Questions related to the different ways of expressing an integer as a sum of integers; or, questions related to the subdivision of a set into smaller disjoint sets; questions related to the subdivision of an interval into smaller intervals that intersect only at the endpoints.

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Partitioning real numbers with sum $1$ to sets

If the sum of a finite number of positive real numbers is $1$ and each of them is less than $x$, then those real numbers can be partitioned into $50$ sets (some of which may be empty) such that the ...
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2answers
19 views

What is an algorithm for describing the partition of this equivalence relation?

Let $\mathbb{R}$ be the set of real numbers,$ f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ ℝ $ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid ...
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1answer
21 views

Quick question for proof on unimodal sequence formula in Enumerative Combinatorics

I am looking at page 238 of Stanley's Enumerative Combinatorics where he says that $\#V_n = \#D_n - \#V_n^1$ because every element in $V_n^1$ appears twice as a value of $\Gamma_1$. Can someone ...
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103 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
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33 views

What are the combinatorial numbers appearing in these repeated derivatives?

Let $f$ be a $C^\infty$-function and define $g(x) = \exp(f(x))$. I am interested in the higher derivatives $g^{(1)}, g^{(2)}, \ldots$ of $g$. Let $\lambda$ be a partition of $n$, i.e. a tuple of ...
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12 views

Edge-partition graph into maximum distinctive paths

Is anyone aware of a graph partitioning algorithm in which each partition is a path and where the criteria for partitioning is to maximise the difference between partitions where two paths are ...
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1answer
237 views

Integer parts of multiples of irrationals

Let $\alpha>0$ and define $$S(\alpha)=\big\{\lfloor n \alpha \rfloor: n\in\Bbb Z^+ \big\}.$$ Here $\lfloor x\rfloor$ is the integer part of $x$ and $\mathbb Z^+$ the set of positive integers. ...
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3answers
70 views

Show that ~ creates a partition of $M_2(\mathbb{R})$

Let $M_2 (\mathbb{R})$ be the set of 2x2 matrices over $\mathbb{R}$: $$ M_2 (\mathbb{R}) = \biggl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \; \biggm| \; \text{with } a,b,c,d \in ...
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8answers
3k views

How many ways can $133$ be written as sum of only $1s$ and $2s$

Since last week I have been working on a way, how to sum $1$ and $2$ to have $133$. So for instance we can have $133$ $1s$ or $61$ $s$2 and one and so on. Looking back to the example: if we sum: $1 + ...
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1answer
58 views

Covering a rectangle of size $n\times1$ with dominos

A rectangle of size $n\times1$ is given. (a) In how many ways the rectangle can be covered with dominoes of size $1\times1$ and $2\times1$? (b) In how many ways the rectangle can be covered with ...
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22 views

Number Partitioning of summands

So, I need to partition the number 133 in 1, 2 and 3. Like $$133 = 128*1 + 1*2 + 1*3$$ $$133 = 126*1 + 2*2 + 1*3$$ $$133 = 125*1 + 1*2 + 2*3$$ Where I always must use at least one 1, 2 or 3. I ...
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2answers
38 views

Number of Partitions proof

How do I prove that the # of partitions of n into at most k parts equals the # of partitions of n+k into exactly k parts? I was trying to improve my ability of bijective-proofs, unfortunately I was ...
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9 views

Indexing for partitions

I'm using spatial hashing for broad-phase collision detection and I'm trying to squeeze some more performance out of it. Currently, it creates a new hashset for every partition which works well ...
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1answer
17 views

Combining an outcome of a score

Hey I was wondering how many are the possibilities of combining the scored points of a result such $133:75$ from a basketball game? Considering that there are fouls($1$ point), normal($2$ points) and ...
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1answer
36 views

partitions are the elements of the free abelian monoid on $\aleph_0$ generators

although the following insight may not be particularly profound, it is always of interest when we see a canonical isomorphism between objects of apparently different types. it is a commonplace that ...
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2answers
170 views

Minimizing over partitions $f(\lambda) = \sum \limits_{i = 1}^N |\lambda_i|^4/(\sum \limits_{i = 1}^N |\lambda_i|^2)^2$

I'm trying to characterize the behavior of the the quantity: $$A = \frac{\sum \limits_{i = 1}^N x_i^4}{(\sum \limits_{i = 1}^N x_i^2)^2},$$ subject to the constraints that $$ \sum \limits_{i = 1}^N ...
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1answer
23 views

Proving that two equivalence classes are disjoint?

I am having trouble with the following proof: Define the relation $R$ on $\mathbb{Z}$ by $nRm$ if $n-m$ is divisible by $2$. Prove that the equivalence class for $0^{(\bar{0})}$ and the equivalence ...
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1answer
47 views

Circles partitioning the plane

What is the equation for the maximum number of regions into which N circles can partition a plane? Is there a name for this equation? A single circle partitions the plane into two regions, inside ...
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1answer
53 views

Partition on a Closed Set A= [2,3]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
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3answers
97 views

Multiplication partitioning into k distinct elements

Let's say I have a list with the prime factors of a number $n$ and their corresponding exponents. Is there a formula to calculate how many multiplications with $k$ distinct factors of $n$ are ...
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1answer
35 views

Partition and equivalence relation

Consider the equivalence relation between non-empty subsets $A , B$ of $\{ 1,2,3, 4,\dots,100\}$ defined by the condition: the greatest element of $A$ is the same as the greatest element of $B .$ ...
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1answer
21 views

Prove an identity with integer partitions

I have already proven this identity: $\prod_i (1+st^i) = 1 + \sum_r \frac{s^rt^{r(r+1)/2}}{(1-t)(1-t^2)\cdots(1-t^r)}$ I expanded the product, grouped the s terms, and then made an argument about ...
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2answers
84 views

Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.

I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$ where $B_n$ is the $n$th Bell number. I found this result by exploring the series on wolframalpha and looking up the ...
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0answers
27 views

Determine the number of partition of 20 into at most 5 parts.

Im stuck on these questions, I can kind of compute them on maxima, but I have to figure out why and how to get to a particular generating function or method. Determine the number of partition of 20 ...
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22 views

Can you partition a rectangle into exactly 3 congruent non-rectangular parts?

Recently I came upon the following result: Theorem (*): Let $n$ be a positive integer not equal to $1,3,5,7,9$. Then it is possible to partition a rectangle into exactly $n$ congruent non-rectangular ...
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1answer
63 views

Exponential generating function of partitions of set [n]

Find the exponential generating function of the partitions of the set [n], all of whose classes have a prime number of elements. The only thing I came up with was $$\sum\limits_{\textrm{t prime}} = ...
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1answer
99 views

Jee Main 2015 Question. Probabilty

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is: (1) $22 \times(\frac{1}{3})^{11}$ (2) $\frac{55}{3} \times ...
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0answers
54 views

Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
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1answer
48 views

How many solutions of equation

How many solutions of equation $x_1+x_2+x_3+x_4=n$ in $N_0$ such that $x_1\leq x_2\leq x_3 \leq x_4$? I found solutions of $x_1+x_2+x_3=n$ in $N_0$ , $x_1\leq x_2\leq x_3 $ in the following way : ...
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22 views

Notation for the set of all integer partitions

I'm working on a project that involves that set $P = \{\{n_1, \ldots, n_k\} \mid k \in \mathbb{N}, n_i \in \mathbb{N} \text{ and } n_1 + \cdots +n_k = n\}$ of all integer partitions of a number $n$. ...
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1answer
36 views

Counting unordered partitions on nested concentric disks

The idea is to think of each layer outside of the [core] as a rotatable disk and then only count a single member from each of the resulting equivalence classes, which I think can be done by requiring ...
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2answers
370 views

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 ...
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1answer
74 views

Introduction to proofs: proving a set is a partition.

I've been really trying to understand how some of these proofs work; I've spent a majority of my time studying the material for this class, but I'm still performing poorly in it. It doesn't help that ...
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1answer
17 views

Number of ways to build a collection of numbers where $sum = k$, each $0 < n_i <= d_i$ for some corresponding $d_i$, and sum of all $d_i >= k$

I apologize for any (mis|ab)use of notation since I'm not a mathematician. My background is software engineering and computer science. I ran into this problem while trying to figure out the ...
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1answer
66 views

Partition problem for consecutive $k$th powers with equal sums (another family)

This is the partition problem as applied to a special set, namely the first $n$ $k$th powers. Assume the notation, $$[a_1,a_2,\dots,a_n]^k = a_1^k+a_2^k+\dots+a_n^k$$ I. Family 1 The following ...
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88 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
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1answer
114 views

Characters of the symmetric group corresponding to partitions into two parts

Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
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2answers
60 views

Sums of Consecutive Cubes (Trouble Interpreting Question)

Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums. Any suggestions on what techniques should be used to ...
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0answers
13 views

On t-core partitions

How exactly can one define what is known as a t-core partition? I know (vaguely) that it involves the definition of what is known as "Hook numbers". Anyone cares to provide a link or explain it?
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4answers
29 views

Number of sequences formed of $k$ pairwise disjoint subsets of a set of $n$ elements is $(k+1)^n$.

Let $S=\{1,2,\dots,n\}$ and $P(S)$ the family of the $2^n$ subsets of $S$. Prove that the number of sequences $(S_1,S_2, \dots, S_k )$ formed by the subsets of $S$ that verify that $S_i \cap S_j = ...
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1answer
53 views

Does and where to does $\lim_{n\to\infty}\sum_{m} \prod_k \frac{1}{\lambda_{k,m}!}$ converge?

Given $n$ you get a number of partitions of $n$ and let's denote $\lambda_{k,m}$ to be the $k$th part of the $m$th partition. Now I built the following sum, that stimulated the following question: $$ ...
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Terminology: Opposite of “refinement”

Let A be a partition of a set, and B a refinement of A. Fill in the blanks: A is a __________ of B. I know that A is coarser than B, but how does one turn that into a noun?
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2answers
23 views

Proving a family of sets is a partition of the set of integers.

I am trying to show that for $E_n = \{10n, 10n + 1, 10n + 2 . . . , 10n + 9\}$, $\{E_n\}$, $n ∈Z$ is a partition of $ Z$. Should this be broken down into cases or is there a more general way to ...
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0answers
57 views

Recurrence relation / recursive formula / closed formula for partition numbers (1,2,3,5,7,11,15,22,30,42,…)

I have already read this: Number partition - prove recursive formula But the formula from the above link requires a parameter k which is the required number of ...
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0answers
76 views

Elementary proof of Ramanujan's “most beautiful identity”

Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity. (for example, H. H. Chan’s proof, M. Hirschhorn's ...
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18 views

Find largest regions bounded by a set of planes

Suppose we are given a set of planes that partition the unit cube into a large number of regions. Is there a computationally efficient way to find the region with the largest volume?
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47 views

Proof of Ramanujan's Identities of Euler's Function

Consider Euler's Function defined as (and not to be confused with the totient!) $$ \phi(x) = (1-x)(1 - x^2) (1 - x^3 ) .... = \prod_{i=1}^{\infty} \left[(1 - x)^i \right] = (1;x)_{\infty}$$ ...
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1answer
58 views

Sample, randomly & uniformly the partitioning of $n$ objects into $K$ groups

I wish to sample randomly and uniformly from the set of partitions of $n$ objects into $K$ groups where the number to be assigned to each group, $n_k$ ($k = 1, 2, \dots, K$) is known. I know that the ...
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0answers
47 views

Counting the number of partitions that are a distance d away from a fixed partition.

Given a positive integer $N \in \mathbb{Z}^{\geq 0}$ let $Partitions(N)$ denote the set of all partitions of $N$, where a tuple $\left(f_1,\ldots,f_N \right)$ is a partition of $N$ if $\sum_{i=1}^N ...
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1answer
46 views

Partitioning a set to get a sum

I have a set of numbers: 2,2,4,4,4,4,4,4,6,6,6,6,6,6 I want to enumerate the possible ways to partition this set into 4 groups, each of which sum to 16. How can I approach this short of brute force? ...