# Tagged Questions

Questions related to the different ways of expressing an integer as a sum of integers; or, questions related to the subdivision of a set into smaller disjoint sets; questions related to the subdivision of an interval into smaller intervals that intersect only at the endpoints.

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### Finding asymptotycs of partition function

I have been stuck in this problem and have no idea of how to solve it. I have a hint from the book but don't really see how to use it. Any suggestion or hint would be really appreciated. Thanks! ...
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### Generating function for the partition function [duplicate]

Could someone explain what is the reasoning behind the following equality? Or maybe direct me to a proof of the following equality? $$\sum_{n=0}^{\infty}p(n)x^n = \prod_{k=1}^{\infty}(1-x^k)^{-1}$$ ...
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### Formula for the number of partitions of 2N elements [duplicate]

We have a set $S$ of $2N$ distinct elements. I want to partition it into $N$ parts each containing 2 elements. My motivation is partitioning a group of people into pairs. What is the formula that ...
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### How many partitions of $n$ are there?

Considering a partition to be an ordered $n$-tuple $(m_1, m_2, m_3, ..., m_n)$ with all the numbers $m_i$ natural, $m_1 \le m_2 \le m_3 \le ... \le m_n$, and $m_1+m_2+...+m_n=n$: how many of those $n$-...
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### $\forall n \in N$ edges of $K_{2n+1}$ can be partitioned to hamiltonian cycles.

A hamiltonian cycle is a cycle which visits each vertex of the graph exactly once. A hamiltonian path is a path which visits each vertex of the graph exactly once. We need to prove that: 1-...
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### Partition $\lambda/\mu$ notation?

When talking about two partitions $\lambda$ and $\mu$, what does the operation $$\lambda/\mu$$ mean? When Macdonald introduces partitions in the first chapter of "Symmetric functions and Hall ...
2answers
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### If $n$ is a natural number $\geq 2$, how many ways are there to partition $n$ with natural numbers $\geq 2$?

The partitions are "ordered", meaning that for 5, from (2,3), (3,2) and (5) all are valid. So for the first few numbers one gets: ...
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27 views

### Using Young Diagram to show the number of partitions of n is equal to number of partitions of 2n into n parts

Show that the number of partitions of the integer $n$ is equal to the number of partitions of $2n$ into $n$ parts using a Young Diagram. I can't seem to figure out any way to create a bijection ...
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### Pairs of Numbers such that the sum of their digits is Equal

How many pairs of numbers $(n,m)$ whose digits add up to the same sum, where $n\ne m$ and $(n,m)=(m,n)$ such that $m,n\le k$ , are there for a given $k$? Observing this in base 10 we are looking at ...
1answer
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### Partition function and Fibonacci n-th number upperbound

i need to proof this upperbound: "Call $p(n)$ the number of partitions of n (integer) and $F_n$ the $n-th$ Fibonacci number. Show that $p(n)\leq p(n-1) + p(n-2)$ and that $p(n)\leq F_n$ for $n\geq 2$"...
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### Is there a standard representation for the composition of an integer?

In the same way as "$\vdash$" represents integer partition, what would be the symbol for composition (ordered partition)?
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### How to find last digit of number of partitions?

Is there simpler way to find last digit or last k digits (in any base) of p(n) than calculating full number?
1answer
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### Can partition be defined for numbers other than positive integers?

An integer partition is a decomposition of a positive integer into positive integers that add up the original number. Apart from $\mathbb N_{> 0}$, does this definition apply to other numeric sets, ...
1answer
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### Partitioning the edges of $K_n$ into $\lfloor \frac n6 \rfloor$ planar subgraphs

Why is it impossible to partition the edges in $K_n$ into $\lfloor \frac n6 \rfloor$ planar subgraphs for $n \ge 6$? I'm just stuck at the beginning and can't figure out how to go about this problem. ...
2answers
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1answer
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### Trouble with integer partition proof

I am reading Keller & Trotter: Applied Combinatorics, pg. 155, and I am having trouble with an intermediate step in a proof. The proof deals with integer partitions: And the part I can't ...
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### What is a Fixed Partition?

I have two partition, P= {$x_0,x_1,...,x_n$} and Q= {$y_0, y_1,...,y_n$}. P is a general partition of the interval of [a,b] and Q is a fixed partition of [a,b]. What does it mean for Q to be fixed? ...
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### Non-Uniform grid

Let say I have $v_0\in [v_1,v_m]$ (say $v_0=0.04\in [0.004,0.24]$) I would like to find a $1$-to-$1$ map that map $[0,1]$ to $[v_1,v_m]$ and more cluster points around $v_0$ from two sides. It seem ...
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33 views

### How many ways can N objects be split into groups of 3?

I've found that if there are $N$ objects split up into two groups of $n_1$, and $n_2=N-n_1$ then this can be divided into $$\frac{N!}{n_1!(N-n_1)!}$$ groups. How can I extend this onto splitting $N$ ...
2answers
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### Understanding the definition of refinement of a partition $P$.

Definition. Let $P$, $P'$ be partitions at $[a,b]$. We say that $P'$ is a refinement of $P$ if $P'$ $\supseteq$ $P$. I did not understand really of the definition of refinement. Can you give me a ...
1answer
26 views