All Questions
Tagged with integer-partitions generating-functions
210
questions
3
votes
1
answer
85
views
Showing $\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}$
I want to show
\begin{align}
\prod_{n\geq 1} (1+q^{2n}) = 1 + \sum_{n\geq 1} \frac{q^{n(n+1)}}{\prod_{i=1}^n (1-q^{2i})}
\end{align}
I know one proof via self-conjugation of partition functions with ...
- 6,490
-1
votes
1
answer
43
views
Number of partitions of $n$ in which parts can occur at most 7 times
I have got the following answer without detailing the whole answer as I am looking precisely for the notation used here as I am new to graph theory, so please pardon me. After a series of operations, ...
- 1,189
0
votes
1
answer
47
views
Infinite product formula for $\sum_{n \geq 0} p_e(n)\cdot x^n$
If $n$ is an integer and $p_e(n)$ is the number of partitions for $n$ such that all parts are even, what would be an infinite product formula for $\sum_{n \geq 0} p_e(n)\cdot x^n$
- 101
1
vote
1
answer
82
views
Number of partitions of an integer with extra conditions
In how many ways can you write number 15 as a sum of positive integers if number 1 can appear 3 times at most in the sum and number 3 can only appear even number of times?
I've tried doing this as $$(...
- 41
1
vote
0
answers
58
views
Partition Generating Function and Euler Transform
Various sources on the web give the result that if {ai} and {bi} satisfy $$1+\sum_{n\geq 1}^n b_n z^n = \prod_{i\geq1}\frac{1}{(1-z^i)^{a_i}}$$ then we have that
$$1+B(z)=exp\sum_{k\geq1}\frac{A(z^k)}...
- 11
1
vote
1
answer
62
views
Deriving a partition identity from some given identities
I am trying questions from Apostol Introduction to ANT of Chapter partitions and need help in deducing this identity.
Problem is question 6(a) which will use some information from 2 and 5(b).
...
user775699
2
votes
1
answer
93
views
2 questions related to generating function of partition function in number theory
I am self studying chapter partitions (chapter number-14) from Apostol Introduction to analytic number theory.
I had studied that chapter earlier also and had questions but as I don't have anyone to ...
user775699
2
votes
0
answers
92
views
Generating function for reverse plane partitions
The MacMahon function is a generating function over total boxes $n=|\pi|$ for the total number $p_n$ of 3d plane partitions $\pi$:
$$ \prod_{k=1}^{\infty} \frac{1}{(1-x^k)^k} = \sum_{n}p_n x^n$$
Is ...
- 85
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
100
views
Polynomials and Partitions with restrictions
(a) Let
$$P(x)=\sum_{n=0}^{\infty} p_nx^n=1+x+2x^2+3x^3+5x^4+7x^5+11x^6+\cdots$$
be the partition generating function, and let $Q(x)=\sum_{n=0}^{\infty} q_nx^n$, where $q_n$ is the number of ...
- 334
0
votes
1
answer
90
views
finding a general expression to the inequality $x_1+x_2+x_3+x_4\leq n$ with generating function
First of all, the integer solutions of $x_1+x_2+x_3+x_4=n$ where $0\leq x_1,$ $2\leq x_2,$ $x_3$ is a multiple of $4$, and $0\leq x_4\leq 3$ can be represented by:
\begin{align}
x_1:&\ (1+x+x^2+x^...
- 645
0
votes
1
answer
635
views
Generating function of composition with odd part fixed
I have met the following question: consider the set of compositions with any number of parts where each odd part is equal to 1. E.g $(1,2,1,3,1,4,1,\ldots)$. Then I am asked to find the generating ...
1
vote
1
answer
77
views
bijections between sets
Let $P_n$ be the set of compositions of $n$ where each part is at least $2, Q_n$ be the set of compositions of $n$ where each part is odd, and $R_n$ be the set of compositions where each part is $1$ ...
user747916
1
vote
1
answer
83
views
Generating Functions Combinatorial argument
Show that any number of partitions of $r+k$ into $k$ parts is equal to the number of partitions of
$$r+\binom{k+1}{2}$$ into $k$ distinct parts for $r \geq k$.
I would like to see a proof for this,...
- 45
1
vote
3
answers
421
views
Partitions using only powers of two on $1000.$
How many ways are there to write $1000$ as a sum of powers of $2,$ ($2^0$ counts), where each power of two can be used a maximum of $3$ times. Furthermore, $1+2+4+4$ is the same as $4+2+4+1$. These ...
- 551