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Questions tagged [integer-partitions]

Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.

In number theory and combinatorics, a partition of a positive integer $n$ is a way of writing $n$ as a sum of positive integers. Two sums that differ only in the order of their summands (also called parts) are considered the same partition. For example, all of the partitions of $4$ are $1 + 1 + 1 + 1$, $2 + 1 + 1$, $2 + 2$, $3 + 1$, or $4$.

The number of partitions of $n$ is given by the partition function $p(n)$. For the example above, $p(4) = 5$.

Partitions can be visualized graphically with Ferrers diagrams.

Partitions have applications in symmetric polynomials, the symmetric group, and group representations.

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Partitioning a natural number $n$ in order to get the maximum product sequence of its addends

Suppose we have a natural number $n \ge 0$. Given natural numbers $\alpha_1,\ldots,\alpha_k$ such that $k\le n$ $\sum_i \alpha_i = n$ what is the maximum value that $\Pi_i \alpha_i$ can take? I'm quite sure that there is a theorem telling me the…
Paolo Parisen T.
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Partitions of 13 and 14 into either four or five smaller integers

There are exactly 18 partitions of the integer 13 into 4 parts, as on the left of the table, and also 18 partitions into 5 parts, as on the right of the table: $$\begin{array}{c|c} 10+1+1+1 & 9+1+1+1+1 \\ 9+2+1+1 & 8+2+1+1+1 \\ 8+3+1+1 & 7+3+1+1+1…
MJD
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Number of Sets of Partitions

I looked at the partitions of numbers, like let's say $n=5$. You get $$ \begin{eqnarray} 5&=&5\\ \hline &=&4+1\\ &=&3+2\\ \hline &=&3+1+1\\ &=&1+2+2\\ \hline &=&2+1+1+1\\ \hline &=&1+1+1+1+1\\ \end{eqnarray} $$ where I grouped the partitions…
draks ...
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How many partitions of 12 that fit the requirements?

How many partitions of $12$ are there that have at least four parts, such that the largest, second-largest, third-largest, and fourth-largest parts are respectively greater than or equal to $4,3,2,1$? For example, $12=4+4+2+2$ works.
asdf
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Number of partitions of $2n$ with no element greater than $n$

The number of partitions of $2n$ into partitions with no element greater than $n$ (copied and slightly adapted from http://mathworld.wolfram.com/PartitionFunctionq.html), so I'm looking for a nice formula of $q(2n,n)$. By asking…
draks ...
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Ordered integer partition

Possible Duplicate: Proving an equality involving compositions of an integer A sequence of natural numbers $\langle a_1,\ldots,a_k \rangle$ is an ordered partition of $n$ if $\sum_{i=1}^k a_i=n$. Prove that for $n\ge 4$, the number of…
ray
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Can partition be defined for numbers other than positive integers?

An integer partition is a decomposition of a positive integer into positive integers that add up the original number. Apart from $\mathbb N_{> 0}$, does this definition apply to other numeric sets, like $\mathbb R$ or $\mathbb Q$?
nightcod3r
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For which $k$ are there most partitions of $n$ into $k$ parts?

Let $P(n,k)$ denote the number of partitions of $n$ into $k$ parts. I would like to know for given $n$, which $k$ does maximize $P(n,k)$? Additionally, information on the maximum of $P(n,k)$, for fixed $n$, would also be of interest. I'm interested…
Xorra
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Minimum number of questions needed to uniquely determine an integer partition

This came up in an algebra class today, but I'll phrase it a bit differently. Let's say Alice and Bob are playing a game. Alice thinks of an integer partition, and tells Bob the sum of the partition's parts $n$. Bob can then ask Alice some questions…
Sp3000
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Second difference of partitions

Let $p(n)$ count the number of integer partitions of $n$. I am interested in the second shifted difference $$p(n+j) - 2p(n) + p(n-j)$$ for positive integers $j$. For $j=1$, these are related to the Dyson ranks of partitions. Are there known…
J123
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Integer partitioning under constraints on number of parts and maximum elements

I have an use case which requires storing and then looking up the integer partitions of numbers. So before I go and actually store all the integer partitions, I am actually trying to check if they are feasible.I have 3 questions associated with…
Vinanth S Bharadwaj
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Hook-length for partitions

Let $\lambda=(\lambda_1,...,\lambda_r,...)$ be a partition (i.e. $\lambda_i\ge \lambda_{i+1}$ and there are only finitely many non-zero terms.) Let $\lambda'$ be a conjugate partition, i.e. $\lambda_i'=\mathrm{card}\{j:\ \lambda_j\ge i\}$. Let…
mikis
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Sum of the hook-lengths of a partition $\lambda$

Given is a partition $\lambda$, and the $\lambda$ also denote it's Young Diagram, and $\lambda'$ is the conjugate/transpose. Then the hook-length of $\lambda$ at $x = (i,j)$ is defined to be $$h(x) = h(i,j) = \lambda_i + \lambda'_j - i - j + 1$$We…
Naweed G. Seldon
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is there any formula that describes the frequency distribution of numbers(from 1 to N) across all partitions of N?

I was interested in frequency of numbers across all partitions of a particular number N. say 5 = 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1 frequencies of 1 to 5 are 11,3,2,1,1 respectively. is there any formula describing their frequency…
Ramprashanth
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Partition function with restrictions

What is the number of ways of partitioning a positive number $k\leq mn$ using non-increasing parts such that the number of parts can be at most $n$ and the value of each part can be at most $m$?
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