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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27 views

When can $S \times S$ be partitioned by triples?

Let $S$ be a finite set of $n$ elements. For which $n$ can the Cartesian Product $S\times S$ be partitioned in such a way that elements of the partition are of one of the following forms: $\{(a,b), (...
-1
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1answer
29 views

Number of solutions of the two equations

Find the number of integral solutions of the equation: $a+b+c=m$ with $0\gt a\gt b\gt c$ And the generalized version: $a_1 + a_2 + \cdots + a_k = m$ with $ 0\gt a_1\gt a_2\gt \cdots \gt a_k$
2
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2answers
66 views

Pairs of irreducible fractions that add up to a given irreducible fraction

Given the irreducible fraction $\frac a b$, with $a, b \in \mathbb N$, what is the expression that enumerates all the irreducible fractions of integers that add up to $\frac a b$? Namely, an ...
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0answers
42 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{...
2
votes
1answer
33 views

Permutation of Disjoint Sets of a Symmetric Group

Problem Description: Consider a symmetric group $S_n$ acting on $n$ objects. We partition $S_n$ into two sets $A, B$ such that $A \cap B= \emptyset$ and $A \cup B = S_n$. In other words, $S_n$ is ...
2
votes
2answers
22 views

Proving that R is a partial Order.

Define the relation $\Bbb R \times \Bbb R$ by $(a,b) \; R$ $ (x,y)$ iff $a \le x$ and $b \le y$ , prove that R is a partial ordering for $\Bbb R\times\Bbb R $ . A partial order is if R is reflexive ...
0
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1answer
43 views

Describe the equivalence relation of the following set with the given partition.

Describe the equivalence relation of the following set with the given partition. $ \Bbb N $ , $ \{\{ 1 \}, \{2,3 \}, \{4,5,6,7\},\{8,9,10,11,12,13,14,15\}....\} . $ What this question has me ...
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2answers
18 views

Checking reflexive, symmetric and transitive properties of $\neq$ on $\mathbb{N}$

QS: Indicate if the relation on the given set are reflexive on a given set, which are symmetric, and which are transitive. $\not = \text{on } \Bbb N$ So for this problem I am trying to ...
6
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0answers
74 views

Inequality for hook numbers in Young diagrams

Consider a Young diagram $\lambda = (\lambda_1,\ldots,\lambda_\ell)$. For a square $(i,j) \in \lambda$ define hook numbers $h_{ij} = \lambda_i + \lambda_j' -i - j +1$ and complementary hook numbers $...
4
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0answers
100 views

Almost-Linear Sequence of Positive Integers Excluding a Near-Quadratic Integer Sequence

I remember that I have seen a similar problem to this one here: The set of natural numbers that don't belong to a set (which is a duplicate of $m$ doesn't come in the sequence $a_n=[n+\sqrt{n}+...
5
votes
0answers
52 views

Towers of coins puzzle [duplicate]

Let $n$ be a natural number. You are given $\frac{n(n+1)}{2}$ coins which are arranged in towers. Every turn you pick up the top coin of each tower and gather all these into a new tower. Prove that ...
0
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1answer
40 views

Elementary proof of MacMahon's generating function for plane partitions

Recall Macmahon's elegant and beautiful generating function for plane partitions $$ \sum_{n=0}^{\infty} pp(n) q^n = \frac{1}{(1 - q)^1(1 - q^2)^2(1-q^3)^3\cdots}= \prod_{j=1}^{\infty}\frac{1}{(1-q^j)^...
0
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1answer
24 views

Longest strict partition of a positive integer?

I have just finished my 1st year in Computer Science. Since the holidays began, I've been studying algorithms and different computer science problems. The following problem has proved quite difficult ...
2
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0answers
25 views

enumerating all permutations of partitions

Given a set of $n$ items, I would like to enumerate all the permutations of all the partitions of those $n$ items. For example, in the case of $n=3$ (with Bell number 5), there are 13 permutations of ...
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0answers
31 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 ...
2
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0answers
49 views

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:=\operatorname{lcm}(d_1,\ldots,d_n) $ and $ d:=\sum_{i=1}^n d_i $. (1) Alice claims that whenever $ \...
0
votes
1answer
7 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 ...
0
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0answers
23 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 ...
2
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0answers
47 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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vote
2answers
49 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 $\binom{2n}{2}$...
4
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1answer
80 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
67 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
votes
3answers
335 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 $\bar p_k(n)$ ...
0
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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
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
156 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
18 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 ...
7
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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
17 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 ...
16
votes
1answer
231 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 $...
2
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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(...
0
votes
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\...
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 ...
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1answer
23 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 \frac{1}{4n\...
2
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0answers
35 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 $f$ ...
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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
votes
1answer
481 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
418 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
80 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 ...
0
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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 ...
0
votes
2answers
45 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. ...