# Tagged Questions

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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### (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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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) $$x=\sum_{i=1}^ks_{t_i} (t_i\in\{1,\dots,n\})$$ ...
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### 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 $n\geq 8$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...