1
vote
0answers
59 views

Integer partitions without rotated solutions?

I'm searching for an algorithm to determine a list of all integer partitions of a number $n$ into a fixed number $m$ of summands (say $n=6$ and $m=4$), for instance to be stored into a list of ...
4
votes
1answer
87 views

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as ...
0
votes
1answer
49 views

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, ...
2
votes
1answer
72 views

The number of partitions of $n$ and the $n$th Fibonacci number.

I'm very sorry if this is a duplicate in any way. There's a lot of material out there on connections between these sequences so it's a possibility . . . Let $P_n$ be the number of partitions of $n$ ...
1
vote
0answers
16 views

asymptotic approximation for number of partitions of integer that do contain 1 nor 2

Hardy and Ramanujan provided a famous asymptotic approximation to $P(n)$ the number of partitions of an integer $n$ when $n$ gets large. I wonder if there is an asymptotic approximation to ...
0
votes
1answer
52 views

Partition numbers with restriction on the greatest part *and* on the number of positive parts

I’m looking at partition numbers. OEIS A008284 says that the number of partitions of $n$ in which the greatest part is $k$, $1 \le k \le n$, is equal to the number of partitions of $n$ into $k$ ...
0
votes
0answers
26 views

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 ...
3
votes
1answer
70 views

Existence of a prime partition

I'm interested in finding out whether there exists a prime partition of a given positive integer $N>1$ such that the partition has specific number of parts. For instance, as given in another ...
4
votes
1answer
52 views

A conjecture on partitions

While trying to prove a result in group theory I came up with the following conjecture on partitions: Let $b(i,j)$ be the number of partitions of $i$ with greatest part exactly equal to $j$ , for ...
2
votes
0answers
35 views

Partitions and divisor functions: what is known about their relations?

If $i\geq 1$ is an integer, we have the following integer valued functions (for any integer $n\geq 0$): \begin{align} p_i(n)&=\textrm{the number of }i\textrm{-dimensional partitions of ...
1
vote
1answer
63 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 ...
1
vote
1answer
61 views

Partitions of n with certain conditions

Let $p$ be prime and $n$ be any integer. Suppose $t=(n^{a_n}, \dots, 2^{a_2}, 1^{a_1}) \vdash n$, (i.e. $t$ is a partition of $n$, where we group repeated integers, so, for example, $2^{a_2}$ means ...
4
votes
1answer
82 views

Which positive integers can be written in the following form?

I was investigating a generalisation of this problem and found that it reduced to finding where the expression $$\frac{p(p+2m+1)}{2}$$ is an integer, where $p\ge 2$ and $m \ge 0$. Since exactly one of ...
2
votes
2answers
93 views

How to use pentagonal number theorem to determine partitions of n

Under the heading Pentagonal Number Theorem > Relation With Partitions, Wikipedia gives the equation $$ p(n) = \sum_{k} (-1)^{k-1}p(n-g_k) $$ where the summation is over all nonzero integers k ...
10
votes
1answer
207 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $S(\alpha)=\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \}$. (Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers.) Question. Is ...
3
votes
2answers
119 views

Partition Identity Proof

Hey guys I am trying to prove the following identity. $p(n)\leq p(n-1)+p(n-2)$ for every ${n}\geq 1$. I worked on breaking it down into steps. I think that the best way to go about with this is in ...
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 ...
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 ...
1
vote
1answer
39 views

About the parity of the partition function.

I am reading this Kolberg's article, where he proofs that the partition function takes both even an odd values infinitely often. http://www.mscand.dk/article.php?id=1555 Although I'm sure it's ...
1
vote
0answers
39 views

Discriminating integer partitions

Given a fixed positive integer, say $n$, letn $P_n$ be the set all partitions of $n$, where each partition itself is a set i.e. order is discarded and each part is less than 5. Can we establish a ...
3
votes
1answer
64 views

Divisibility in the partition function

There are several formulas for the calculation of the partition function $p(n)$ of an integer $n$. The last has been found by Ken Ono in 2011. My question is: using these formulas is it possible to ...
3
votes
2answers
423 views

How many combinations of $3$ natural numbers are there that add up to $30$?

How many combinations of $3$ natural numbers are there that add up to $30$? The answer is $75$ but I need the approach. Although I know that we can use $_{(n-1)}C_{(r-1)}$ i.e. $_{29}C_2 = 406$ but ...
4
votes
0answers
131 views

Which unordered partition of $n$ gives rise to the largest number of ordered partitions?

A quick look at the wikipedia article on partitions of $n \in \mathbb{N}$ shows that the number of ordered partitions is $2^{n-1}$, and the number of unordered partitions is asymptotically $ \sim ...
4
votes
1answer
77 views

Partitions of a prime power into powers of the same prime

Fix a prime $p$, and $k$ a natural number. The question is then: How many partitions of $p^k$ are there into powers of $p$? So, for instance, if $p = 2$ and $k = 2$, there are 4, namely (4), (2, 2), ...
3
votes
1answer
109 views

What will be time complexity using dynamic programming

If I were to find a set of 10 positive integers whose sum = 87248 and the sum of their squares = 447804117. Using an efficient dynamic programming, what will be time complexity of this kind of ...
1
vote
2answers
76 views

Compute all the sets of 87248 into 10 parts

How many sets are possible? I have to compute all the sets of 87248 into 10 parts (Additional conditions which may be useful are: integers can be repeated, every integer is less than or equal to ...
2
votes
1answer
121 views

sum factors of natural numbers

Using natural numbers 1,2,...n, in how many ways can the number n be formed from the sum of one or more smaller natural numbers? I thought it would be an easy problem but i couldn't figure it out. ...
3
votes
1answer
152 views

Prove that $10\mid A000793(n\ge16)$

Prove that if $n\ge16,$then $10\mid g(n),$where $g(n)$ is the largest LCM of partitions of $n$. For more information,see http://oeis.org/A000793 Here is the list of $g(n)$ for $n>0,$ ...
3
votes
1answer
86 views

Is partition function increasing function?

I have some exercises which require knowing the number of partitions of particular numbers, so I used some python code which I found on internet to compute the values of the partition function for the ...
0
votes
1answer
29 views

Make a partition that contains a set of points??

I am given a set of $M$ points in a segment (the edges are also points in this set) I would like to partition the segment (with equidistant points), in such a way that my partition contains all these ...
2
votes
1answer
104 views

Related to the partition of $n$ where $p(n,2)$ denotes the number of partitions $\geq 2$

I was trying the following question from the Number theory book of Zuckerman. If $p(n,2)$ denotes the number of partition of n with parts $\geq2$, prove that $p(n,2)>p(n-1,2)$ for all $n\geq 8$, ...
6
votes
2answers
140 views

For every partition $\pi$ of a fixed integer $n$, $\sum{F(\pi)}=\sum{G(\pi)}$

I need to prove the following question. For every partition $\pi$ of a fixed integer $n$, define $F(\pi)$=number of occurrences of 1 as a summand, and $G(\pi)$=no. of distinct summands in the ...
3
votes
1answer
216 views

bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n

Suppose $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is a partition of $2n$ where $n\in\mathbb N$ satisfying the following conditions: (1) $\lambda_k=1$. (2) $\lambda_i−\lambda_{i+1}\leq 1$ for ...
0
votes
2answers
105 views

Get all possible partitions of number

Is there a way to get all possible partitions of an integer? Possibly specifying max and/or min summand. I'm interested in partitions themselves, not just partition count.
5
votes
1answer
86 views

Elementary proof of a bound on the order of the partition function

I am interested in the asymptotic order of the partition function $p(n)$. The paper Asymptotic Formulae in Combinatory Analysis proves there are constants $A$,$B$ such that $e^{A\sqrt{n}} < p(n) ...
2
votes
1answer
660 views

Partitions of an integer into k parts.

I am interested in knowing whether an exact formula (analogous to the Hardy-Ramanujan-Rademacher formula for $p(n)$) for the number of partitions of a positive integer into k parts is known. I tried ...
2
votes
3answers
103 views

Number decomposition

Recently I encountered a problem I was not familiar with. So hope someone can help me for this. Here is the problem. Given any odd integer, how many different ways of decomposition into sum of three ...
1
vote
1answer
110 views

Nonnegative integers with unique restricted partitions

I have come over the following problem in elementary number theory: suppose that a positive integer $k$ is given. Is it possible to find, for every such $k$, a nonnegative integer $N(k)$ and a set of ...
1
vote
3answers
358 views

Finite or infinite set?

Due to my not-so-advanced math skills, this question may take a few attempts to state clearly: Consider the unordered pair (2-tuple) partitions of n (e.g. with n=4, we have {{4,0},{3,1},{2,2}}). ...
4
votes
2answers
312 views

Lower bounds for the partition function

In this question we consider the partition function $p(n)$ - that is, the number of ways to express $n$ as a sum of positive integers. One easy exercise is to show that $$ p(n) \geq 2^{\lfloor ...
3
votes
2answers
628 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 ...
2
votes
1answer
198 views

How many circular distinct compositions of $n$ into $k$ parts at most $g$

how many circular distinct sequences (e.g. $(4124)\equiv(4412$)) are there that sum up to $n$, have $k$ elements may be non-negative integers at most $g$? In other words: we're looking for the number ...
0
votes
1answer
36 views

The “theory of compound partitions”

I was reading a sort of mini-bio on Sylvester the other day and a "Theory of Compound Partitions" was mentioned in the discussion of his research interests. I wanted to ask, is this the same or the ...
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 ...
6
votes
1answer
192 views

When is the sum of first $n$ numbers equal to the sum of the next $k$ numbers?

When is the sum $1+2+\cdots + n = (n+1) + (n+2) + \cdots +(n+k)$? The easiest solution $(n,k)$ is $(2,1)$. For example, $1+2 = 3$. Do any others exist? Roots of $(n+k)^2 + (n+k) = 2n^2 +2n$ give ...
4
votes
1answer
242 views

Using Durfee squares to prove partition identities?

I was surprised to learn of Durfee squares, which can be visually explained as the largest square contained within a partition's Ferrers diagram. Moreover, partition identities have always amused me ...
3
votes
1answer
556 views

Generating Function for Even Number of Odd Parts

How I would write the generating function for a partition of a positive integer n with an even number of odd parts? Any hints or suggestions will be greatly appreciated.
2
votes
1answer
179 views

number of additive partition

I have a question related with number of additive partition or method similar like this: $$p(5)=1+4=2+3=1+1+1+1+1=1+1+1+2=1+2+2=1+1+3$$ For a given number $n$,if we are trying to calculate ...
3
votes
2answers
619 views

Same number of partitions of a certain type?

Is there a quick explanation of why the number of partitions of $n$ such that no parts are divisible by $d$ is the same as the number of partitions of $n$ where no part is repeated $d$ or more times, ...
1
vote
2answers
110 views

Every number $n^k$ can be written as a sum of $n$ distinct odd numbers

I wish to prove that for $n,k\in\mathbb{N} > 1$, we can always write $n^k$ as a sum of $n$ odd positive integers. I have an idea of how to approach this, but my method seems to cumbersome. I am ...