# 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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### How to calculate least moves of fruit.

I have a question, I'll try to abstract from the real problem to not lose people. What I'm really looking for is the name of the algorithm or class of problem to find my solution. I feel that this is ...
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### Partition of the 3d space with circles?

Does it exist a partition of the 3d space with circles of positive radium? I know the answer is no for a plane, but I can not transpose my demonstration to the space and I have no clue on how to do ...
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### How to prove this identity about Sylvestered partitions of n into m parts such that …

Let us say a partition $\lambda$ is $Sylvestered$ if the smallest part to appear an even number of times ($0$ is an even number) is even. For each set of positive integers $T$, define $S(T\,;m,\,n)$ ...
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### Two different coins on a chessboard

Two different coins are placed on squares of a standard 8x8 chessboard; they may both be placed on the same square. Let us call two arrangements of these coins on the chess board equivalent if we can ...
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### Let $P_1(r;n)$ denote the number of partitions of n into parts that are either even and not congruent to 4r-2(mod 4r)

Let $P_1(r;n)$ denote the number of partitions of n into parts that are either even and not congruent to 4r-2(mod 4r) or odd and congruent to 2r-1 or 4r-1(mod 4r). Let $P_2(r;n)$ denote the number of ...
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### Represent $N$ as the sum of exactly $K$ distinct positive integers

You are given two integers $N$ and $K$. Find all ways to represent $N$ as the sum of exactly $K$ distinct positive integers $x_1,x_2, \ldots,x_K$ — in other words. $xi_>0$ for each valid $i$; ...
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### Extend of Stanley's problem about product of integer compositions

I am trying to extend the problem from Stanley's book. Has expressed the following problem: For positive integer $k,n$. Show that: \begin{align} \sum x_1 x_2 \cdots x_k = \binom{n+k-1}{2k-1} \end{...
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### Nine objects in non-empty boxes [closed]

In how many ways 9 identical objects can be put in non-empty boxes of arbitrary size? Is solution integer partition of 9? That is 30?
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### Question about the partitions of a natural number

There is a function that counts the number of partitions of with $n$ digits? I am aware of the partition function studied by Ramanujan, but what I want is a subset of the partitions that are counted ...
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### Uniqueness of a P-measurable random variable

"Let $X$ be a discrete random variable on a discrete sample space. Let P be a partition of that sample space. a) Show that $E[X|P]$ is the unique P-measurable random variable $Y$, ie $Y$ = $E[X|P]$, ...
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### 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 $y_1=x_1-1$ ...
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### Generating functions (parts equal to $2$) [closed]

What is the generating function for partitions into parts equal to $2, 5$ or $7$? I know the function of distinct parts, but what if they don't have to be distinct?
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### Set as a union of 3 disjoint sets ,with equal sum

The problem is to find in which value of n the {1,2,3,...n} set can be parted in 3 subsets such that each one has equal sum. I have looked for a lot for this answer, but I can't solve it to the end. ...
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### How many possibilities of writing a natural number $M$ as a sum of $N$ natural numbers between $0$ and $M$?

How many possibilities are there of writing a natural number $M$ as a sum of $N$ natural numbers between $0$ and $M$? For example, I need to write $4$, using $4$ numbers between $0$ and $4$. The ...
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### Alternative reference for number of restricted partitions

I am looking for the number of partitions of some number $n$ into $k$ parts. Following the Wikipedia article on partitions, I ended up with Andrew's book [1]. Judging by Google's preview Chapter 3 ...
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### 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.
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### 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 - ...
### Show that $p(n,k)=p(n-1, k-1)+p^2(n, k)$, Partition Theory
I'm struggling to prove this as I'm not sure how to do so with words/equations as opposed to visually. $p^2(n,k)$ denotes the number of partitions of n having exactly k parts with each part greater ...
General problem: Using the elements of some list of length $m* n$, create a list with $m$ sub-lists, each of length $n$. In my case, $m= 10 > n=3$. The final output should be a list ("lis1") ...