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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What does s(n) = s(n) mean?

I'm trying to create python analogue of wolframalpha's PartitionsQ function which calculates number of distinct partitions of integer. I found documentation about this function http://mathworld....
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Decompose the permutation module $M^{(2, 2)}$ into irreducible representations.

My current approach is to take some elements of $M^{(2, 2)}$ and examine the submodules generated by them, in the hopes of finding a basis for them. Each submodule will correspond to an irreducible ...
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To show that $\sum_{x \in \lambda}(h(x)^2 - c(x)^2)=|\lambda|^2$, $h(x)$ is hook-length & $c(x)$ content of $x$, a block in the diagram of $\lambda$

I'm self-reading Macdonald's Symmetric Functions and Hall Polynomials, and he gives quite a few examples without any proof, and I'm unable to trace the line of thought' this is one of them. I have ...
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On an example in Macdonald's Symmetric Functions and Hall Polynomials on Paritions and their Frobenius Notation [duplicate]

I am following Macdonald's Symmetric Functions and Hall Polynomials, and in his example-section. He stated the following without any proof, and I'm trying to understand how he reached the expression. ...
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How to turn number into sum of unique primes?

I have to find algorithm which find prime number less than $n$ which is sum of the largest amount of unique primes, for example for $n=81$, the answer is $79 = 3 + 5 + 7 + 11 + 13 + 17 + 23$. I have ...
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Is there any proof of this identity?

$$\prod\limits_{i=1}^{\infty}\frac{1}{1-yx^{i}}=\sum\limits_{k=0}^{\infty}\frac{y^k x^k}{(1-x)(1-x^2)...(1-x^k)}$$ I know its combinatorial proof through "inspection",but is it true when $x,y$ are ...
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How to count all restricted partitions of the number $155$ into a sum of $10$ natural numbers between $[0,30]$? [closed]

How to count all restricted partitions of the number $155$ into a sum of $10$ natural numbers between $[0,30]$? I really have no clue what to do with this one. Thanks for any help!
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Equivalence of partitions of integers problem

Let $m$ and $n$ be fixed natural numbers, both at least 2. Let $X \in Z_m$ and $Y \in Z_n$. Prove that $X \subseteq Y$ if and only if $n | m$. [ Hint: $X$ and $Y$ are subsets of $Z$ since they are ...
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Partitions reference

In this paper, the author mentions before Theorem 23 that the result was proven by Elmo Moore in 1968; the associated reference says "unpublished result". I would be grateful to anyone who knows a ...
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If I have an integer, how many numbers are there whose digits sum to the integer?

Suppose I have an integer, $m$, and another integer, $n$, then is there a way of working out how many numbers of length $n$ exist such that the sum of their individual digits is $m$? Another way of ...
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Number of ways to partition $\{1,2,3, \dots, N\}$ into tuples where the size of no tuple exceeds $3$.

While it seems to me that the general answer is not going to be a neat formula, I really only need this for $N=4$ and $N=5$. I'm getting $61$ and $321$ respectively, but I'm not sure. Please help.
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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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Partition Generating Functions [duplicate]

Prove the following identity by counting two sets of partitions in two different ways. \begin{equation}\prod_{i\geq 0}(1+x^{2i+1})=1 + \sum_{n\gt 0}x^{n^2}\prod_{j=1}^n\frac{1}{1-x^{2j}}.\end{...
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Number of partitions of a $\mathbb{N}^n$ vector

Using power series, we can find an formula for the partitions of an integer. Can we do the same for the partitions of a vector? (I tried to apply the same technique, but failed due to the fact that ...
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disjoint Partition of sets

Hello I'm a bit confused by what this definition means $$DT_n = \{ \{C,D \} \mid C,D ⊆ S_n \text{ and } C \cup D = S_n \text{ and } C \cap D = \emptyset \}$$ where $S_n = \{1,2....n\}$ and n is a ...
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Coin Combinations for any given scenario.

I am trying to work out the number of scenarios I can cover with a given set of coin combinations so I can decide when I have the optimal amount of change to carry. For the sake of the example, lets ...
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For every $k \in \mathbb{N}$ there exists a $q \in \mathbb{R}$ such that $q^0 = 1$ and $q^k = x$

Let $x > 1$ Prove that: For every $k \in \mathbb{N}$ there exists a $q \in \mathbb{R}$ such that $q^0 = 1$ and $q^k = x$ I have considered a proof by induction, but don't think it works. I also ...
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Proof for this partition graph

Consider the below graph for 7. Each node represents a unique partition of 7. I notice that we can reach to any node from the bottom by partitioning only even integer except the bottom node. I have ...
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Counting set for generating functions

Give a proof of the following identity by counting two sets of partitions in two different ways. \begin{equation*} \prod_{i\geq 0}(1+x^{2i+1})=1 + \sum_{n\geq 1}x^{n^2}\prod_{j=1}^n\...
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Number of Integer Solutions for $x_1 + x_2 + x_3 + x_4 = 15$ where $-5 \le x_i \le 10$

I am trying to find the number of Integer Solutions for $x_1 + x_2 + x_3 + x_4 = 15$ where $-5 \le x_{i_{\in }} \le 10$ I know if $x_i$s are all non-negative integers, it is a number partition ...
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Counting Semistandard Young Tableaux For Triangular Shapes?

If $k \leq n$ I denote the Young diagram with shape $(n,n-1,n-2,\ldots,1)$ by $\lambda^{n,n-1,\ldots,1}$. I write $f^{\lambda_n^{n,n-1,\ldots,1}}$ to count the number of semistandard Young tableaux ...
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Intuition on why this problem doesn’t have an analytic solution

I recently overheard some people trying to work out a solution for a problem informally stated as follows: In how many ways can you put $n$ indistinguishable particles into $n$ boxes with the ...
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Counting the number of semistandard Young Tableaux with maximum entry $n.$ Reference/Formula request

Question: If $k \leq n$ let $\lambda_k$ be a Young diagram with square $k \times k$ shape. I write $\#_{\lambda_{k}^n}$ to count the number of semistandard Young tableaux with shape $\lambda_k$ and ...
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How to construct an example for defined Partition

Suppose $m \geq n \geq 1$ are two integers. An ordered $n$-tuple of integers $\pi = (m_{1}, \dots , m_{n})$, $m _{i} \geq 1$ is called an $n$-partition of $m$ if $m_{1}+ \dots m_{n} = m$. The set ...
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Mock theta function

I am unable to find a formal definition of the order of the mock theta function. Can you explain briefly, what is the order of the mock theta function?
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A generating function $G(x)=-\frac{\frac{1}{x^5}(1+\frac{1}{x})(1-\frac{1}{x^2})}{((1-\frac{1}{x})(1-\frac{1}{x^3}))^2}$ related to partitions of $6n$

Fix a sequence $a_n={n+2\choose 2}$ of triangular numbers with the initial condition $a_0=1$,such that $1,3,6,10,15,21,\dots$ given by $F(x)=\frac{1}{(1-x)^3}=\sum_{n=0}^{\infty} a_n x^n\tag1$ ...
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Counting number of integer solutions to $a_1 + a_2 + a_3 + \ldots = n$ where all $a$'s must be in certain range

For a given $(n,m,k)$.. Using values in the range $(0..k)$, how many different $m$-combos exist which sum to n? ex. for $(n,m,k)$ = $(3,3,2)$, there are 7 possible combinations. For $(5,4,2)$ ...
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Use a generating series to prove that the number of partitions of $n$ in

Use a generating series to prove that the number of partitions of n in which only the odd parts can be repeated is equal to the number of partitions of n in which no part can be repeated more that ...
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Number of integer partitions

Let's $N$ be a positive integer and $P$ - set of all possible partitions of the $N$, where $p = (a_1,a_2,...,a_n)$ with $a_1\le a_2 \le ... \le a_n$ and $a_1+a_2+...+a_n = N$. Let's $A$ be the number ...
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Restricted partitions including zero, without repeated numbers

I'd like to try to develop some formal maths for listing the degeneracies of spinless fermion states in a harmonic oscillator. For those who don't know much quantum physics, I'm essentially trying to ...
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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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partitions of positive integer $n$ with respect to a multiset

Recently, I think on a new problem related to partitions. Let $n$ be a non-negative integer and $\mathbb{A}=\{a_1,\ldots,a_k\}$ be a multiset with $k$ not necessarily distinct positive integers. We ...
The partition function for $n$, $P(n)$ gives the number of partitions that exist for $n$. I've been trying to find a function that gives the number of partitions where order matters, e.g. $1+2+3$ is ...
Dominance ordering on partitions of $n$.
Denote the collection of partitions of $n$ by $\mathcal{P}(n)$, with the property that for $\lambda = (\lambda_1,\cdots,\lambda_s)\in\mathcal{P}(n)$ we have \lambda_1\geq \cdots \geq \lambda_s \geq ...