Questions tagged [integer-partitions]

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

871 questions
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Demonstration with Ferrers Diagram (Integer Partitions)

Show, using the Ferrers Diagram, that a perfect square number $n$ will always have a partition with $𝑟$ odd and distinct parts ranging from $1$ to $2𝑟-1$, where $𝑟 = √n$. Explain why the parts are ...
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Partitions with multiplicity locally restricted

I am interested in finding any result, paper, taxonomy, generating function, bijection... on what I call "Partitions with multiplicity locally restricted" that is partitions in which multiplicity of ...
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An interesting way of partitioning with inner ordered combinations

Assume $K$ labeled blocks $s_1, s_2, \dots, s_K$ ($s_1 < s_2 < \dots < s_K$) that arrive sequentially and need to be accomodated as they arrive in $N$ containers (partitions with ...
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partition function - using each number once and using only odd numbers

1)I was asked to find a partition function , where each number appears only once. for example, for n=2 - 1+1 is not good but 2 is. I think the function is : $\prod\limits_{k=1}^{\infty}(1+q)^k$,...
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Which integer partitions correspond to the most set partitions?

Which integer partitions of n correspond to the most distinct set partitions? For small n where it is feasible to calculate these values for every integer partition and compare, this is a ...
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Let $f(n)$ the number of partitions on $n$ with distinct parts. Let $g(n)$ be the number of partitions with odd parts only. Show $f(n)=g(n)$ [duplicate]

I have no clue how to start. Any hint would be appreciated!
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For what $n$ can $\{1, 2,\ldots, n\}$ be partitioned into equal-sized sets $A$, $B$ such that $\sum_{k\in A}k^p=\sum_{k\in B}k^p$ for $p=1, 2, 3$?

This is a recent problem in American Mathematical Monthly. The deadline for this question just passed: $\textbf{Problem:}$ For which positive integers $n$ can $\{1,2,3,...,n\}$ be partitioned into ...
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Find B(x) such that $A(x) = P(x) \cdot B(x)$

A(x) is enumerator (generating function) of partitions of number such that contain exactly $1$ (but maybe multi times) of $2,3,5$. P(x) is enumerator of all partitions. Find compact pattern for $B(x)$ ...
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Smallest rectangle inscribing Young tableaux

I am interested in knowing the name of any of this characteristics of a Young tableaux (Ferrer's diagram): Smallest rectangle that contains it or the area of such a rectangle. For instance: Partition ...
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Clarification of the proof of Euler's identity regarding the generating function for partitions.

In reference to this question which I asked here couple of days back but didn't get any answer I am posting this question to clarify whether we can able to extend Euler's identity regarding the ...
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Converse of a proof

Let $\pi$ be the partition of $n=a_1+a_2+...+a_r$, where $a_1\geq a_2 \geq ....\geq a_r\gt0$ and $\pi$’ be the partition, $n=b_1+b_2+....+b_s”.$Prove that $\pi$’ is the conjugate of $\pi$ if and only ...
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Forms of sum $\frac{n(n+1)}{2}$ with natural numbers.

How many ways are there of sum $\frac{n(n+1)}{2}$ with $n$ addends? Knowing than $i$ appears at most $n+1-i$ times. I really haven't had any important ideas. When speaking of a fixed number of ...
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Understanding the following steps

I asked the following question . Let $\pi$ be the partition of $n=a_1+a_2+...+a_r$, where $a_1\geq a_2 \geq ....\geq a_r\gt0$. Prove that number of partition $\pi$ of $n$ with $a_r=1$ and $a_j-a_{j+1}$...
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Number of partitions of a number into distinct parts.

Let $\pi$ be the partition of $n=a_1+a_2+...+a_r$, where $a_1\geq a_2 \geq ....\geq a_r\gt0$. Prove that number of partition $\pi$ of $n$ with $a_r=1$ and $a_j-a_{j+1}$ = 0 or 1 for $1\leq j\leq r-1$ ...
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Showing two partitions to be equal

I was posed with the following problem " Let $F(n)$ denote the number of partitions of $n$ with every part appearing at least twice and $G(n)$ denote the number of partitions of $n$ into parts larger ...
Let's imagine you want to get a number $A$ from a set of number $M$. You might use only +/- between the numbers and your job is to determine, whether it's possible to get the result $A$ or not. So ...