ of $[1,3]$ for which ...
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2answers
76 views

Subsets of $\{1, 2,\ldots, n\}$ which add up to $n$

Problem: Given a number $n,$ we want to find out the subsets of $\{1,2,\ldots,n\}$ that add up to the given number $n.$ Example: If $n=6,$ then the output is: $\{1,5\}, \{2,4\}, \{1,2,3\}.$ ...
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1answer
2k views

For what coinage systems does a greedy algorithm not work in providing change?

For the United States coinage system, a greedy algorithm nicely allows for an algorithm that provides change in the least amount of coins. However, for a coinage system with 12 cent coins, a greedy ...
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1answer
137 views

Equality of times blocks of size $k$ appear in a partition with the number of distinct parts appearing $\geq k$ times?

I've been reading up on combinatorics in my spare time recently. Suppose $\lambda$ is a partition of $n$, and $f_k(\lambda)$ the number of times that $k$ appears as a part of $\lambda$. Also, let ...
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0answers
139 views

Given $x = \sum_{i \in k} a_i$ from $\{ a_1, \ldots, a_n \}$, partition $x$ as $\sum_{j \in k} a_j$

Let me put this into an example. We have an array of the first $r$ (example $r=7$) natural numbers; $1, 2, 3, 4, 5, 6, 7$ and given $k=3$, so you get to select any three integers from the pool ...
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1answer
601 views

Partition function without repetitions of parts and largest part $k$

Um, well, I think the title pretty much says it all. Nevertheless, allow me to explain. I am aware of a certain partition function $Q(n, k)$ that is supposed to remove duplication of parts and $k$ ...
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2answers
184 views

Partition of a Nonempty Set $X$

Let $X$ be a nonempty set, and $\{A_\alpha : \alpha\in I\}$ be a partition of $X$. If $B\subseteq X$ such that $A_\alpha\cap B\neq\emptyset$ for every $\alpha\in I$, is $\{A_\alpha\cap B : \alpha\in ...
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1answer
335 views

How do I approach this combinatorics problem about composition?

The question is from Bogart's A composition of the integer k into n parts is a list of n positive integers that add to k. How many compositions are there of an integer k into n parts. To ...
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2answers
117 views

Verifying That A Collection Is A Partition of $\mathbb R^3$

Can someone possibly explain this to me, I'm having difficulties visualizing it: For each $r\in\mathbb{R}$, let $A_r=\{(x, y, z)\in\mathbb{R}^3 : x+y+z=r\}$. How can I tell if this is a partition of ...
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1answer
323 views

Using Durfee squares to prove partition identities?

I was surprised to learn of Durfee squares, which can be visually explained as the largest square contained within a partition's Ferrers diagram. Moreover, partition identities have always amused me ...
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1answer
169 views

Combinatorial interpretation of $ n P_n = \sum_{j=1}^n \, \sigma(j) P_{n-j}$?

By logarithmic differentiation, one can deduce that \[ n P_n = \sum_{j=1}^n \, \sigma(j) P_{n-j} \] where $P_n$ is the $n$-th partition number, and $\sigma(j)$ the sum of the divisors of $j$. I see ...
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2answers
513 views

Why does $S(n,k)=\sum 1^{a_1-1}2^{a_2-1}\cdots k^{a_k-1}$?

I've been trying to get back into some combinatorics, and in my reading, I find that $$ S(n,k)=\sum 1^{a_1-1}2^{a_2-1}\cdots k^{a_k-1} $$ where the sum is taken over all compositions ...
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1answer
557 views

Partitions of $\{1,\dots,n\}$ with no consecutive integers in each block is counted by $B(n-1)$?

I'm trying to understand why $B(n-1)$ also counts the number of partitions of $[n]$ where not two consecutive integers appear in the same block. Now the bell number $B(n-1)$ counts the number of ...
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1answer
308 views

For integer partitions, why does $e(n)-o(n)=k(n)$?

I denote by $e(n)$ is the number of partitions of $n$ with an even number of even parts, $o(n)$ the number of those with an odd number of even parts, and $k(n)$ the number of those that are ...
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1answer
442 views

Restricted Integer Partitions

Two Integer Partition Problems Let $P(n,k,m)$ be the number of partitions of $n$ into $k$ parts with all parts $\leq m$. So $P(10,3,4) = 2$, i.e., (4,4,2); (4,3,3). I need help proving the ...
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3answers
500 views

Partition an integer $n$ by limitation on size of the partition

According to my previous question, is there any idea about how I can count those decompositions with exactly $i$ members? for example there are $\lfloor \frac{n}{2} \rfloor$ for decompositions of $n$ ...
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1answer
547 views

How to find the coefficient of a term in this expression

How to determine the coefficient of z3q100 in I stumbled upon this problem while trying to solve this type of partition problem: Find the number of integer solutions to x + y + z = 100 such that 3 ...
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1answer
211 views

dual representations and partitions

Suppose we have a partition $\mu$ of $n$. There is an associated polynomial irreducible representation $\phi_{\mu}$ of $GL_n(\mathbb C)$. How do I obtain a new representation of $GL_n(\mathbb C)$ ...
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284 views

Is there a connection between the 3D random walk constant and the partition function?

In thinking about this question, I took a look at Pólya's random walk constants and was struck by the fact that an expression for the constant for a three-dimensional random walk, ...
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2answers
131 views

Generating sequences of numeric partitions

Definition: A tuple $\lambda = (\lambda_1, \ldots, \lambda_k)$ of natural numbers is called a numeric partition of $n$ if $1 \leq \lambda_1 \leq \cdots \leq \lambda_k$ and $\lambda_1 + \cdots + ...
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227 views

Partitions in which no part is a square?

I asked a similar question earlier about partitions, and have a suspicion about another way to count partitions. Is it true that the number of partitions of $n$ in which each part $d$ is repeated ...
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2answers
997 views

Same number of partitions of a certain type?

Is there a quick explanation of why the number of partitions of $n$ such that no parts are divisible by $d$ is the same as the number of partitions of $n$ where no part is repeated $d$ or more times, ...
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2answers
132 views

Every number $n^k$ can be written as a sum of $n$ distinct odd numbers

I wish to prove that for $n,k\in\mathbb{N} > 1$, we can always write $n^k$ as a sum of $n$ odd positive integers. I have an idea of how to approach this, but my method seems to cumbersome. I am ...
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2answers
256 views

Number of ways to represent a number as a sum of only $1$’s and $2$’s and $3$’s

How do I find the number of ways to represent a number as a sum of only $1$’s and $2$’s and $3$’s? I think the title is self-explanatory. E.g., if I have to represent $13$ as a sum of only ...
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2answers
383 views

How can I reduce a number?

I'm trying to work on a program and I think I've hit a math problem (if it's not, please let me know, sorry). Basically what I'm doing is taking a number and using a universe of numbers and I'm ...
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1answer
183 views

Number of perturbations of the Jordan form

I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation. For example, if a Jordan form consists of a single cell $2 \times ...
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1answer
605 views

Feeding real or even complex numbers to the integer partition function $p(n)$?

Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — ...
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1answer
450 views

The coin change problem in the quantitative way

Today,I came across this problem: Suppose you have a currency, named miso, in three denominations, $1, 10$ and $50$. In how many ways can $107$ miso be given in this currency if you have ...
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3answers
189 views

A formula on partitions

Suppose that $\lambda,\mu$ are integer partitions, with conjugates $\lambda^*,\mu^*$. Could you help me to prove the following formula, please? ...
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4answers
273 views

Summing up Problem (Combinations)

This is a part of a bigger problem I was solving. Problem: $N$ is a positive integer. There are $k$ number of other positive integers ($\le N$) In how many ways can you make $N$ by summing up any ...
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510 views

Proving that exactly half the partitions of $n$ into powers of 2 have an even number of parts

Could you help me how to prove that exactly half the partitions of $n$ into powers of 2 have an even number of parts, please?
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1answer
420 views

Exponential generating function for restricted compositions

I wanted to know if it is possible to use exponential generating functions to evaluate composition of N using K distinct numbers (where the supply of numbers is infinite)? For e.g if N=10 and ...
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2answers
138 views

Infinitely many $n$ such that $p(n)$ is odd/even?

We denote by $p(n)$ the number of partitions of $n$. There are infinitely many integers $m$ such that $p(m)$ is even, and infinitely many integers $n$ such that $p(n)$ is odd. It might be proved ...
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2answers
613 views

Some other ways to prove a partitions problem

Show that the number of partitions of the integer $n$ into three parts equals the number of partitions of $2n$ into three parts of size $< n$. I can only prove it by building a bijection ...
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3answers
241 views

Two series relations, each one implies the other - from Andrews' partition book

That's my first question here, and i was encouraged to post because my question in MathOverflow (HERE) was beautifully and fast answered. But my questions in not at research level... As i said there, ...
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1answer
143 views

Partition and power of 2s

I'm trying to find a partition of natural number N whereby we can generate a partition for all natural numbers less than N from the partition elements. The number of elements in the partition should ...
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1answer
114 views

HAKMEM 18(B): Cubic Partitions

Taken from HAKMEM 18. Quoting... A partition of $N$ is a finite string of non-increasing integers that add up to $N$. Thus 7 3 3 2 1 1 1 is a partition of 18. Sometimes an infinite string of ...
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1answer
220 views

Seeking a textbook proof of a formula for the number of set partitions whose parts induce a given integer partition

Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq ...
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3answers
567 views

An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$

The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening): $$ B_n^2 \leq B_{n-1}B_{n+1} $$ for $n \geq ...