# Questions tagged [integer-partitions]

Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.

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### Number of partitions contained within Young shape $\lambda$

It is well known that the number of partitions contained within an $m\times n$ rectangle is $\binom{m+n}{n}$. Furthermore, it is not difficult to calculate the number of partitions contained within a ...
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### Partitions of $0$ and $1$ by integers in the interval $[-N,\ldots,N]$

Background Often one comes across the problem of trying to find the number of ways to partition a positive integer into a sum of nonnegative integers. There are three ways to partition the number 3, ...
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If you look at one of the comments on A053445 you'll see it mentions that the second difference of partition numbers is equivalent to the number of integer partitions with two largest parts equal and ...
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### Proof of these identities

The following is a screenshot from a paper by Daniel B. Grunberg called On asymptotics, Stirling numbers, Gamma function, and polylogs. I only offer the page as a reference to explain equation 3.1. ...
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### Towers of coins puzzle [duplicate]

Let $n$ be a natural number. You are given $\frac{n(n+1)}{2}$ coins which are arranged in towers. Every turn you pick up the top coin of each tower and gather all these into a new tower. Prove that ...
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### 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 $n$. ...
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### How To Apply and Understand the Generating Function for Number Partitioning

The function p(n) counts the number of ways a number can be made up of smaller numbers. For example, the p(5) = 7 because you ...
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### How many permutations in S(n) have this particular type?

I'm working through the textbook A Course in Enumeration. In the section about permutations and Stirling numbers, I'm having trouble understanding problem 1.45. It is: We fix $n \in \mathbb{N}$, and ...
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### Partition of number's squares

The problem is to divide $\{k^2\}_{k=1}^{1000}$ into two groups of 500 numbers each, such that they have equal sum. I know that $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6},$$ but it isn't enough ...
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### 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 perhaps?...
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### A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?

Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why, $$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$ \sum_{n=0}^\infty P(n) q_4^{n+1}\approx\...
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Prove that if $n\ge16,$then $10\mid g(n),$where $g(n)$ is the largest LCM of partitions of $n$. For more information,see http://oeis.org/A000793 Here is the list of $g(n)$ for $n>0,$ $g(15)=105,g(... 3answers 6k views ### The number of ways to write a positive integer as the sum of distinct parts with a fixed length I am a topologist and not terribly familiar with the combo literature so please forgive me if this is standard. I'm hoping for some sort of reference for this. Given a positive integer$n$, I wish ... 1answer 553 views ### Exponential generating function for restricted compositions I wanted to know if it is possible to use exponential generating functions to evaluate composition of N using K distinct numbers (where the supply of numbers is infinite)? For e.g if N=10 and a1=2,a2=... 2answers 692 views ### Learning about partitions and modular forms I'm interested in learning about partitions and modular forms. I already know algebra and analysis (complex and real). Can any one suggest me books or other materials from where I can learn these ... 1answer 142 views ### Is there a partition of an open square into closed segments (not reduced to a point)? Let$C$be an open square (for example$]0, 1[ \times ]0, 1[$)of the plane$\mathbf R^2$. Is there a partition of$C$into closed segments (not reduced to a point) ? 2answers 145 views ### Partition bijections How do I prove bijectively that the number of partitions of$n$with largest part$k$equals the number of partitions of n with exactly$k$parts. 2answers 6k views ### Median of medians algorithm I am referring to the algorithm presented here used to find a good pivot: http://en.wikipedia.org/wiki/Selection_algorithm#Linear_general_selection_algorithm_-_Median_of_Medians_algorithm My ... 3answers 918 views ### Number of partitions of$2n$with no element greater than$n$The number of partitions of$2n$into partitions with no element greater than$n$(copied and slightly adapted from http://mathworld.wolfram.com/PartitionFunctionq.html), so I'm looking for a nice ... 4answers 219 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?$\sum_{i,j}\mathrm{min}(\lambda_i,\mu_j)=\sum_k\...
I want to know in how many ways can we represent a number $x$ as a sum of numbers from a given set of numbers $\{a_1.a_2,a_3,...\}$. Each number can be taken more than once. For example, if $x=4$ and ...
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 between ...