All Questions
Tagged with integer-partitions generating-functions
210
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Prove that number of partitions of $n$ into parts $2,5,11$ modulo $12$ is the same as the number of partitions into distinct parts $2,4,5$ modulo 6
Let $A(n)$ denote the number of partitions of the positive integer $n$ into parts congruent to $2$, $5$, or $11$ modulo $12$. Let $B(n)$ denote the number of partitions of $n$ into distinct parts ...
- 1,248
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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
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OGF for partitions of integer r-N into even parts less than or equal to 2m (N is some integer that may depend on m).
Let $a_r$ be the number of partitions of the integer r-N into even parts, the largest of which is less than or equal to 2m, where N is some integer that may depend on m. Find the ordinary generative ...
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Integer partition generating functions problem
This is a homework question so I'd prefer to not receive a full solution but rather a hint.
The question is that $r_3(n)$ denotes the number of partitions of an integer $n$ into parts that are not ...
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2
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Integer partitions using generating functions
For each natural number $ n $ we consider the equation
$$x_{1}+2x_{2}+\dots+nx_{n}=n$$
Where $x_{1},\dots,x_{n}$ are nonnegative integers. Prove that this equation has the same number of solutions ...
user1009942
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3
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141
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Number of integer solutions of linear equation.
I have the following problem.
Assume we have an unlimited number of blocks of 1cm, 2cm and 3cm height. Ignoring the position of the blocks, how many towers of 15cm height can we build?
I know I must ...
- 51
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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
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Demonstration for the equal number of odd and unequal partitions of an integer
I'm having some problems trying to resolve one exercise from The art and craft of problem solving by Paul Zeitz. What this problem asks you is to prove that $F(x)$ is equal to 1 for all $x$, where
$$F(...
- 73
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Recurrence relations in the generating function of binary partitions
Let $b(n)$ denote the number of binary partitions of $n$, that is, the number of partitions of $n$ as the sum of powers of $2$. Define
\begin{equation*}
F(x) = \sum_{n=0}^\infty b(n)x^n
= \prod_{n=0}^\...
- 2,238
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What is the closed form solution to the sum of inverse products of parts in all compositions of n?
My question is exactly as in the title:
What is the generating function or closed form solution to the sum of inverse products of parts in all compositions of $n$?
This question was inspired by just ...
- 171
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What is the generating function for the number of partitions of an integer in which each part is used an even number of times?
What is the generating function for the number of partitions of an integer in which each part is used an even number of times? I'm trying to prove the number of partitions of an integer in which each ...
- 1
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83
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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 ...
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Let g_n equal the number of lists of any length taken from {1,3,4} having elements that sum to n.
For example, g_3 = 2 because the lists are (3) abd (1,1,1). Also g_4 = 4 because the lsits are (4), (3,1), (1,3), and (1,1,1,1). Define g_0 = 1.
(a) Find g_1, g_2, and g_5 by complete enumeration.
(b) ...
- 61
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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
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Using generating functions to count number of ways to plan a semester
This is the question:
The semester of a college consists of n days. In how many ways can we separate the semester into sessions if each session has to consist of at least five days?
My work:
If $A(x)$ ...
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