All Questions
29
questions
1
vote
1
answer
53
views
The product of components in the partition of a number.
Let $f(n,k)$ be the sum of expressions of the form $x_1 \cdot x_2 \cdot \ldots \cdot x_k$, where the sum counts over all solutions of the equation $x_1 + \ldots + x_k = n$ in natural numbers. Find ...
- 675
2
votes
2
answers
168
views
Finding the generating function.
Find the generating function for the number of solutions for the equation $x_1+x_2+x_3+x_4 = n$, where $x_1,x_2,x_3,x_4\geq1$, and $x_1 < x_2$.
My attempt so far: I have tried putting a $y$ value ...
- 165
1
vote
1
answer
86
views
Number of partitions with distinct even parts/parts with multiplicity $\leq 3$
I am supposed to solve a problem regarding partitions of $n \in \mathbb{N}$ into:
distinct even parts
parts with multiplicity $\leq 3$
I am supposed to prove that 1. and 2. are equal.
So I tried ...
- 333
0
votes
0
answers
75
views
Question about number of generating functions
I know that the generating function for the number of integer partitions of $n$ into distinct parts is
$$\sum_{n \ge 0} p_d(n)x^n = \prod_{i \ge 1}(1+x^i)$$
I'm trying to use this generating function ...
- 27
1
vote
2
answers
327
views
Partitions into distinct even summands and partitions into (not necessarily distinct) summands of the form $4k-2,k\in\Bbb N$
Prove that the number of ways to partition $n\in\Bbb N$ into distinct even summands is equal to the number of ways of partitioning $n$ into (not necessarily) distinct summands of the form $4k-2,k\in\...
- 4,670
0
votes
1
answer
83
views
Partitions of $n$ where every element of the partition is different from 1 is $p(n)-p(n-1)$
I am trying to prove that $p(n|$ every element in the partition is different of $1)=p(n)-p(n-1)$, and I am quite lost...
I have tried first giving a biyection between some sets, trying to draw an ...
- 763
7
votes
3
answers
234
views
Prove that $\prod_{i\geq 1}\frac{1}{1-xy^{2i-1}} = \sum_{n\geq 0} \frac{(xy)^{n}}{\prod_{i=1}^{n}\left( 1-y^{2i} \right)}.$
Prove that
$$\prod_{i\geq 1}\frac{1}{1-xy^{2i-1}} = \sum_{n\geq 0} \frac{(xy)^{n}}{\prod_{i=1}^{n}\left( 1-y^{2i} \right)}.$$
Here I am trying the following
\begin{align*}
\prod_{i\geq 1}\frac{1}{1-xy^...
- 376
1
vote
1
answer
819
views
Partitions of $n$ into even number of parts versus into odd number of parts
I'm under the impression that if $n$ is even then the number of partitions of $n$ into an even number of parts exceeds the number of partitions of $n$ into an odd number of parts. And the opposite if $...
- 462
2
votes
2
answers
128
views
Counting the number of partitions such that every part is divisible by $k$.
I would like to do this using generating functions. I'm comfortable with the generating function for the number of partitions of $n$: $$(1+x+x^2+\dots)(1+x^2+x^4+\dots)(1+x^3+x^6+\dots)\cdot\dots,$$ ...
- 462
0
votes
3
answers
256
views
How many ways can you pay $100 when the order in which you pay the coins matters?
Suppose there is a vending machine where you have to pay $100. You
have an unlimited amount of 1 dollar, 2 dollar and 5 dollar coins. How many ways
can you pay when the order in which you pay the ...
1
vote
1
answer
78
views
Computing problems and generating functions
Q. Find the generating function for the sequence $\{a_n\}$, where $a_n$ is the number of solutions to the equation: $a+b+c=k$ when $a, b, c$ are non-negative integers such that $a\ge2, 0\le b\le3$ and ...
2
votes
1
answer
62
views
Question about coefficients of generating functions
Theorem: Let $n> 0 \in \mathbb Z.$ Let $p_n$ stand for the number of integer partitions of $n$ and let $k$ be the number of consecutive integers in a partition. Then $p_n + \sum_{k \ge 1}(-1)^k(p_{...
- 23
0
votes
1
answer
75
views
Double Product in the proof of partitions of $n$
So I am supposed to show that the number of partitions of $n$ for which no part appears more than twice is equal to the number of partitions of n for which no part is divisible by 3. and there is one ...
- 1,073
0
votes
1
answer
46
views
Give an argument using generating functions
Let $l_n$ be the number of partitions of $n$ which have exactly two parts of size $1$.
Let $L(x) = \sum l_n x^{n}$ be the generating function for $(l_n)$. (Let $l_0 = 1$ and $l_1 = l_2 = l_3 = 0.)$
...
- 41
1
vote
0
answers
272
views
Generating function for Durfee Squares
Let the Durfee Square of length $d$ of a partition of $n$ be the largest square that can be fit into the Ferrer's shape. Call the blocks in the Ferrer's shape on the right of the partition's Durfee ...
- 41