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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5answers
106 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) ...
3
votes
0answers
23 views

Iterate Over 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 + ...
5
votes
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 ...
1
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1answer
21 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 $\mu=(4,2,1,0)$ ...
5
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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 ...
2
votes
0answers
33 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
votes
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
votes
1answer
412 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 ...
1
vote
1answer
384 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 ...
1
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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 ...
0
votes
0answers
28 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 ...
1
vote
4answers
72 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 ...
1
vote
2answers
56 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 ...
0
votes
0answers
35 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. ...
1
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0answers
37 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 ...
0
votes
2answers
36 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?
4
votes
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. ...
0
votes
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 ...
-1
votes
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 ...
0
votes
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
26 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
votes
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
votes
1answer
32 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 ...
0
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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
39 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 ...
0
votes
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?
0
votes
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 ...
1
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1answer
13 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 ...
-1
votes
1answer
45 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
21 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 ...
0
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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 ...
1
vote
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 ...
1
vote
1answer
20 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 ...
0
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0answers
74 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) ...
1
vote
0answers
54 views

partitions of finite set in same-size parts having at most one element in common

Given $g \ge 2$, $k \ge 1$ and a population of $p = kg$ workers, I'm trying to figure out the longest series of work shifts such that: during each shift, all workers work in $k$ teams of g people; ...
1
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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 ...
1
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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 ...
0
votes
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 ...
-2
votes
1answer
38 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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1answer
51 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 - ...
1
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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 ...
0
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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 ...
3
votes
4answers
54 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 ...
1
vote
1answer
37 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 ...
1
vote
2answers
48 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 ...
3
votes
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 ...
4
votes
2answers
160 views

A formal name for “smallest” and “largest” partition

Consider a set $A=\{1,2,3,4,5\}$, is there any terminology for the following partitions of $A$ ? (1) $A=\{ \{1\},\{2\},\{3\},\{4\},\{5\} \}$ (2) $A=\{\{1,2,3,4,5\}\}$.
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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! ...
1
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1answer
41 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? $$\sum_{n=0}^{\infty}p(n)x^n = \prod_{k=1}^{\infty}(1-x^k)^{-1}$$ ...