Questions tagged [integer-partitions]

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

0
votes
1answer
48 views

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....
1
vote
0answers
122 views

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 ...
2
votes
1answer
146 views

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 ...
0
votes
0answers
30 views

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. ...
3
votes
1answer
187 views

Sum of the hook-lengths of a partition $\lambda$

Given is a partition $\lambda$, and the $\lambda$ also denote it's Young Diagram, and $\lambda'$ is the conjugate/transpose. Then the hook-length of $\lambda$ at $x = (i,j)$ is defined to be $$h(x) = ...
1
vote
0answers
24 views

Asymptotic behavior of the number of ways a real plane curve of degree $n$ can intersect a real line

Let $C$ and $L$ be plane algebraic curves of degree $n$ and $1$ defined over the real numbers. For $k=1,\ldots n$, let $h(k)$ be the number of real points where $L$ and $C$ meet with multiplicity $...
1
vote
1answer
102 views

On a theorem (1.7) in Macdonald's Symmetric Functions and Hall Polynomials

I'm using Macdonald's Symmetric Functions and Hall Polynomials as a reference, and here's what's given: Theorem (1.7, Page 3): Let $\lambda$ be a partition and let $m \geq \lambda_1$, $n \geq \...
2
votes
1answer
176 views

Number of positive integral solutions of $a+b+c+d+e=20$ such that $a<b<c<d<e$ and $(a,b,c,d,e)$ is distinct

This is from a previous question paper for an entrance exam I am preparing for. https://www.allen.ac.in/apps/exam-2014/jee-advanced-2014/pdf/JEE-Main-Advanced-P-I-Maths-Paper-with-solution.pdf (Link ...
12
votes
2answers
195 views

Permutation induced by a partition

Let $\lambda$ be a partition of length $n$ and suppose its largest diagonal block, the Durfee square of $\lambda$, has size $r$. By this I mean that $\lambda = (\lambda_1,\ldots,\lambda_n)$ is a non-...
1
vote
1answer
43 views

What form does the Law of Total Probability take if the partition you use is generated by the random variable Y?

The Law of Iterated Expection Looks like E{E(X|Y)} when the partition you use is generated by the random variable Y rather than Ω. What happens when you use such a partition on the Law of Total ...
4
votes
0answers
110 views

Evaluate $ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty $

This identity is taken from a physics paper [1] stated without proof, on page 43. $$ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty =...
1
vote
1answer
89 views

Counting number of ways to sum integers and half integers to a specific integer modulo N

I came across this while working on a Physics problem. I want to count the number of ways I can get a nonnegative integer $k \in$ {$0,1, \cdots, N-1 $} where $$k \equiv \sum_{i=1}^{N} x_{i} n_{i}\...
2
votes
1answer
58 views

Expand $a^5 + b^5 + c^5$ in terms of Schur polynomials

How to expand certain sums of powers in terms of Schur polyomials. I have been gaining proficiency with symmetric polynomials, today I would like to expand: $$ a^5 + b^5 + c^5 = \sum_{\lambda_1 + \...
3
votes
1answer
398 views

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 ...
1
vote
1answer
85 views

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 ...
2
votes
3answers
185 views

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!
0
votes
1answer
25 views

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 ...
2
votes
0answers
42 views

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 ...
1
vote
0answers
63 views

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 ...
0
votes
0answers
65 views

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.
-1
votes
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 ...
2
votes
0answers
83 views

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{...
1
vote
0answers
22 views

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 ...
0
votes
2answers
190 views

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 ...
1
vote
1answer
371 views

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 ...
1
vote
2answers
53 views

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 ...
0
votes
1answer
58 views

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 ...
2
votes
2answers
254 views

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\...
1
vote
1answer
46 views

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 $-5 \le x_{i_{\in [4]}} \le 10$ I know if $x_i$s are all non-negative integers, it is a number partition ...
2
votes
0answers
153 views

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 ...
1
vote
1answer
71 views

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 ...
0
votes
1answer
164 views

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 ...
1
vote
1answer
50 views

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 ...
1
vote
1answer
58 views

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?
0
votes
0answers
153 views

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 $F(x)=\frac{1}{(1-x)^3}=\sum_{n=0}^{\infty} a_n x^n\tag1$ ...
1
vote
0answers
66 views

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 $(5,4,2)$ ...
0
votes
0answers
104 views

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 ...
0
votes
1answer
25 views

Converse of Ramanujan's Congruences

Of Ramanujan's famous congruences for the partition function, $p(5k+4)\equiv0\mod 5$, $p(7k+5)\equiv0\mod7$, and so on, does the converse also hold? For example, if $p(n)\equiv0\mod5$, does that mean $...
1
vote
1answer
310 views

Ordered Integer Partition of fixed size

I'm trying to find how many ways there are to partition n into k partitions with max value s. I've found many solutions that does this for unordered partitions. They use recursion and the say that p(n,...
0
votes
1answer
20 views

Generating function for certain partition functions.

what is the generating function of partitions of n into positive integers where the integers come in two kinds and the second kind has weight 23 times the weight of the first kind. Does the ...
2
votes
1answer
83 views

What is a combinatorial proof for $p_k(n) \leq (n-k+1)^{k-1}$

Suppose $p_k(n)$ is the number of partitions of the integer $n$ into $k$ parts. For example, the partition of 5 into 2 parts is; $p_2(5) = 2$, since the partitions are (4,1) and (3,2). What is a good ...
1
vote
1answer
333 views

Generating function for number of partitions with only distinct even parts

What is a generating function for the sequence $\{a_n\}_{n\ge1}$ where $a_n$ is the number of partitions of $n$ with only distinct even parts, and how would you show that it was found? The only ...
1
vote
1answer
70 views

Algorithm related to partitions

I hope this is not a foolish question... Define $T(n,k)$ = number of partitions of $n$ in which the greatest part is $k$, $1 \le k \le n$. The triangle for $n$ = {1,..,10} looks like: 1: [1] 2: [1, ...
0
votes
1answer
61 views

Is the partition function 5-adically differentiable at 1/24?

Ramanujan's congruences, as extended by Watson and Atkin, show that in the $\ell$-adic metric for $\ell\in\{5,7,11\}$, the partition function is continuous at $\frac{1}{24}$, having a limit $\lim_{n\...
2
votes
1answer
160 views

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 ...
3
votes
1answer
267 views

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 ...
-1
votes
1answer
85 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{...
1
vote
1answer
190 views

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 ...
0
votes
1answer
28 views

Equation to approximate a Partition-like function

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 ...
1
vote
0answers
101 views

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 ...