All Questions
7
questions
-3
votes
4
answers
129
views
Find a minimal set whose elements determine explicitly all integer solutions to $x + y + z = 2n$
Is there a way to exactly parameterise all the solutions to the equation $x + y + z = 2n$, for $z$ less than or equal to $y$, less than or equal to $x$, for positive integers $x,y,z$?
For example, for ...
- 67
0
votes
1
answer
109
views
Find number of solutions for equation: $~x+y+z=n~$ where $~x,~y,~z~$ are non-negative whole numbers and $~x\le y\le z~$.
Find number of solutions for equation: $~x+y+z=n~$ where $~x,~y,~z~$ are non-negative whole numbers and $~x\le y\le z~$.
First I used substitution $~y=x+k,~ z=y+k~$ where $~k\ge 0~$(that is $y=x+k, z=...
- 533
0
votes
0
answers
181
views
A generating function $G(x)=-\frac{\frac{1}{x^5}(1+\frac{1}{x})(1-\frac{1}{x^2})}{((1-\frac{1}{x})(1-\frac{1}{x^3}))^2}$ related to partitions of $6n$
Fix a sequence $a_n={n+2\choose 2}$ of triangular numbers with the initial condition $a_0=1$,such that
$1,3,6,10,15,21,\dots$
given by
$F(x)=\frac{1}{(1-x)^3}=\sum_{n=0}^{\infty} a_n x^n\tag1$
...
- 2,813
7
votes
1
answer
233
views
What is the significance of this identity relating to partitions?
I was watching a talk given by Prof. Richard Kenyon of Brown University, and I was confused by an equation briefly displayed at the bottom of one slide at 15:05 in the video.
$$1 + x + x^3 + x^6 + \...
- 1,866
0
votes
2
answers
101
views
Number of partitions of $n$ formed by combinations of $2$ and $4$
I'm trying to find the number of partitions of a natural number that are a combination of $2$ and $4$.
For example: $$6 = 2+2+2 = 2+4
\Rightarrow p_6 = 2$$
So I start by defining $p_n$ as the ...
- 330
1
vote
0
answers
77
views
Generating function for writing an even number as a sum of at most k squares
I would like to find the exact number of ways in which $n$ can be represented as a sum of at most $k$ squares such that each term is less than or equal to say, $N$. A generating function for this ...
- 517
2
votes
1
answer
100
views
Explain this generating function
I have a task:
Explain equation:
$$\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 ...
- 21