All Questions
Tagged with integer-partitions generating-functions
210
questions
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Generating function and currency
We assume that we have a country's currency that contains three coins worth 1, 3, and 4. How many ways can we get an amount of $n$ using these three pieces?
In others words what is the number of ...
- 37
1
vote
1
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49
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Generating function of partitions of $n$ in $k$ prime parts.
I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$.
I know ...
2
votes
1
answer
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About the product $\prod_{k=1}^n (1-x^k)$
In this question asked by S. Huntsman, he asks about an expression for the product:
$$\prod_{k=1}^n (1-x^k)$$
Where the first answer made by Mariano Suárez-Álvarez states that given the Pentagonal ...
2
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1
answer
38
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Prove $\sum_{j=0}^{n} q^{j^{2}}\binom{n}{j}_{q^{2}}$ generates the self-conjugate partitions with part at most $n$.
Prove $\sum_{j=0}^{n} q^{j^{2}}\binom{n}{j}_{q^{2}}$ generates the self-conjugate partitions with part at most $n$, and that it equals $(1+q)(1+q^{3})\cdot\cdot\cdot(1+q^{2n-1})$.
For the first part, ...
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Need help with part of a proof that $p(5n+4)\equiv 0$ mod $5$
Some definitions:
$p(n)$ denotes the number of partitions of $n$.
Let $f(q)$ and $h(q)$ be polynomials in $q$, so $f(q)=\sum_0^\infty a_n q^n$ and $h(q)=\sum_0^\infty b_n q^n$. Then, we say that $f(q)\...
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34
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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
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43
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Find generating function for the number of partitions which are not divisible by $3$. [duplicate]
I'm trying to find the generating function for the number of partitions into parts, which are not divisible by $3,$ weighted by the sum of the parts. My idea is that we get the following generating ...
- 121
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1
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81
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Integer partitions with summands bounded in size and number
This book says it's easy, but to me, it's not. :(
As for 'at most k summands', in terms of Combinatorics, by using MSET(),
$$ MSET_{\le k}(Positive Integer) = P^{1,2,3,...k}(z) = \prod_{m=1}^{k} \frac{...
- 185
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60
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Generating Function for Modified Multinomial Coefficients
The multinomial coefficients can be used to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + ...} \right)^n}$ in the basis of monomial symmetric polynomials (MSP). For example,
$$\...
- 51
4
votes
1
answer
339
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Why do Bell Polynomial coefficients show up here?
The multinomial theorem allows us to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + {x_4} + ...} \right)^n}$. I am interested in the coefficients when expanding ${\left( {\sum\...
- 51
1
vote
0
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47
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Congrunces of partitions into distinct parts
Let $P_{d}(n)$ denote the number of partitions of n into distinct parts. The generating function of $P_{d}(n)$ s given by:
$$ \sum_{n \geq 0}P_{d}(n)q^{n}= \prod_{n \geq 0} (1+q^{n}).$$
Now let $P_{2,...
- 21
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1
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76
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Generating function of number of partitions of $n$ into all distinct parts
I am trying to grasp this example from the book A Walk Through Combinatorics:
Show that $\sum_{n \ge 0} p_d(n)x^n = \prod_{i \ge 1}(1+x^i)$ where
$p_d$ stands for partitions of $n$ into all distinct ...
- 309
1
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1
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319
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Generating function of number of partitions of $n$ into parts at most $k$
I am trying to grasp the intuition behind this example.
Show that $\sum_{n \geq 0} p_{\leq k}(n)x^n = \prod_{i=1}^k \frac{1}{1-x^i}$
where $p_{\leq k}(n)$ denotes the number of partitions of the ...
- 309
3
votes
1
answer
111
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Finding the number of integer composition using only a specific pair of numbers [closed]
I want to find the number of integer compositions of 19 using numbers 2 and 3.
If I wanted to find the number of integer compositions of 19 using 1 and 2, I could write it as $F(n)=F(n-1)+F(n-2)$ ...
- 309
1
vote
1
answer
53
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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
4
votes
1
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134
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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
2
votes
2
answers
168
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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
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1
answer
143
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Generating function of ordered odd partitions of $n$.
Let the number of ordered partitions of $n$ with odd parts be $f(n)$. Find the generating function $f(n)$ .
My try : For $n=1$ we have $f(1)=1$, for $n=2$, $f(2)=1$, for $n=3$, $f(3)=2$, for $n=4$, $...
user1137804
1
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1
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Identity about generating function related to binary expression of integers
For any nonnegative integer $n$, let $\mu(n)$ be $1$ if the binary expression of $n$ contains even number of ones; and $-1$ if the binary expression of $n$ contains odd number of ones. For example, $\...
user1089451
1
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1
answer
96
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Representation of number as a sums and differences of natural numbers
Lets consider all the combinations of:
$$1+2+3+4=10,\ \ 1+2+3-4=2,\ \ 1+2-3+4=4,\ \ 1+2-3-4=-4, $$
$$1-2+3+4=6,\ \ 1-2+3-4=-2,\ \ 1-2-3+4=0,\ \ 1-2-3-4=-8,$$
$$-1+2+3+4=8,\ \ -1+2+3-4=0,\ \ -1+2-3+4=2,...
- 1,818
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Generating function of sequence related to binary expression of integer as follows
For any $n \in \Bbb N$, let $O(n)$ be $1$ if the binary expression of $n$ contains even number of ones; and $-1$ if the binary expression of $n$ contains odd number of ones, and $Z(n)$ be $1$ if the ...
user1089451
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0
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178
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Generating function of integer partitions at most m parts
The generating function of partition integer with the maximum number is m is $$
\frac{x^m}{(1-x)(1-x^2)\cdots(1-x^m)}
$$
It's quite easy to understand if you expand it $$
(1+x+x^2+\cdots)(1+x^2+x^4+\...
2
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178
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two-variable generating function for all number partitions
I'm stuck to solve the problems below:
(a) Let $p(n, k)$ be the number of partitions of $n$ into exactly $k$ parts. Show that
$$
\sum_{n, k \geq 0} p(n, k) x^{n} t^{k}=\prod_{i \geq 1} \frac{1}{1-x^{i}...
- 143
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3
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73
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Partitions without repetition
I want to know how many partitions without repetition 19 has. I know I should see the coefficient of $x^{19}$ in $$\prod_{k=1}^\infty(1-x^k),$$
but i'm having trouble finding it. Ay hint?
- 153
1
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1
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86
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Number of partitions with distinct even parts/parts with multiplicity $\leq 3$
I am supposed to solve a problem regarding partitions of $n \in \mathbb{N}$ into:
distinct even parts
parts with multiplicity $\leq 3$
I am supposed to prove that 1. and 2. are equal.
So I tried ...
- 333
7
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1
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167
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How to prove the following resummation identity for Erdős–Borwein constant?
Question:
How to prove
$$\sum_{m=1}^{\infty}\left(1-\prod_{j=m}^{\infty}(1-q^j)\right) = \sum_{n=1}^{\infty}\frac{q^n}{1-q^n} \tag{1}$$
for all $q \in \mathbb{C}$ such that $\left|q\right| < 1$?
...
- 3,132
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Coefficients in Gaussian polynomials [duplicate]
Let
$$
\binom{n}{k}_{\!q} = \frac{(1-q^n) \cdots (1-q^{n-k+1})}{(1-q) \cdots (1-q^k)}
$$
be the Gaussian polynomials. For example,
$$
\binom{6}{3}_{\!q} = 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9.
$$
...
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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
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Infinite product expression of partition function
I'm working on a problem (specifically, I'm using an exam paper without course notes to prepare for a course starting in September),
Define the partition function $P(q)$ and give its infinite product ...
- 2,281
1
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Counting Partitions of $k$ into $j$ distinct parts, with sizes restricted by a sequence.
This question has come up in my research, but since I usually do not do combinatorics, I am struggling to find out any information regarding these sequences of numbers.
Let $L_n = L_1, L_2\dots$ be a ...
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