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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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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1answer
73 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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1answer
37 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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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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0answers
21 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
4 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
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 ...
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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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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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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(...
8
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2answers
147 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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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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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 $f$ ...
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 ...
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0answers
29 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
77 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
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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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
25 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 ...
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
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
38 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 ...
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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
26 views

Generating functions, Schur's identity

Let $S=\{n\in \mathbb{Z}_+ \mid n \equiv 1, 5 \,\,(\text{mod 6})\}.$ Let $a(n)$ be the number of partitions of $n$ into parts belonging to $S,$ and $b(n)$ be the number of partitions of $n$ into ...
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1answer
44 views

Partitions of a number with greatest product

For $n\in\mathbb{N}$ choose $k_1,\dots,k_l\in\mathbb{N}$ so that $\sum_{i=1}^{l}k_i = n$. Set $k = \prod_{i=1}^{l}k_i$. What is the largest $k$ that one can get? Is there an explicit formula? What ...
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1answer
17 views

Describing a partition as a function

Let P be a partition such that P $=\{t_0,...,t_n\}$ over the interval $[a, b]$, then to refer to a point $i$ in the partition $P$, some would say $t_i$. So my question would be then, in this case ...
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0answers
26 views

Partitions of $n$ vs $(n-k_0)! $

Let $p(n)$ denote the partitions of $n$. It's easy to prove that $p(n) < n!$ (for n>2). I want to prove that if $n \ge 6$ then $(n-2)! > p(n)$. Or more generally let $k_0\in \mathbb{N}$ be a ...
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0answers
75 views

Partitions into non-negative powers of $2$

Let $c(n)$ denote the number of partitions of $n$ into non-negative powers of $2.$ (Thus $c(5)=4$ since $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1).$ (a). Prove that $1+\sum\limits_{n=1}^{\infty} c(n) q^n=\prod\...
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0answers
19 views

Inclusion/exclusion argument for partitions

My question regards Frobenius partitions, or $F$-partitions for short, of a number $n$. A short explanation of the concept is linked below. Specifically, my question is as follows. $F$-partitions of ...
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1answer
40 views

Number of partitions into parts not greater than 9 [closed]

I'm looking for a closed-form formula for the number of partitions of integer $n$ into integer parts less than or equal to 9. Thanks.
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0answers
23 views

Understanding the expansion of product notation.

I have a question regarding the expansion of product notation in the picture below. Equation 3.1 in the attached picture is $$\prod_{n=1}^{\infty}(1+zq^n)(1+z^{-1}q^{n-1})=\frac{1}{\prod_{n=1}^{\...
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1answer
21 views

Index set of dyadic partition

Imagine if you have a line from 0 to 1, and you begin partitioning it dyadically. The first point will be at 0, the second at 1 and the third at 0.5, the fourth at 0.25, the fifth at 0.75 etc. Let's ...
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1answer
15 views

Determine if given relation is an equivalence relation, and describe the partition?

Can someone walk me through how to do these types of problems in my Discreet Mathematics II Textbook? (ex.): Determine whether the given relation is an equivalence relation on the set. Describe the ...
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1answer
56 views

Number of different groups given a list of repeating digits

Suppose that you are given the list[1,1,2,2] . The different groups that can be formed with this list are - ...
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4answers
56 views

Equivalence relation on $\mathbb{N}$ with infinitely many equivalence classes

Does there exist an equivalence relation, defined on $\mathbb{N}$, with infinitely many equivalence classes, all of which contain infinitely many elements? I see no reason for such a relation not to ...
3
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0answers
24 views

How to call a partition of $X$ which consists of all singleton subsets of $X$? [duplicate]

In other words, if $X$ is a set, then how do we call $Y=\{\{x\}:x\in X\}$? $\{X\}$ is already named the trivial partition, so that cannot be it. Complete partition and total partition did not yield ...
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1answer
39 views

Decomposition of $\mathbb N$ into mutually disjoint infinite subsets

$$\mathbb N =\bigcup_{j\in \mathbb N}\Delta_j $$ where each $\Delta_j$ is an infinite subset of $\mathbb N$ and $\Delta_j\cap \Delta_i=\Phi \ for\ i\neq j.$ Now what I need is a few examples of such ...
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2answers
50 views

Formula for Number of possible N element sequences such that their Sum is S?

How many ways I can choose a $N$ element sequence such their cumulative is $S$? Is there any formula for it? Values of $N$ will be greater Than $0$. Here are few examples Let $ N=4$ and $S=5$. Their ...
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0answers
22 views

Finding asymptotycs of partition function

I have been stuck in this problem and have no idea of how to solve it. I have a hint from the book but don't really see how to use it. Any suggestion or hint would be really appreciated. Thanks! ...
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1answer
44 views

Generating function for the partition function [duplicate]

Could someone explain what is the reasoning behind the following equality? Or maybe direct me to a proof of the following equality? $$\sum_{n=0}^{\infty}p(n)x^n = \prod_{k=1}^{\infty}(1-x^k)^{-1}$$ ...