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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Conjecture: Ramsey Number R(m,n)=(2m-1)*p(2m-6+n,m)+{1,m,m+1}, for 3<=m<=n
Today(2023-11-22), I have a conjecture on Ramsey numbers.
Fence Conjecture(栅栏猜想): Ramsey Number
R(m,n)=(2m-1)*p(2m-6+n,m)+{1,m,m+1}, for 3<=m<=n.
Here p(n,k) denotes both the number of ...
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What is the 11th unordered combination of natural numbers that add upto 6 in the partition function?
So, I was making unordered combinations of natural numbers which add upto a certain natural number. I was able to go till 6 when I got to know about the partition function. I was pleased to see that ...
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Generating function of number of partitions of $n$ into all distinct parts
I am trying to grasp this example from the book A Walk Through Combinatorics:
Show that $\sum_{n \ge 0} p_d(n)x^n = \prod_{i \ge 1}(1+x^i)$ where
$p_d$ stands for partitions of $n$ into all distinct ...
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Generating function of number of partitions of $n$ into parts at most $k$
I am trying to grasp the intuition behind this example.
Show that $\sum_{n \geq 0} p_{\leq k}(n)x^n = \prod_{i=1}^k \frac{1}{1-x^i}$
where $p_{\leq k}(n)$ denotes the number of partitions of the ...
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6 digit numbers no repetitions with equal part sums
I am interested in six digit numbers where the sum of the first digits numbers is equal to the last three digits but no digit repeats.
Examples:
$143260$ (since $1+4+3=2+6+0$)
$091325$ (since $0+9+1=...
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How do I prove the integer partition problem in combinatorial mathematics?
Question: Show that the number of partitions of $n$ into $k$th powers $(k>1)$ in which no part appears more than $k-1$ times is always equal to 1.
This is actually an exercise from the book Integer ...
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A question about partition function of integers
First I would like to mention that I am not too well versed with the properties of the partition function of an integer so I apologise if this question is too elementary or daft. Recall we define the ...
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Do we have recurrences for evaluating the Partition Function on Graphs?
Inspired by this question about defining the partition function on non integers, I was thinking about what sorts of other objects can a partition function be defined on.
I noticed that if we have an ...
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Can we partition the reciprocals of $\mathbb{N}$ so that each sum equals 1
Let $S = \{1, 1/2,1/3,\dots\}$
Can we find a partition $P$ of $S$ so that each part sums to 1, e.g.
$$P_1 = {1}$$
$$P_2 = { 1/2,1/5,1/7,1/10,1/14,1/70}$$
$$P_3 = {1/3,1/4,1/6,1/9,1/12,1/18}$$
$$P_4 = \...
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Identity involving Sum of Inverse of Product of Integer partitions [closed]
Is there a way to prove the following identity
\begin{equation}
\sum_{l = 1}^{k} \left( \frac{(-s)^l}{l!} \sum_{n_1 + n_2 + \ldots n_l = k} \frac{1}{n_1n_2 \ldots n_k} \right)= (-1)^k {s \choose k} \,...
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Is there any mathematical operation that can be reversed to two unique numbers?
Is there any mathematical operation “op”, such that when applied to two integers a, b: a op b = n. We can use that n and reverse the operation, to get n = a op b?
For example, the sum is not ...
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Find closed form of real valued function
Let $f:\mathbb{N}\rightarrow\mathbb{R}, h:\mathbb{N}\rightarrow\mathbb{R}$ be two functions satisfying $f(0)=h(0)=1$ and:
$$f(n) = \sum_{i=0}^{n}h(i)h(n-i)$$
Find a closed form for $h(n)$ in terms of $...
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All the different ways to add a set of lengths - need explanation of the answer
I wish to make an integer length n of boards laid down end to end. Each board is an integer length and among boards of one length there are unique types. For example, boards of length one come in two ...
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Let $P(n)$ be a number of all integer partitions of n. Then, $P(1) + P(2) + ... P(n) < P(2n)$
Let $P(n)$ be a number of all integer partitions of n. Then, show that $P(1) + P(2) + ... P(n) < P(2n)$.
The solution from the textbook:
If $\pi$ is a partition of $i$ for $i \leq n$, then its ...
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An identity related to the series $\sum_{n\geq 0}p(5n+4)x^n$ in Ramanujan's lost notebook
While browsing through Ramanujan's original manuscript titled "The Lost Notebook" (the link is a PDF file with 379 scanned pages, so instead of a click it is preferable to download) I found ...
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Finding the number of integer composition using only a specific pair of numbers [closed]
I want to find the number of integer compositions of 19 using numbers 2 and 3.
If I wanted to find the number of integer compositions of 19 using 1 and 2, I could write it as $F(n)=F(n-1)+F(n-2)$ ...
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Partition of a number $n$ whose each part is coprime with this number
I'm trying to solve the following problem: given an integer $n$, under which conditions of $n$ the following statement is true:
For any $1 < k \leq n$, there is always a partition of $k$ parts of $...
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Finding groupings of numbers that add up to values in a target set
I am wondering if there is an efficient way to partition a set of numbers into subsets that sum to values in a set of target values.
Specifically, let's assume that we are given $k$ distinct values (i....
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On partition of a number $n$ into positive integers
I need to prove this inequality, but I do not have a good background in algebra, if you can guide me:
We have:
$$
p_1+p_2+\ldots+p_k = q_1+q_2+\ldots+q_k+\ldots+q_t = n
$$
and
$$
p_1 + 2p_2 + \ldots +...
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Partition of n into k parts with at most m
I ran into a problem in evaluating a sum over kronecker delta. I want to evaluate
$$\sum_{\ell_1,...,\ell_{2m}=1}^s\delta_{\ell_1+\ell_3+...+\ell_{2m-1},\ell_2+\ell_4+...+\ell_{2m}}$$
My approach was ...
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Construct bijection between partitions of $[n-1]$ with $k-1$ parts, and partitions of $[n]$ with $k$ parts with no parts contains consecutive integers
So... the title is just I want to ask for. This is exercise #104 of Chapter 1, in Lessons In Enumerative Combinatorics, GTM 290.
With previous exercise, I proved that number of $k-1$-part paritions of ...
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Number of positive integral solution of $\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10$
I want to find the number of positive integer solutions of the equations given by
$$\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10.$$
I know the case that, for any pair of ...
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How to fill a set of container by partition of set?
Let $\{A,B,C,D\}$ be a table with $4$ containers of sizes respectively $5,5,3,2$. Let $\pi=\{B_1,B_2,\cdots B_k\}$ a partition of a set, where $k\in \mathbb{N}$. I wonder how to enumerate the ...
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The product of components in the partition of a number.
Let $f(n,k)$ be the sum of expressions of the form $x_1 \cdot x_2 \cdot \ldots \cdot x_k$, where the sum counts over all solutions of the equation $x_1 + \ldots + x_k = n$ in natural numbers. Find ...
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Sum of square of parts, and sum of binomials over integer partition
Let $n$ be positive integer. Consider its integer partitions denoting as $(m_1,\cdots,m_k)$, where $m_1+\cdots+m_k=n$ and the order does not matter.
I am interested in the following two quantity
(1) $$...
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Upper bound a sum of over partitions of integer
Let $n,L$ are positive integers. $(m_1,\cdots,m_k)$ is partition of $n$, i.e. $m_1+\cdots+m_k=n$ or $(m_1,\cdots,m_k) \vdash n$.
Note that a partition of n is a representation of n as a sum of ...
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Whether a sum over all partitions of n decays exponentially with n
Given $L$ as an absolute constant (does not depend on $n$), and $\beta$ is a function of $n$. Consider
$$\sum_{m_1+\cdots+m_k=n\\ 1\leq k\leq L\\ \text{all }m_i \text{ are positive integers}}\binom{L}{...
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The same number of distinct elements and ones in the partitions of the number.
For a given partition $\pi$ of the number $n$, let $A(\pi)$ denote the number of ones in $\pi$, and $B(\pi)$ the number of distinct parts in $\boldsymbol{\pi}$.
EXAMPLE: for the partition $\pi: 1+1+2+...
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Skew diagram horizontal $m$-strip defintion and board strip question
In Symmetric Functions and Hall Polynomials by Manin, Manin claims that for a skew diagram $\theta = \lambda - \mu$ to be a horizontal $m$-strip, "the sequences $\lambda$ and $\mu$ are interlaced,...
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Homogeneous and inhomogeneous recurrences for a sequence
Some integer sequences have both homogeneous and inhomogeneous recurrence relations. If you know one, are there techniques for figuring out the other? (Or seeing if the other exists?) Below is an ...
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