All Questions
Tagged with integer-partitions discrete-mathematics
16
questions
1
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1
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Generating functions of partition numbers
I don't understand at all why:
\begin{equation}
\sum\limits_{n=0}^\infty p_n x^n = \prod\limits_{k=1}^\infty (1-x^k)^{-1}
\end{equation}
Where $p_n$ is the number of partitions of $n$. Specifically ...
- 7,418
4
votes
4
answers
1k
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# of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers?
How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?
user223957
5
votes
3
answers
140
views
Plain integer partitions of $n$ using $r$ parts
Division of number $n$ on parts $a_1,...,a_r$ where $a_1 \le ... \le a_r$ we call a plain if $a_1 = 1$ and $a_i - a_{i-1} \le 1$ for $2 \le i \le r$. Find enumerator (generating function) for plain ...
user677339
3
votes
2
answers
2k
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Help understanding solution to growth of partition function
I'm currently a Combinatorics student trying to parse through this solution. I do not understand the proof currently. Any help understanding it is greatly appreciated.
Question Let the number of ...
- 31
-1
votes
1
answer
90
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Find a bijection to prove a fact about integer partitions
Prove that for fixed positive integers $k$ and $n$, the number of partitions of $n$ is equal to the number of partitions of $2n + k$ into $n + k$ parts.
I tried this with different values of n but ...
- 41
9
votes
1
answer
899
views
Expected Value for the Number of Parts of a Random Partition (Considering Only a Portion of the Partition Spectrum)
Let $n$ be a positive integer. If we take the set of all partitions of $n$ and choose a random partition from it (uniformly), then the expected value of the number of parts of this partition is a ...
- 113
7
votes
7
answers
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$a+b+c+d+e=79$ with constraints
How many non-negative integer solutions are there to $a+b+c+d+e=79$ with the constraints $a\ge7$, $b\le34$ and $3\le c\le41$?
I get that for $a\ge7$ you do $79-7=72$, $\binom{72+5-1}{5-1}=\binom{76}4$...
- 81
4
votes
2
answers
596
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Number of ways to represent a number as a sum of only $1$’s and $2$’s and $3$’s
How do I find the number of ways to represent a number as a sum of only $1$’s and $2$’s and $3$’s?
I think the title is self-explanatory.
E.g., if I have to represent $13$ as a sum of only $1$'s ...
- 223
3
votes
4
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1k
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Two hard number partition problems
For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$. Also define $p(0)=1.$
Problem 1.
Prove that $p(n)-p(n-1)...
- 1,517
2
votes
1
answer
836
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Find the total number of partitions of $12$ having unequal positive parts.
$P(k, n) = P(k - 1, n - 1) + P(k - n, n)$
Let $P^\star(k,n)$ denote the number of partitions of $k$ having exactly $n$ positive parts, all of which are unequal. For example, $P(8, 3)$ implies $8$ can ...
- 137
2
votes
1
answer
2k
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Partition function without repetitions of parts and largest part $k$
Um, well, I think the title pretty much says it all.
Nevertheless, allow me to explain.
I am aware of a certain partition function $Q(n, k)$ that is supposed to remove duplication of parts and $k$ ...
- 135
1
vote
3
answers
365
views
Decomposition by subtraction
In how many ways one can decompose an integer $n$ to smaller integers at least 3? for example 13 has the following decompositions:
\begin{gather*}
13\\
3,10\\
4,9\\
5,8\\
6,7\\
3,3,7\\
3,4,6\\
3,5,5\\...
- 197
1
vote
1
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78
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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 ...
1
vote
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
0
votes
3
answers
735
views
Partition an integer $n$ by limitation on size of the partition
According to my previous question, is there any idea about how I can count those decompositions with exactly $i$ members? for example there are $\lfloor \frac{n}{2} \rfloor$ for decompositions of $n$ ...
- 197