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 a set of integers (with Alice and Bob)

Let $ d_1,\ldots,d_n \in \mathbb{N}_{\ge 2} $ (not necessarily distinct) be given. Define $ D:=lcm(d_1,\ldots,d_n) $ and $ d:=\sum_{i=1}^n d_i $. (1) Alice claims that whenever $ \lbrace d/D\rbrace \...
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1answer
6 views

finite partitions of the square that separate all equipotent sets of points

This question asked whether there exists a finite partition of $[0, 1]^2$ and a finite set of points in $[0, 1]^2$ that can't be affinely transformed to fall into one part of the partition. I would ...
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0answers
22 views

$y=x/(1+a(x))$, $\quad$ $x=y/(1+b(y))$. What is known about $a\mapsto b$?

\begin{align} y & = f(x) = \frac x {\displaystyle 1 + \sum_{n=1}^\infty a_n \frac{x^n}{n!}} \\[15pt] x & = f^{-1}(y) = \frac y {\displaystyle 1 + \sum_{n=1}^\infty b_n \frac{y^n}{n!}} \end{...
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0answers
21 views

Partition identity with generating functions

I'd like to show that: The number of partitions of $n$ such that parts appear 2,3 or 5 times is equal to the number of partitions of $n$ into parts congruent to $\pm 2$, $\pm 3$, $6$ mod $12$. The ...
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0answers
39 views

The number of partitions of $n$…Subbarao

(Subbarao) The number of partitions of $n$ in which each part appears two, three, or five times equals the number of partitions of $n$ into parts congruent to $2, 3, 6, 9,$ or $10$ modulo $12$. ...
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3answers
40 views

Number of ways to partition a set with $2n$ elements

In how many ways can I partition $S = \{1,2,\cdots,2n\}$ into $n$ disjoint $2$ element subsets. Suppose if I subsets of $S$ were $S_{1},S_{2},\cdots,S_{n}$, then I can choose $S_{1}$ in $\binom{2n}{2}$...
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1answer
76 views

Pattern in numbers

I bumped into this mathematical calculation while driving my car. With lot of traffic jams and lot of time to kill, I did some scrambling of the registration numbers of the cars in front of me. I ...
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3answers
66 views

Number of non-negative distinct integer solutions of $x+y+z+w=10$

I understand that there are already many questions relating to this, but my question is regarding some concept of mine that should be working but doesn't produce the right result. So, I follow an ...
5
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3answers
332 views

Asymptotics for partitions of $n$ with largest part at most $k$ (or into at most $k$ parts)

Let $\bar p_k(n)$ be the number of partitions of $n$ with largest part at most $k$ (equivalently, into at most $k$ parts). Is there an elementary formula for the asymptotic behavior of $\bar p_k(n)$ ...
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1answer
38 views

Upper and Lower Bound on Partition Function

The partition function $p(n)$ counts the number of ways an integer can be expressed as a sum. For example, $p(4)=5$ as $$4=3+1=2+2=2+1+1=1+1+1+1$$ Hardy and Ramanujan were able to develop a converging ...
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4answers
3k views

Partition of a set, definition not clear

From wikipedia: Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and: The union of the elements of P is equal to X. (The elements of P are said ...
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0answers
23 views

Relation of relative numbers of (restricted) ways to distribute identical / distinct objects into distinct bins

If want to know if the following inequality holds for general values of $s \leq n \ll m$. $$\frac{C_0(n,m,s)}{C_0(n,m)} \leq \frac{p(n,m,s)}{m^n}$$ $C_0(n,m) = \binom{n+m-1}{m-1}$ is the number of ...
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0answers
12 views

Is this right? Showing that if $S\subset \mathbb{R}^n$ arbitrary and $f$ a $C^r$ function then there is $A$ open such that $f$ is $C^r(A)$.

Let $f : S \to \mathbb{R}$ a $C^r$ function, where $S$ is any subset of $\mathbb{R}^n$. We say that $f$ is differentiable on $x_o \in S$ if there is $U_{x_0}$ and $g : U_{x_0} \to \mathbb{R}$ such ...
8
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2answers
148 views

Integer Partition Refinement in Sage

A partition of an integer $n$ is a non-decreasing list of positive integers summing to $n$. For example, $3$ can be partitioned as $1 + 1 + 1$, $1 + 2$ or just $3$, but $2 + 1$ is indistinct from $1 + ...
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0answers
17 views

Differentiating between equivalence relations and partitions

What is the purpose of defining equivalence relations and partitions separately when they are the same? Are they used for different purposes?
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0answers
42 views

An interesting problem on partitions and odd/even numbers

A number k is expressed as a sum of n non-negative numbers. Let us call the sequence of numbers that are a part of the sum, S. Each element of S is replaced by the remainder it leaves when divided by ...
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2answers
2k views

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'...
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1answer
16 views

discrete convexity arising in a simple discrete optimization problem

Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
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1answer
230 views

Partition of ${1, 2, … , n}$ into subsets with equal sums.

The following is one of the old contest problems (22nd All Soviet Union Math Contest, 1988). Let $m, n, k$ be positive integers such that $m \ge n$ and $1 + 2 + ... + n = mk$. Prove that the numbers $...
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0answers
38 views

How do I calculate the variation of a function?

I am trying to understand how to calculate the variation of a function. In this regard, the book that I am reading offers the following definition - $$V_g([a,b] = sup \sum_{i=0}^n |f(x_{i+1}) - f(...
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5answers
108 views

Does {$\Bbb Z_0$,$\Bbb Z_1$, $\Bbb Z_2 ,\cdots$, $\Bbb Z_{m-1}$} form a partition of $\Bbb Z$?

"Definition 5. Let X be a nonempty set. By a partition P of X we mean a set of nonempty subsets of X such that (a) If $A, B \in \mathscr P$ and $A \neq B$, then $A \cap B = \emptyset$, (b) $\bigcup\...
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3answers
6k views

Number of equivalence relations on a finite set

I need a hint for computing the number of ways in which all the equivalent classes on a set of $n$ elements can be realized. For example, if the set has 2 elements ${a,b}$, then there are 2 possible ...
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1answer
22 views

Partitions: reading skew diagrams?

Consider the following two partitions. Partition $\lambda=4^4=(4,4,4,4)$ o o o o o o o o o o o o o o o o And partition $\mu=(4,2,1,0)$ ...
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1answer
2k views

Hardy Ramanujan Asymptotic Formula for the Partition Number

I am needing to use the asymptotic formula for the partition number, $p(n)$ (see here for details about partitions). The asymptotic formula always seems to be written as, $ p(n) \sim \frac{1}{4n\...
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0answers
34 views

Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
1
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1answer
35 views

Partitions of an interval and convergence of nets

Let $\mathscr{T}$ be the set of partitions $\tau = (\tau_0 = 0 < \tau_1 < \dots < \tau_N = 1)$ of the interval $[0,1]$ (where $N$ is not fixed). This becomes a directed set by setting $\tau &...
4
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1answer
32 views

Balanced partition of $\{\ln 3, \ln 4,\dots,\ln n\}$

For a positive integer $n\ge 3$, let $A_n=\{\ln 3, \ln 4,\dots,\ln n\}$. Does there exist $N$ such that for all $n>N$, the set $A_n$ can be partitioned into two sets so that their sums differ by no ...
3
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1answer
446 views

Demystifying the asymptotic expression for the partition function

A partition of an integer $n$ is a way of writing $n$ as a sum of integers. The partition function $p(n)$ counts the number of distinct partitions of $n$. In 1918, Hardy and Ramanujan proved the ...
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1answer
400 views

Identity using q-Pochhammer symbols

Prove - $$∑_{n=0}^{∞} \frac{(a;q)_n}{(q;q)_n} q^{n\choose 2} q^n={(−q;q)_∞}{(aq;q^2)_∞}.$$ where $(a;q)$ are the q-Pochhammer symbols. I know that the RHS is the product of generating functions of ...
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2answers
45 views

Riemann Sum to show convergence help?

I have the function $f(x)=\sin\left(\dfrac{\pi x}{2}\right)$ on the partition of $[0,1]$ given by $$P_{n}: 0 < \frac{1}{n} < \frac{2}{n} < ... < \frac{n-1}{n} < 1$$ I have shown that $$...
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0answers
30 views

Partition Theorem and Markov Chains

Suppose a Markov chain has $s$ states, $S = {1, 2, . . . , s}$, with PTM $P =$ ($p_{ij}$). That is, $p_{ij} = P[X_{n+1} = j | X_n = i]$. Use the Partition Theorem to verify that if $X_n ∼ ν$, then $X_{...
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4answers
78 views

Is there any Algorithm to Write a Number $N$ as a Sum of $M$ Natural Numbers?

I have a number $N$ (for example $N=64$). Is there any algorithm to find all the feasible ways for writing the number $N$ as a sum of $M$ positive numbers? (For example $M=16$) --- Repetition is ...
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2answers
57 views

evaluate the $\int_{0}^1 x\,dx$ using the definition

From the definition -"$f$" is integrable on [a,b] if there exists a number $A$ so that for any $\epsilon > 0$ there exists a $\delta > 0$ such that if the sequence ${X_i}$ from $i=0,n$ is a ...
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0answers
36 views

How can I correctly catalog this partition problem?

Studying the partition problems, I tried to do an special version to apply it to a kind of model of "orbits and energy levels" (explained below), but I am having problems to properly catalog this. ...
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0answers
38 views

Combinatorics: Counting Set Partitions with Moebius Function

Let $\pi_n$ be the poset of all set partitions of $\{1,...,n\}$ ordered by refinement, $\sigma = \{B_1,...,B_k\}$ be a set partition with blocks $B_i$, and $max(B_i)$ be the maximum value in the block ...
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2answers
39 views

Proving the Hardy-Ramanujan-Rademacher series for $p(n)$

How to prove the series of the Hardy-Ramanujan-Rademacher for the partition for an integer n using the Cauchy residue theorem?
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1answer
1k views

Partition an integer $n$ into exactly $k$ distinct parts

I know how to find the number of partition into distinct parts, which is necessarily equal to the number of ways to divide a number into odd parts. I also know how to partition n into exactly k parts. ...
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0answers
13 views

How can this series of partitions be converted?

I have ascertained from this answer to one of my previous questions that $$ \sum_{N=1}^\infty\sum_{k=1}^{p(N)}\prod_{j=1}^lf(i_{j_k},r_{j_k}) = \prod_{k=1}^\infty \left(1+\sum_{n=1}^\infty f(n,k)\...
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1answer
32 views

Number Partitions

Is this series complete, using '0' is not allowed: $6 = 6$ or $5+1$ or $1+5$ or $4+2$ or $2+4$ or $3+3$ or $4+1+1$ or $1+4+1$ or $1+1+4$ or $1+2+3$ or $1+3+2$ or $2+3+1$ or $2+1+3$ or $3+1+2$ or $3+2+...
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1answer
29 views

Can a sum over all the partitions be reduced to a non-partition sum?

Let's say we have $$\sum_{k=1}^{p(N)}\prod_{j=1}^lf(i_{j_k},r_{j_k})$$ for some two-variable function $f(x,y)$. Let $\lambda_k$ be the $k\text{th}$ partition of the integer $N$ into $l_k$ distinct ...
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0answers
31 views

Relationship between Riemann Zeta function and Prime zeta function

In his paper, Daniel Grunberg shows a relationship between the Stirling Numbers of the first kind and the Harmonic numbers via series of partitions (see Equation 3.1 on Page 5 in the link above). If ...
4
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1answer
131 views

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as $\...
2
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1answer
33 views

Count a Partial Equivalence Relations on a set

A Partial Equivalence Relation is a relation that is symmetric and transitive, and to count the number of equivalence relations on a set exists Bell Numbers, my question is How I can count the number ...
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1answer
71 views

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

Is it possible to define a partition of a closed set, such that the union of the partitions will give $[2,3]$ and their intersection to be empty?
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1answer
40 views

Why are is partitions counting technique wrong?

I recently heard about partitions. I tried to count them using the following technique: 1) Ways to write $5$ as a sum of five positive integers: $$1+1+1+1+1$$ 2) Number of ways to write $5$ a sum of ...
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1answer
39 views

List the partitions of the set $S = \{1, 2, 3\}$.

I write the partition sets of set $S=\{1,2,3\}$ as follows: $\{\{1, 2, 3\}, \{1\}, \{2\}, \{3\}\}$ can someone show me why and how to complete the list?
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1answer
28 views

Problem on partioning

On reading the book 'Aha! Solutions' by Martin Erickson I came to know that the number of partitions of $n$ ($n \in \mathbb{N}$) into three parts is $\left\{ {\frac{{{n^2}}}{{12}}} \right\}$ where $\...
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1answer
14 views

Number of Blocks of Size $i$ in Set Partitions

Given a set of $N$ labeled elements $\{1, 2, ..., n\}$, we know that there are $S(N, k)$ ways to partition the elements into $k$ non-empty subsets (where $S(N, k)$ is the Stirling number of the second ...
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1answer
46 views

Conjecture: Partitioning $\Bbb N$ into parts that sum to $13^i$ [closed]

Recently I was thinking and came up with a conjecture that goes as follows: Conjecture: There exists a $\Omega$ such that $$\Omega = \Bigg\{A_i \ \Bigg| \ \forall i,j:i\not=j, \ A_i\cap ...
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0answers
23 views

Partitioning the set of mappings.

The following is first two steps of an algorithm given from a research paper. I understood the first step. But please explain the second step: what does mean " Rearrange the partition according to ...