All Questions
Tagged with integer-partitions generating-functions
45
questions with no upvoted or accepted answers
4
votes
1
answer
134
views
The total number of all different integers in all partitions of n with smallest part $\geq 2$
I want to show that the total number of all different integers in all partitions of $n$ with smallest part $ \geq 2$ is $p(n-2)$.
Example: partitions of $7$ with smallest part $ \geq 2$ are (7), (5,2),...
- 53
3
votes
0
answers
104
views
Determine in how many ways the amount of 50 can be paid by bank notes of 1, 2, 5, 10, and 20
Determine in how many ways the amount of 50 can be paid by bank notes of 1, 2, 5, 10, and 20
I have used the generating function that
$G(x)=\frac{1}{1-x}\frac{1}{1-x^2}\frac{1}{1-x^5}\frac{1}{1-x^{10}...
- 63
3
votes
0
answers
190
views
Link between partition function and ordered partition function
The partition function $p(n)$ measures the number of partitions of $n$, or the number of ways in which natural numbers can be summed to produce $n$, without regard to order. For example, the ...
- 1,790
3
votes
0
answers
1k
views
Distributing problem using generating functions
For $r\in\mathbb{Z^*}$, let $a_r$ denote the number of ways to distribute $r$ identical objects into $3$ identical boxes, $b_r$ be the number of distributions so that the boxes are to be non-empty, ...
- 671
2
votes
0
answers
45
views
Express number of partitions into prime numbers using partitions into natural numbers.
Let $P(n)$ is number of partitions of $n$ into natural numbers.
$R(n)$ is number of partitions of $n$ into prime numbers.
Is there any expression that relates $P(n)$ , and $R(n)$? I look for ...
- 1,382
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
2
answers
84
views
Prove that $p_2(n) = \left \lfloor{\frac{n}{2}}\right \rfloor+1$ using identity
Prove that $p_2(n) = \left \lfloor{\frac{n}{2}}\right \rfloor+1$ using the identity $$\frac{1}{(1-x)(1-x^2)}=\frac{1}{2}\left(\frac{1}{(1-x)^2}\right)+\frac{1}{2}\left(\frac{1}{1-x^2}\right)$$
where $...
- 85
2
votes
0
answers
53
views
Degree of generating polynomial associated with two partitions
Let $\lambda=(\lambda_1,\ldots,\lambda_k)$ and $\mu=(\mu_1,\ldots,\mu_r)$ be two strictly increasing sequences of non-negative integers ($r$ and $k$
may be $0$, in which case the sequence is empty, ...
- 2,488
2
votes
0
answers
192
views
Is there a generating function for this sequence?
The sequence is:
1, 2, 3, 5, 8, 12, 18, 25, 35, 50, 69, 93, 126, 167, 220, 290, 377, 486, 627, 800, 1017, 1290, 1623, 2032, 2542, 3161, 3917, 4843,...
It is related to partitions of $n$. It is a ...
- 351
2
votes
0
answers
59
views
What are the combinatorial numbers appearing in these repeated derivatives?
Let $f$ be a $C^\infty$-function and define $g(x) = \exp(f(x))$.
I am interested in the higher derivatives $g^{(1)}, g^{(2)}, \ldots$ of $g$.
Let $\lambda$ be a partition of $n$, i.e. a tuple of ...
- 10.7k
2
votes
0
answers
533
views
Proving Identities using Partition and Generating Function
I have a problem with these two questions:
Let $P_E(n)$ be the number of partitions of $n$ with an even number of parts, $P_O(n)$ the number of partitions of n with an odd number of parts, and $...
- 1,502
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
2
votes
0
answers
475
views
Generating Function of Integer Partition Such that at Least One Part is Even
I've been having a few issues coming up with a generating function for an integer partition such that at least one part is even. What I have got so far is:
The generating function with no ...
- 165
1
vote
0
answers
34
views
Proving an Identity on Partitions with Durfee Squares Using $q$-Binomial Coefficients and Generating Functions
Using the Durfee square, prove that
$$
\sum_{j=0}^n\left[\begin{array}{l}
n \\
j
\end{array}\right] \frac{t^j q^{j^2}}{(1-t q) \cdots\left(1-t q^j\right)}=\prod_{i=1}^n \frac{1}{1-t q^i} .
$$
My ...
- 195
1
vote
0
answers
60
views
Generating Function for Modified Multinomial Coefficients
The multinomial coefficients can be used to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + ...} \right)^n}$ in the basis of monomial symmetric polynomials (MSP). For example,
$$\...
- 51