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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Truncation of partitions generating function question

$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $ $$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
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113 views

Different flag signal questions

How many different signals can be created by lining up 9 flags in a vertical column in 3 flags are white, 2 are red, and 4 are blue? Is it 9 choose 3 * 6 choose 2 * 4 choose 4?
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110 views

Proof of Partitions

Let $|n,k|$ denote the number of partitions of $n$ into $k$ distinct parts. Prove $$|n,k| = |n-k,k-1| + |n-k,k|$$ Workings: LHS counts the number of partitions of $n$ into $k$ distinct parts. RHS: ...
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170 views

Special partition of a number $n$

Given any integer $n$, how many ways can it be partitioned in which the number $1$ is not allowed? For instance, if $n=6$, then the partitions obeying the aforementioned rule are $6+0$, $4+2$, $3+3$, ...
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59 views

Sum of positive integers estimating sum of fractions

Given $m$ fractions adding up to an positive integer $n$ For example: $m=3\\n=10=\frac{30}{6}+\frac{20}{6}+\frac{10}{6}$ How can we find $m$ positive integers that sum to $n$ (a partition of $n$), ...
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Factorial Taxicab Number

What is the $i$th number $T_! (n,k,i)$ such that $T_! (n,k,i)$ is the sum of $n \in \mathbb{N}$ distinct positive integer factorials in $k \in \mathbb{N}$ distinct ways (ignoring ordering, parentheses,...
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362 views

Partitions applications in physics

Is there any direct application of all developments related to partitions? I am specially interested in physics but cryptography or other mostly theoretical areas would also be a good answer. By ...
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Difficulty parsing combinatorics exercise

I am working through the wonderful book Proofs and Confirmations by David Bressoud. In the section 2.2, I came across the following exercise, which has me scratching my head. (2.2.8) ~ Let $a(m,n)$ ...
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An identity relating sum of number of partitions to sum of number of parts

I encountered this identity while studying about the Kac determinants in CFT. $$\sum_{\{n_1 + \cdot \cdot \cdot + n_k \}} k = \sum_{pq \le N} P(N-pq)$$ Here $P(N-pq)$ is the number of partitions of ...
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50 views

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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539 views

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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1answer
126 views

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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Partitioning techniques for finding large matrix determinents

I'm in a linear algebra class and we're doing determinants right now. I got this matrix to do: $\begin{matrix} 2 & 1 & 0 & 0 & 0 \\ 3 & -1 & 2 & 0 & 0 \\ 0 & 4 &...
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Number of ways to divide a stick of integer length $N$, take 2

This is a follow up and motivated by this question, Number of ways to divide a stick of integer length $N$, Suppose we have a stick of integer length $N$. I'm looking for (preferably closed-form) ...
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1answer
98 views

How to find random numbers that can sum up to n?

I have a random integer $n$ and another integer called the summary. I want to know how many ways I can sum a subset of numbers from $1$ to $n$ to produce the value of summary. For example, I have $1,...
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761 views

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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1answer
87 views

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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46 views

Number Partitions

Is this series complete, using '0' is not allowed: $6 = 6$ or $5+1$ or $1+5$ or $4+2$ or $2+4$ or $3+3$ or $4+1+1$ or $1+4+1$ or $1+1+4$ or $1+2+3$ or $1+3+2$ or $2+3+1$ or $2+1+3$ or $3+1+2$ or $3+2+...
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55 views

Total Collections of integers that sum to constant

For a range of positive integers $1 - S$, how many collections of $N$ integers are there that their sum is a constant $S$. Example: Integers from $1$ to $100$ Collections of $4$ integers Each ...
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1answer
74 views

Number of ordered set partitions with subset size $\leq 3$

For $n \ge 0$, let $h_n$ be the number of ways of taking $n$ (distinguishable) rabbits, putting them into identical cages with one to three rabbits per cage and then ordering the cages in a row. Find ...
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1answer
134 views

partitions of the number n

I'm having difficult with the following question : show that the number of partitions of n into parts of size 3,5,7,9,... equals to the number of partitions of n into different parts which are not 1,...
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144 views

Prove that this set (involving fractional part of any rational number) is a partition of the set of rationals.

For any rational number $x$, we can writte $x=q+\,n/m$ where $q$ is an integer and $0\le n/m<1$. Call $n/m$ the fractional part of $x$. For each rational $r\in \{x : 0\le x<1\}$ ,let $A_r = \{ x\...
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2answers
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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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1answer
39 views

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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1answer
79 views

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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1answer
71 views

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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1answer
30 views

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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1answer
75 views

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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1answer
141 views

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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2answers
81 views

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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1answer
144 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.
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1answer
73 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 - ...
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1answer
61 views

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 ...
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1answer
59 views

list partitions (python) - why is the index out of range? [closed]

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") ...