2
votes
1answer
63 views

Explain this generating function

I have a task: Explain equation: $$\prod_{n=1}^{\infty}(1+x^nz) = 1 + \sum_{n=m=1}^{\infty}\lambda(n,m)x^nz^m $$ $\lambda(n,m)$ - is number of breakdown $n$ to $m$ different numbers (>0) It's ...
4
votes
1answer
86 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 ...
3
votes
1answer
69 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
34 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 ...
2
votes
0answers
59 views

What is the least integer of additive dimension 4?

Say that $m$ is the additive dimension of $n\in\Bbb N$, and write $m=\operatorname{ad}n$, if $m$ is the greatest integer for which there is an irredundant $m$-element set $M\subset\Bbb N$ that ...
2
votes
1answer
59 views

(i).Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n)$ without using the generating functions for $\pi_m(n)$.

Questions: $\pi_m(n)$ is defined as the number of partitions of n in which each part is no larger than m. (i).Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n)$ without using the generating functions for ...
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 ...
2
votes
1answer
120 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
85 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 ...
6
votes
1answer
166 views

Representations of an integer as the sum of other integers

Given a finite set $S$ of (distinct) integers $s_1, \dots, s_n$ and an integer $x$, I'm looking for all representations (where order is important) $$ x=\sum_{i=1}^ks_{t_i} (t_i\in\{1,\dots,n\}) $$ ...
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 ...
2
votes
0answers
143 views

Partition minimizing maximum of Euler's totient function across terms

Given natural numbers $M$ and $N$, I'd like to find a partition of $2^N$ with $M$ or fewer terms, $t_1 + t_2 + ... + t_M$, such that $\max(\phi(t_1), \phi(t_2), ..., \phi(t_M))$ is minimized, where ...
0
votes
3answers
113 views

Combination Question

I currently have an open question about counting the possible ways of summing numbers. I am still exploring all the ideas provided - those within my level of understanding. This is a question ...
5
votes
2answers
335 views

Find all ways to factor a number

An example of what I'm looking for will probably explain the question best. 24 can be written as: 12 · 2 6 · 2 · 2 3 · 2 · 2 · 2 8 · 3 4 · 2 · 3 6 · 4 I'm familiar with finding all the prime ...
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 ...
10
votes
4answers
1k views

Prime Partition

A prime partition of a number is a set of primes that sum to the number. For instance, {2 3 7} is a prime partition of $12$ because $2 + 3 + 7 = 12$. In fact, there ...
1
vote
1answer
303 views

The coin change problem in the quantitative way

Today,I came across this problem: Suppose you have a currency, named miso, in three denominations, $1, 10$ and $50$. In how many ways can $107$ miso be given in this currency if you have ...
1
vote
1answer
386 views

Number of solutions of Frobenius equation

I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$ where $x,y,z \in \{0,1,2,3,4,5,\dots\}$. My attempt: I noticed that that maximum value of $z$ could ...
4
votes
2answers
307 views

Natural set to express any natural number as sum of two in the set

Any natural number can be expressed as the sum of three triangular numbers, or as four square numbers. The natural analog for expressing numbers as the sum of two others would apparently be the sum ...