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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Number of set partitions of n elements into k sets with subsets of size r not allowed

This is a generalization of the question Number of ways to partition a set with $n$ elements to $k$ subsets where at least one subset has $r$ elements . At the end of answer for this question, there ...
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2answers
457 views

Partitions and Bell numbers

Let $F(n)$ be the number of all partitions of $[n]$ with no singleton blocks. Find the recursive formula for the numbers $F(n)$ in terms of the numbers $F(i)$, with $i ≤ n − 1$ Find a formula for ...
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3answers
1k views

Number of cycles of all even permutations of $[n]$ and number of cycles of all odd permutations differ by $(-1)^n (n-2)!$

I'm trying to solve task 44 of the first chapter of Stanleys Enumerative Combinatorics (found here). Show that the total number of cycles of all even permutations of $[n]$ and the total number ...
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2answers
58 views

What does George Andrews mean by i in “Theory of Partitions”?

From the first page of chapter 1 of George Andrews "Theory of Partitions" (Rather ominous place to get stuck): What do these last two sentences mean? I don't get "where exactly $f_l$ of the ...
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26 views

An identity relating sum of number of partitions to sum of number of parts

I encountered this identity while studying about the Kac determinants in CFT. $$\sum_{\{n_1 + \cdot \cdot \cdot + n_k \}} k = \sum_{pq \le N} P(N-pq)$$ Here $P(N-pq)$ is the number of partitions of ...
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36 views

How to calculate least moves of fruit.

I have a question, I'll try to abstract from the real problem to not lose people. What I'm really looking for is the name of the algorithm or class of problem to find my solution. I feel that this is ...
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3answers
99 views

Ways of Distributing $n$ balls among $k$ boxes, each box containing $L \leq x_i \leq M$ or $0$ Balls

I need to calculate the number of ways of distributing $n$ balls among $k$ boxes, each box may contain no ball, but if it contains any, then it must contain $\geq L$ & $\leq M$ balls. This ...
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2answers
43 views

Question about partitions in intervals of the real numbers.

I have to prove the following: Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any kind, except intervals containing a single element. Prove $ \mathcal{D} $ is countable. ...
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1answer
52 views

How to find Partition of unity in $ \mathbb{S}^n$ with only $2$ functions

How to find Partition of unity in $\mathbb{S}^n$ with only $2$ functions?
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0answers
42 views

intersection of stabilizers of block systems

Lets assume the following situation: $G$ acts regularly on a set $M$. Then there is a bijection between the set of subgroups and the set of blocks containing a fixed element $m \in M$. The blocks ...
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1answer
26 views

Example where a cover is fewer than a partition?

Given a Cartesian product of sets $X \times Y$, A (combinatorial) rectangle is a set $A \times B$ where $A \subseteq X$ and $B \subseteq Y$. Given a function $f : X \times Y \rightarrow \{ 0, 1\}$ one ...
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23 views

generalization of bijection proof

$p(n|\text{odd parts}) = p(n|\text{distinct parts})$ I need bijection proof and generalization of this proof. for $k = 7$ odd parts are : ...
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63 views

OEIS sequence A086449

OEIS sequence A086449 http://oeis.org/A086449 is defined by: $a(0)=1$, $a(2n+1)=a(n)$, $a(2n) = a(n)+a(n-1)+\ldots+a(n-2^m)+\ldots$ $= a(n)+\sum_{i=0}^{\lfloor\lg n\rfloor}a(n-2^i)$ One can show ...
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1answer
49 views

Integer Partition into Powers

Is there any way to count the number of integer partitions of a number N into powers of two such that each size is repeated a power of two times? Ok so the recurrence can be expressed by: $a(0)=1$, ...
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1answer
70 views

Existence of a prime partition

I'm interested in finding out whether there exists a prime partition of a given positive integer $N>1$ such that the partition has specific number of parts. For instance, as given in another ...
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1answer
33 views

A bound on the balanced equipartition of a multi-set of integers

A balanced equipartition of a multi-set of $2n$ integers is a partition into two multi-sets $S_1$ and $S_2$ of size $n$ such that the sum of the integers in $S_1$ is as close as possible as the sum of ...
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0answers
55 views

Generation of n terms that sums to a specific value?

I'm about to conduct an integer program where I am to transport some products from different cities to docks and refinement stations. I am however required to work with simulated data where I only ...
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1answer
26 views

partitioning a set using a non-equivalence relation

I understand that a set can be partitioned into equivalence classes by an equivalence relation (wikipedia article). However can we partition a set into a forest of trees by a relation that's simply ...
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7answers
438 views

Book recommendation for Integer partitions and q series

I have been studying number theory for a little while now, and I would like to learn about integer partitions and q series, but I have never studied anything in the field of combinatorics, so are ...
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1answer
98 views

Convergence of characteristic functions on hypercube

I have a question regarding the following partition of a hypercube $H_{R}(x)$ centered at $x$ with sides of length $R$ in $\mathbb{R}^{n}$: Consider this hypercube $O = H_{R}(x) = ...
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0answers
23 views

What are all the possible sums (and how often do they occur) of a k-subsequence of an n-sequence of integers?

Let $A_n = \{a_1,\dots,a_n\}$ be a sequence of non-decreasing non-negative integers. Let $P(A_n,k)$ be the set of all subsequences of $A_n$ of length $k$. Given $n,k\in\mathbb Z_{\geqslant0}$ with ...
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2answers
196 views

Partitions of Natural Numbers [duplicate]

This is a question from Complex analysis by Stein. The question is Prove that it is not possible to partition $\mathbb N$ into finitely many infinite AP's with distinct common differences.(other ...
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1answer
66 views

Ramsey Theory: Showing the existence of a special set of natural numbers.

Show that there is an infinite set of natural numbers such that the sum of any two elements has an even number of prime factors. My attempt: Define a coloring on the doubletons of $\mathbb{N}$, that ...
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1answer
68 views

How to prove this identity?(perhaps related to partition)

How to prove this identity? $$ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)} = \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$$ Maybe the method using generating functions is good.
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4answers
203 views

Algorithm to partition sum between buckets in all unique ways

The Problem I need an algorithm that does this: Find all the unique ways to partition a given sum across 'buckets' not caring about order I hope I was clear reasonably coherent in expressing ...
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1answer
27 views

Partition induced by a Relation

Here's the problem: Let $A=\{1,2,3,4,5,6,7,8,9\}$. Define a relation $R$ on set $A$ by $xRy$ if and only if $2\mid(x+y)$ Assuming that $R$ is an equivalence relation, determine the partition of set ...
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1answer
90 views

Finding the amount partitions of a multiset

A multiset $A$ contains $n$ positive integers. The multiplicity of every integer is less or equal to $m$. $A$ is partitioned into $m$ subsequences in such a way that the multiplicity of all elements ...
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43 views

Cauchy's Coefficient Formula

How can I recover the coefficients of a polynomial power expansion of the type (number of partitions): $ \prod_i (1+x^{\alpha_i}z) = A_1 z + A_2 z^2 + A_3 z^3 + A_4 z^4 \ldots $ using Cauchy's ...
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1answer
84 views

Number of ways to add up to a number without repetition (order does not matter)?

I have a number x and want to find how many ways there are to add up to that number using the y numbers from numbers 1-z. for example, for x=15 y=3, z=9, there are 8 ways to add up to 15 using 3 ...
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1answer
37 views
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99 views

Finding Distinct Elements and Permutation in Partitioned Set

I am having a hard time figuring out where to start on a homework problem. The question is: A set of $nk$ elements is partitioned into $k$ subsets in two ways, each subset having size $n$: one ...
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How many partitions of $N$ are there into $n$ non-negative parts $c_k$ such that $\sum_{k=1}^n c_k = N$ and $\sum_{k=1}^n kc_k = M$??

So when coming up with a recursive solution to a counting problem of placing 1's into an $N \times N$ matrix ($N$ even) so that every row and every column has exactly $N/2$ 1's, my recursive ...
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1answer
34 views

Construct matrix of ones and zeros based on sequences

We are given ($a_1,a_2,....,a_m$) and ($b_1, b_2,....,b_n$) sequences with non-negative integers. Decide whether it's possible and if it is construct a matrix $\Re^{m x n}$ of ones("1") and ...
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2answers
322 views

Lower bounds for the partition function

In this question we consider the partition function $p(n)$ - that is, the number of ways to express $n$ as a sum of positive integers. One easy exercise is to show that $$ p(n) \geq 2^{\lfloor ...
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0answers
81 views

Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
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1answer
32 views

List 4 partitions of the alphabet containing 3 sets. How many different partitions can be made?

The first part of the question is easy; there are many right answers but I put down: $$(a), (b), (c..z)$$ $$(a,b), (c), (d..z)$$ $$(a,b,c), (d), (e..z)$$ $$(a,b,c,d), (e), (f..z)$$ The part I am not ...
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1answer
43 views

Scalar products and partitions of Hypercubes

My questions relate to scalar products defined in $\mathbb{R}^{n}$ and partitions of hypercubes. Take $s \in \mathbb{R}$, $\xi, \eta \in \mathbb{R}^{n}$. My first question is why is it possible to ...
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1answer
61 views

Real Analysis Riemann integrals with piece wise function

this is part of my homework assignment and I have been stuck on it for a few days. Let f be the function on [0,1] given by f(x)= { 1 if x does not = 1/2 and 2 if x=1/2 Prove f is Riemann integrable ...
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29 views

Definition of a Regular Partition of a Closed Generalized Rectangle in $\mathbb R^n$

What the heck does this definition of a regular partition $P$ of $R$ mean? I follow what it is saying until we get to the last part, "the $k_1\cdot k_2\cdot \cdots \cdot k_n$ subrectangles of the ...
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0answers
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Limits and integration

I have the following quick question: Consider bounded open domain $O \subset \mathbb{R}^{n}$ assume that we partition $O$ into $O_{1}^{m}$ and $O_{2}^{m}$ such that $O_{1}^{m},O_{2}^{m} \subset O$, ...
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0answers
28 views

Mapping a set of sets to a partitioning.

I've been experimenting with the following idea, and I wondered if there's a name for it: Suppose $S_0, S_1, ... S_{n-1}$ is an array of $n$ sets of elements in $U$. Now for any element $e \in U$ we ...
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2answers
42 views

Alternative reference for number of restricted partitions

I am looking for the number of partitions of some number $n$ into $k$ parts. Following the Wikipedia article on partitions, I ended up with Andrew's book [1]. Judging by Google's preview Chapter 3 ...
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Partition of the node set of a graph into connected subsets

What word is most commonly used in graph theory for a partition of the node set of an undirected graph into connected subsets? More rigorously: Given an undirected graph $(V,E)$, a partition $S ...
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35 views

Partition of the 3d space with circles?

Does it exist a partition of the 3d space with circles of positive radium? I know the answer is no for a plane, but I can not transpose my demonstration to the space and I have no clue on how to do ...
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3answers
119 views

integral $ \int e^{x} \sqrt{e^{x} - 1} dx $

i want to determinate the following integral \int e^{x} \sqrt{e^{x} - 1} dx my try and steps were as follow $$ \int e^{x} \sqrt{e^{x} - 1} dx $$ let $ u = \sqrt{e^{x} - 1} $ and $ v' = e^{x} $ ...
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133 views

Integral $\int \sqrt{x^{2}+a^{2}}$

i want to determinate the following integral $$\int \sqrt{x^{2}+a^{2}}dx \space | \space a> 0$$ i used integration by partition and u-substitution but i came to no result
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2answers
347 views

Integrate $ \int \sinh 2x \cosh x dx$

I want to integrate $$\int \sinh 2x \cosh x dx$$ my steps was as follow: $$ \int \sinh 2x \cosh x dx = \\ \sinh 2x . \sinh x - 2 \int \cosh 2x \sinh x dx = \\ \sinh 2x . \sinh x - 2 (\cosh 2x . ...
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1answer
34 views

How to talk about overlapping partitions of a set?

If I have a set $A$ and a number of sets $A_\sigma$ for each $\sigma \in \Sigma$ such that $$A = \bigcup_{\sigma \in \Sigma}A_\sigma$$ Is there a concise and eloquent way of saying this in plain ...
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18 views

How to prove this identity about Sylvestered partitions of n into m parts such that …

Let us say a partition $\lambda$ is $Sylvestered$ if the smallest part to appear an even number of times ($0$ is an even number) is even. For each set of positive integers $T$, define $S(T\,;m,\,n)$ ...
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1answer
17 views

Regular expression for a particular language

Several years ago I came across a paper that defined a regular expression (or collection of regular expressions?) for a specific language. The language is the language of set partitions enumerated by ...