All Questions
11
questions
1
vote
1
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294
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Minimum number of partitions required for obtaining all numbers from $1$ to $n$
I took part in a coding contest organized by a club in my college for freshers, In one of the problems it was asked to find out the minimum number of partitions of a number $n$, so that all the ...
- 13
1
vote
0
answers
44
views
Partition an integer $n$ into parts so as to maximize the product of the parts [duplicate]
We are given an integer $n\geq 3$. Our goal is to partition $n$ into $k$ parts $p_1, p_2, \ldots, p_k$ with $p_1 + p_2 + \ldots + p_k = n$ (for an arbitrary $k$ we can choose) so as to maximize the ...
- 281
2
votes
1
answer
62
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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
2
votes
1
answer
165
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About the proof of Euler’s Pentagonal Number Theorem on Wiki
Euler’s Pentagonal Number Theorem on Wikipedia
For convenience, here below is the statement:
Let $n$ be a nonnegative integer, let $q_e(n)$ be the number of partitions of $n$ into even number of ...
- 23
9
votes
1
answer
899
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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
5
votes
0
answers
278
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Expected Value for the Number of Parts of a Partition of n
Given a positive integer $n$, I want to know the expected value for the number of parts of a random partition of $n$.
I am aware that a similar question has been asked already: Expected number of ...
- 113
2
votes
2
answers
2k
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Number of partitions of a number
I've been trying to find out how many partitions of a number I can make if I restrict the partition size, The specific problem I am trying to tackle is,
How many ways can you partition the number '9' ...
2
votes
1
answer
371
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Proof of an integer partitions inequality
I came across an interesting problem the other day.
Let $P_n$ be the number of partitions of a positive integer $n$. For instance $P_4$ = $5$, as there are five ways of partitioning $4$:
$4$
$3+1$
$...
- 233
2
votes
3
answers
689
views
Integer Partitions asymptotic behaviour
Let $ P(n) $ be the number of partitions of number $n$.
Prove that $ P(n)$, grows faster than any polynomial from $n$.
I am looking for an elementary (rather bijective) proof of the fact.
- 571
0
votes
1
answer
76
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Why are is partitions counting technique wrong?
I recently heard about partitions. I tried to count them using the following technique:
1) Ways to write $5$ as a sum of five positive integers:
$$1+1+1+1+1$$
2) Number of ways to write $5$ a sum of ...
- 467
1
vote
0
answers
130
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Generating functions and integer partitioning [duplicate]
Show that the number of partitions of a positive integer n where no
summand appears more than twice is equal to the number of partitions
of n where no summand is divisible by 3
So I begin by ...
- 6,294