Questions tagged [integer-partitions]
Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.
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Link Between Integer Partition and Partition of a Set
Definition.
Let $A$ be a non-empty set. A collection $P$ of subsets $A_1,A_2,A_3,\ldots$ of $A$ is called a partition of $A$ if the following three conditions hold:
$\emptyset \notin P$,
$A_1 \cup ...
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Is the set of all rational additive partitions of a rational number countable?
We usually call additive partitions the set, we call it $P$, of all the ways to write a positive integer $n$ as a sum of positive integers. Formally:
\begin{equation}
P_n = \left\{ (a_1 ,...,a_n)\in\...
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Combinatorial interpretation of $\prod_{\lambda\vdash n} \prod\lambda_{k} = \prod_{\lambda\vdash n} \prod m^{(\lambda)}_i!$
Notation:
(i) $\uparrow^{G}_{H}$ denotes the induced representation (or its corresponding character) of $H$ to $G$, and $\downarrow^{G}_{H}$ denotes restriction similarly.
(ii) $\lambda = \...
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Can we use Burnside's lemma to count partitions of an integer?
I am currently learning group theory and find that Burnside Lemma can be used to calculate partition number, although not that efficiently.
Suppose we want to distribute $n$ different balls into $n$ ...
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A number partition problem
I have encountered the following interesting integer partitioning problem.
Let $n,k,t \in \mathbb{N}$ be given parameters and let $S_1,\ldots, S_t$ be a partition of the numbers $1,2,\ldots,n$ such ...
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Weighted sum over integer partitions involving hook lengths
I am trying to compute the following quantity:
$$ g_n(x) = \sum_{\lambda \vdash n} \prod_{h \in \mathcal{H}(\lambda)} \frac{1}{h^2} \exp\left[x\sum_i \binom{\lambda_i}{2} - \binom{\lambda_i'}{2}\right]...
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Unsolved problems for partition function
In number theory, the partition function $p(n)$ represents the number of possible partitions of a non-negative integer $n$. For instance, $p(4) = 5$ because the integer $4$ has the five partitions $1 +...
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A summation formula for number of ways $n$ identical objects can be put in $m$ identical bins
A famous counting problem is to calculate the number of ways $n$ identical objects can be put into $m$ identical bins. I know that this problem is somewhat equivalent to Partition problem. There is no ...
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the bracket partition function ? $3 = 1 + 1 + 1 = 1 + 2 = 1 + (1 + 1) $
I want to know in how many ways we can write a positive integer by using strict positive integers, addition and brackets. The order of addition does not matter.
For instance
$$3 = 1 + 1 + 1 = 1 + 2 = ...
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Understanding a Proof Related to Binary Partitions
In Section 10.2 Rödseth's Theorem for Binary Partitions of The Theory of Partitions by G. Andrews, we find the following:
For $q \in \Bbb C$ with $|q|<1$,
\begin{align}
\mathcal{F}_2(q) &= \...
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Minimizing the magnitude of the sum of a vector of complex numbers with an integer constraint
Let $h_{1}, ..., h_{N} \in \mathbb{C} $
Consider minimizing the function below:
$ \min \left| \sum\limits_{i=1}^N h_{i} x_{i} \right| $
with the constraints $x_{i}^2 = 1$
i.e., $x_{i}$ can only take ...
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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 ...
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Finding "Hamiltonian paths" in fixed-size integer partitions
For $ p_k(n) $, the partitions of $ n $ with exactly $ k $ parts, it's possible to order them such that each adjacent pair of partitions differ only by one, i.e. one can be transposed to the other by ...
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No. of partitions of $m$ with bounded $k$ parts [duplicate]
Let $\bar{n} = (n_1,\dots, n_k)$ be a fixed tuple of positive integers. Define $\mathcal P_m ({\bar{n}}) $ to be the set of all partitions of $m$ with exactly $k$ parts and the i-th part is bounded ...
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Books for developing an intuitive understanding of the partitions of numbers
I understand from the fundamental theorem of arithmetic that every number can be written as a product of its prime factors,but I’m curious about partitions,how numbers can be broken up into sums and ...
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Combinatorial proof of binary partition function $b(n)$ is always even
For all integer $n$, let $b(n)$ be the number of partition of $n$ into power of two.
For instance, $b(4)=4$, since
\begin{align*}
4 &= 2^2 \\
&= 2^1+2^1 \\
&= 2^1+2^0+2^0 \\
&= 2^0+2^0+...
user1089451
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How many Young diagrams with length of at most $p$ rows and $q$ columns
How many Young diagrams are there with length of at most $p$ rows and $q$ columns, without restrictions on the weight of the diagrams?
The previous 2 questions asked for
Total number of Young ...
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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
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Congruence properties of binary partition function
For every positive integer $n$, denote $b(n)$ be the number of binary partition of $n$, i.e., the number of partition of $n$ into power of two, where the power is decreasing.
For instance, $b(5)=4$ ...
user1089451
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Number of ways to compose a number such that there is a maximum pile size and for each pile size there are a finite number of valid piles of that size
So, I want to count all of the different ways that a number $n$ can be written as the sum of numbers that are less than or equal to $k$ where the order of the numbers matters. Now, this is just ...
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When are the partition numbers squares?
I'm unsure if this question is even interesting. I am playing around with partition numbers $p(n) :=$ # partitions of $n$, and I noticed that $p(n)$ never really is a square number, except for of ...
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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
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Is there a pattern to the number of unique ways to sum to a number?
I don’t think there is a proper name for these so I will refer to them as “phactors”. Basically, a phactor is a way to sum up to a number using positive real integers that are non zero and not equal ...
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Compute a certain sum [duplicate]
How would I find a formula for $$S(n,r) = \sum_{i_1+\ldots+i_r = n,~(i_k)\in\mathbb{N}^r} ~~~i_1\ldots i_r ~~~~.$$
It's easy to find that it satisfies $$ S(n,r+1) = \sum_{j=0}^n(n-j)S(j,r),$$ which ...
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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,...
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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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Sum of Schur functions incorporating the length statistic
Macdonald's Symmetric Functions and Hall Polynomials includes
the following identities (listed as exercises on pages 76 and 78
respectively):
\begin{equation}
\begin{array}{ll}
\text{Ex.} \,(4) \quad \...
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Summing over tuples $(b_1, \ldots, b_n)$ with $\sum ib_i = n$ and $b_j = k$.
Let $T_n$ denote the set of $n$-tuples $\left(b_1, \ldots, b_n \right)$ of non-negative integers such that $\sum_{i=1}ib_i=n.$ I am trying to simplify the sum
\begin{align*}
\sum_{\underset{b_{j}=k}{\...
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Does the partition function $p(n)$ generate infinite number of primes
Wikipedia says
As of June 2022, the largest known prime number among the values of $p(n)$ is $p(1289844341)$, with $40,000$ decimal digits
citing [1].
Is it known whether the partition function ...
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Could this yield a formula for the Partition numbers?
Background: Lately, I have fallen down the rabbit hole of partition numbers. Specifically the partition function, $p(n)$. It's well known that no closed-form expression (with only finitely many ...
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