1
vote
0answers
36 views

Count the number of ways n different-sided dice can add up to a given number

I am trying to find a way to count the number of ways n different-sided dice can add up to a given number. For example, 2 dice, 4- and 6-sided, can add up to 8 in 3 different ways: ...
0
votes
1answer
58 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 ...
3
votes
1answer
67 views

How to prove this identity?(perhaps related to partition)

How to prove this identity? $$ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)} = \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$$ Maybe the method using generating functions is good.
1
vote
1answer
62 views

Proof that the series for the generating function of the partition function converges?

For $|q| < 1$, the generating function of the partition function $p(n)$ is given by $$ \sum_{n=0}^\infty p(n) q^n = \prod_{k=1}^\infty {1 \over 1-q^k}. \tag{1} $$ I have an intuitive ...
0
votes
0answers
219 views

Counting distinct restricted integer partitions of $n$ into exactly $k$ distinct parts less or equal then $M$

How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$? The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at ...
0
votes
0answers
38 views

The Generating Functions for $p(S_d,n)$

Prove: $$\sum_{n=0}^{\infty}p(S_2,n)q^n=\prod_{j=1}^{\infty}\frac{1}{(1-q^{5j-4})(1-q^{5j-1})}$$ I have been trying to figure out where to even start this problem and I have no idea what to do. Can ...
1
vote
2answers
49 views

Number of 1's among all partitions of an integer

I am trying find a recurrence relation for the number of 1's among all partitions of an integer. The OEIS database has an entry mentioning this particular sequence but does not give a recurrence ...
2
votes
0answers
58 views

Distributing problem using generating functions

For $r\in\mathbb{Z^*}$, let $a_r$ denote the number of ways to distribute $r$ identical objects into $3$ identical boxes, $b_r$ be the number of distributions so that the boxes are to be non-empty, ...
0
votes
0answers
53 views

The generating function for partitions: an alternate (and false) representation

The combinatorics book I'm going through asked me to find the generating function whose coefficients $p_n$ give the number of integer partitions of $n$. I reasoned the following answer: the generating ...
0
votes
1answer
42 views

Number of partitions of integer into parts repeated <= 2 times

The generating function for the number of such partitions is $$ G(q) = \prod_{i=0}^{\infty}(1+q^i+q^{2i}) $$ - that much I understand. Is there any way to transform it into a form ...
0
votes
0answers
36 views

The number of 2's in all partitions of n

Let $a_{n}$ be the number of "2"'s that appear in all partitions of $n\geq 0$. The sequence begins like $(0,0,1,1,3,4,\cdots)$. I am tasked to show that it's OGF is $$\dfrac{x^{2}P(x)}{(1-x)^{2}}$$ ...
4
votes
1answer
110 views

How to extract coefficient of $x^n$ in an infinite product generating function?

Are there methods for obtaining the coefficient of $x^n$ in a generating function like $$\prod_{i=1}^\infty Q(x),$$ where $Q(x)$ is a rational function? This arises when we want to count partitions of ...
1
vote
1answer
207 views

Generating function of the number of integer partitions of $n$ into all distinct parts

Let $p_d (n)$ denote the number of integer partitions of $n$ into all distinct parts. I am given the following equation, but I can't figure out why it holds: $$\sum_{n \ge 0} p_d(n)x^n = \prod_{i \ge ...
2
votes
1answer
125 views

Number of even parts of a partition

Fix a positive integer $n$. For a partition $\lambda$ of $n$, let $e(\lambda)$ be the number of even parts in $\lambda$. Using generating functions or bijections, we can show the statistic ...
0
votes
1answer
315 views

Generating function of partition with restriction [duplicate]

Let $c(m,n)$ denotes number of partitions $n$ into parts not greater than $m$, where order of elements does matter (so they are not classic partitions). Prove that: $$\sum_{n\ge 0}c(m,n)x^n = ...
1
vote
0answers
61 views

Restricted Partitions of $n$ [duplicate]

The original question was to find the number of ways to split an integer, $n$, into any number of partitions where each of the parts belong to the set $\lbrace 1,3,4,9\rbrace$. Assuming I did this ...
1
vote
1answer
117 views

Generating functions of partition numbers

I don't understand at all why: \begin{equation} \sum\limits_{n=0}^\infty p_n x^n = \prod\limits_{k=1}^\infty (1-x^k)^{-1} \end{equation} Where $p_n$ is the number of partitions of $n$. Specifically ...
7
votes
3answers
498 views

Combinatorics: Generating Function related to compositions of a number

My goal is to find the coefficients of the generating function for the following situation: The number $f(n)$ is the sum over all compositions of $n$ into $3$ parts of the product of those parts. Fo ...
-2
votes
1answer
156 views

From a Generating Function find $R(x)$ as an infinite product of Quotients

Let $r(n)$ be the number of partitions of $n$ so that no multiple of $3$ appears as a part. For example, $r(8) = 13$. Let $R(x) =\sum_0^\infty r(n)x^ n $ be the generating function for $r(n)$. Find ...
-2
votes
1answer
203 views

Find a form for $Q(x)$ as an infinite product of polynomials

Let $q(n)$ be the number of partitions of $n$ so that no part appears three or more times. For example, $q(8) = 13$ Let $Q(x) = \sum\limits_{n=0}^\infty q(n) x^n$ be the generating function for ...
6
votes
3answers
217 views

How to prove it? (one of the Rogers-Ramanujan identities)

Prove the following identity (one of the Rogers-Ramanujan identities) on formal power series by interpreting each side as a generating function for partitions: ...
1
vote
1answer
85 views

Determining Stirling number

In the first part of the question I was asked to find the exponential generating function for $s_{n,r}$, the number of ways to distribute $r$ distinct objects into $n$ (a fixed constant) distinct ...
0
votes
1answer
105 views

Generating function: number of partitions that add up to at most $n$

Find a generating function $a_n$, the number of partitions that add up to at most $n$. So I know that if it were asking the number of partitions of the integer $n$, I would have my generating ...
1
vote
1answer
176 views

Combinatorial proof involving partitions and generating functions

Show that any number of partitions of $2r + k$ into $r + k$ parts is the same for any $k$. I've tried this, but I haven't come up with anything; hence why I have nothing written here. But in any ...
7
votes
1answer
91 views

Generating function for $r^\binom{n}{2}$

I'm trying to find a closed form of the generating function $$ G(x) = \sum_{n \ge 0} r^\binom{n}{2} x^n $$ for a real number $0 < r < 1$. I found that $G(x) = 1 + xG(rx)$. Any hints where to ...
3
votes
3answers
269 views

Further clarification needed on proof invovling generating functions and partitions (or alternative proof)

Show with generating functions that every positive integer can be written as a unique sum of distinct powers of $2$. There are 2 parts to the proof that I don't understand. I will point them out ...
2
votes
1answer
558 views

Finding a generating-function using partitions

Find a generating function for a , the number of partitions of r into (a.) Even integers (b.) Distinct odd integers. I am at a loss of starting this.
1
vote
1answer
95 views

Sum of $\prod 1/n_i$ where $n_1,\ldots,n_k$ are divisions of $m$ into $k$ parts.

Fix $m$ and $k$ natural numbers. Let $A_{m,k}$ be the set of all partitions divisions of $m$ into $k$ parts. That is: $$A_{m,k} = \left\{ (n_1,\ldots,n_k) : n_i >0, \sum_{i=1}^k n_i = m \right\} ...
4
votes
1answer
126 views

Showing two generating functions to be equal

Let $\mathcal{A}$ be the set of partitions in which each part may occur 0, 1, 4, or 5 times and let $\mathcal{B}$ be the set of partitions which have no parts congruent to 2mod4, and in which parts ...
6
votes
1answer
232 views

Creating generating functions for integer partitions

Say I have a generating function $\Phi_\mathcal{A}$ for the set of partitions $\mathcal{A}$ which have no parts congruent to 2 mod 4, and I have the generating function for $\Phi_\mathcal{B}$ for the ...
3
votes
2answers
619 views

Keep getting generating function wrong (making change for a dollar) [duplicate]

Possible Duplicate: Making Change for a Dollar (and other number partitioning problems) I am working on the classic coin problem where I would like to calculate the number of ways to make ...
6
votes
3answers
568 views

Putnam Problem: Partitioning integers with generating functions

We were given the following A-1 problem from the 2003 Putnam Competition: Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, $$ n= a_1+a_2+ ...
2
votes
0answers
209 views

Generating Function of Integer Partition Such that at Least One Part is Even

I've been having a few issues coming up with a generating function for an integer partition such that at least one part is even. What I have got so far is: The generating function with no ...
2
votes
3answers
1k views

The number of ways to write a positive integer as the sum of distinct parts with a fixed length

I am a topologist and not terribly familiar with the combo literature so please forgive me if this is standard. I'm hoping for some sort of reference for this. Given a positive integer $n$, I wish ...
4
votes
3answers
1k views

Number of cycles of all even permutations of $[n]$ and number of cycles of all odd permutations differ by $(-1)^n (n-2)!$

I'm trying to solve task 44 of the first chapter of Stanleys Enumerative Combinatorics (found here). Show that the total number of cycles of all even permutations of $[n]$ and the total number ...
1
vote
1answer
355 views

Partition function without repetitions of parts and largest part $k$

Um, well, I think the title pretty much says it all. Nevertheless, allow me to explain. I am aware of a certain partition function $Q(n, k)$ that is supposed to remove duplication of parts and $k$ ...
3
votes
1answer
224 views

For integer partitions, why does $e(n)-o(n)=k(n)$?

I denote by $e(n)$ is the number of partitions of $n$ with an even number of even parts, $o(n)$ the number of those with an odd number of even parts, and $k(n)$ the number of those that are ...
4
votes
1answer
373 views

Exponential generating function for restricted compositions

I wanted to know if it is possible to use exponential generating functions to evaluate composition of N using K distinct numbers (where the supply of numbers is infinite)? For e.g if N=10 and ...
5
votes
3answers
217 views

Two series relations, each one implies the other - from Andrews' partition book

That's my first question here, and i was encouraged to post because my question in MathOverflow (HERE) was beautifully and fast answered. But my questions in not at research level... As i said there, ...
4
votes
1answer
96 views

HAKMEM 18(B): Cubic Partitions

Taken from HAKMEM 18. Quoting... A partition of $N$ is a finite string of non-increasing integers that add up to $N$. Thus 7 3 3 2 1 1 1 is a partition of 18. Sometimes an infinite string of ...
3
votes
2answers
110 views

Generating sequences of numeric partitions

Definition: A tuple $\lambda = (\lambda_1, \ldots, \lambda_k)$ of natural numbers is called a numeric partition of $n$ if $1 \leq \lambda_1 \leq \cdots \leq \lambda_k$ and $\lambda_1 + \cdots + ...
2
votes
1answer
442 views

How to find the coefficient of a term in this expression

How to determine the coefficient of z3q100 in I stumbled upon this problem while trying to solve this type of partition problem: Find the number of integer solutions to x + y + z = 100 such that 3 ...
11
votes
7answers
4k views

Making Change for a Dollar (and other number partitioning problems)

I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quoted ...