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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3
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0answers
43 views

Asymptotics of partitions in at most n parts, bounded by r

For every positive integers $n,r,w$ define $$ p_w(n,r)=\#\{ (i_1,...,i_r) | \, 0\leq i_1 \leq \dots \leq i_r\leq n, \, i_1+\dots+i_r=w\} $$ as the number of partitions of $w$ in at most $r$ piece ...
1
vote
1answer
77 views

Number of 2n-1 equal size partitions up to symmetry

Consider the $K_{2n}$ (or just the set $\{1,\dots,2n\}$) with $S_{2n}$ acting on the vertices. Moreover consider a collection of 2n-1 partitions of the vertices into two equal sized sets (repeated ...
4
votes
1answer
72 views

Which positive integers can be written in the following form?

I was investigating a generalisation of this problem and found that it reduced to finding where the expression $$\frac{p(p+2m+1)}{2}$$ is an integer, where $p\ge 2$ and $m \ge 0$. Since exactly one of ...
2
votes
0answers
41 views

Partial order on the orbits of the variety of commuting nilpotent matrices

The variety of nilpotent $n\times n$ matrices $\mathcal{N}_n$ over an algebraically closed field $k$ is the disjoint union of orbits under the action of conjugation by $GL_n(k)$. These orbits are ...
0
votes
1answer
37 views

Unordered multinomial coefficients

Let $\{n_1,\ldots,n_k\}$ be a partition of the integer $m$, that is $m=n_1+\ldots+n_k$, and denote by $\mathcal{P}_m$ the set of all such partitions. For a partition $\pi\in\mathcal{P}_m$, the ...
0
votes
1answer
30 views

Easy set partition problem

So I have this problem : Let A be a non empty set an P1 and P2 be two random partitions of the set A. Prove that the set is also a partition of A. I know that this is probably very easy to most of ...
4
votes
6answers
412 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 ...
2
votes
1answer
148 views

Dilworth's Lemma

We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that either For all boxes, all pockets in a box must include a ball with the same color ...
7
votes
0answers
132 views

Partitions of 13 and 14 into either four or five smaller integers

There are exactly 18 partitions of the integer 13 into 4 parts, as on the left of the table, and also 18 partitions into 5 parts, as on the right of the table: $$\begin{array}{c|c} 10+1+1+1 & ...
6
votes
0answers
49 views

Partitioning points with a line

Let $A_{m, n} = \{1, 2, \dots, n\} \times \{1, 2, \dots, m\}$. A straight line would partition the points into two sets. How many ways are there to do it? Let $p_{m, n}$ be that number. Apparently ...
2
votes
3answers
939 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 ...
2
votes
1answer
30 views

Partitions of $\alpha$ Variation

Suppose $T>0$ . Does anyone know if there exists a sequence of partitions $(\pi_n)_{n\in\mathbb{N}}$ of the interval $[0,T]$ such that the mesh size goes to $0$, and such that it is of bounded ...
3
votes
1answer
45 views

Prove: Exactly a quarter of 3-part partitions of numbers >2 equal to 0, 2, 10 mod 12 will make a triangle.

Consider perimeters $>2$ equal to $0$, $2$, or $10 \mod(12)$. The sequence starts $10, 12, 14, 22, 24, 26, 34, 36, 38, 46, 48, ...$ and we can look at the three part partitions that make triangles. ...
0
votes
0answers
39 views

Partition of $[a,b]\subset\mathbb{R}$

Is it possible to create a numerable partition of $[a,b]$? Because I think that it isn't possible, because the last point of the partition must be $b$. But I use the function defined in $[0,1]$ that: ...
1
vote
1answer
31 views

Injection from a set to its partition

Let $A$ be a set such that $|A|=2^{\aleph_0}$ and let $S$ be a partition of $A$ such that $S$ is not countable. Is there a way to define an injective function $f:A\longrightarrow S$ in order to prove ...
0
votes
1answer
35 views

Invective function from $(0,1)$ to a partition

Consider the set $(0,1)$ and denote every $a \in (0,1)$ by it's decimal expansion $$ a=0.a_1a_2a_3\ldots $$ Now, define the equivalence relation $a \sim b$ if and only if $a_p = b_p$ for every prime ...
1
vote
0answers
17 views

partitions are the elements of the free abelian monoid on $\aleph_0$ generators

although the following insight may not be particularly profound, it is always of interest when we see a canonical isomorphism between objects of apparently different types. it is a commonplace that ...
0
votes
1answer
23 views

Do cosets of this semigroup partition this ring?

Let $R = \Bbb{Z}^3$ be the usual direct product ring, and let $S = \{ (k^a, k^b, k^c) : k \in \Bbb{Z} - \{0\}\}$ be a subsemigroup of $R$. Then do the cosets $aS$ partition $R$? Let $R^* = S^{-1}R$ ...
2
votes
2answers
58 views

How to use pentagonal number theorem to determine partitions of n

Under the heading Pentagonal Number Theorem > Relation With Partitions, Wikipedia gives the equation $$ p(n) = \sum_{k} (-1)^{k-1}p(n-g_k) $$ where the summation is over all nonzero integers k ...
0
votes
2answers
22 views

Proving partition function equation

Let $\pi_m(n)$ represent the number of partitions of $n$ in which no part is greater than $m$. Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n).$ I know there is a theorem that helps, but I don't ...
3
votes
2answers
102 views

Partition Identity Proof

Hey guys I am trying to prove the following identity. $p(n)\leq p(n-1)+p(n-2)$ for every ${n}\geq 1$. I worked on breaking it down into steps. I think that the best way to go about with this is in ...
1
vote
3answers
293 views

Proof of the duality of the dominance order on partitions

Could anyone provide me with a nice proof that the dominance order $\leq$ on partitions of an integer $n$ satisfies the following: if $\lambda, \tau$ are 2 partitions of $n$, then $\lambda \leq \tau ...
0
votes
1answer
39 views

integral partition, real analysis

I'm struggling with this question: If we have $f(x) = x^2 $ and $P_n $ which partitions $[1,3]$ into $n$ sub-intervals, each equal in length,how can I write the formulas for $L(f,P_n)$ and $U(f,P_n)$ ...
1
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1answer
41 views

Number of possibilities in a partition problem

Given a set of n items, how many possibilities are there, to distribute these items in two sets with $\dfrac{n}{2}$ items, each? I came up with something like $\dfrac{n!}{\dfrac{n}{2}!}$ but the ...
6
votes
2answers
138 views

For every partition $\pi$ of a fixed integer $n$, $\sum{F(\pi)}=\sum{G(\pi)}$

I need to prove the following question. For every partition $\pi$ of a fixed integer $n$, define $F(\pi)$=number of occurrences of 1 as a summand, and $G(\pi)$=no. of distinct summands in the ...
2
votes
1answer
55 views

(i).Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n)$ without using the generating functions for $\pi_m(n)$.

Questions: $\pi_m(n)$ is defined as the number of partitions of n in which each part is no larger than m. (i).Prove that $\pi_m(n)=\pi_m(n-m)+\pi_{m-1}(n)$ without using the generating functions for ...
0
votes
0answers
36 views

The Generating Functions for $p(S_d,n)$

Prove: $$\sum_{n=0}^{\infty}p(S_2,n)q^n=\prod_{j=1}^{\infty}\frac{1}{(1-q^{5j-4})(1-q^{5j-1})}$$ I have been trying to figure out where to even start this problem and I have no idea what to do. Can ...
1
vote
0answers
25 views

Finding equivalence relations containing specific equivalences

"Find the number of equivalence relations on the set $\{1,2,3,\ldots,7\}$ such that: a) $1\sim2$ and $3\sim4$. b) $1\not\sim2$, $1\not\sim3$ and $3\not\sim2$." Solving this problem is equivalent to ...
0
votes
0answers
26 views

Partitions of a closed interval on the reals

I'm currently trying to go through my textbook in real analysis where the integral is defined. And I'm really confused by something that seems very counter intuitive, and the proof isn't given, and so ...
0
votes
0answers
13 views

Prove that the number of partitions of n into 3 parts is equal to the number of partitions of 2n into 3 parts, each of size less than n. [duplicate]

Prove that the number of partitions of n into 3 parts is equal to the number of partitions of 2n into 3 parts, each of size less than n. For the partitions of n into 3 parts, we have the first row ...
2
votes
0answers
82 views

Partitioning an n-set into k same-size subsets [duplicate]

"How many ways can you partition a set of size $n$ into $k$ parts of the same size?" I've tried solving in the following way but I'm not sure if it's correct, feedback would be appreciated. I'm new ...
1
vote
2answers
45 views

Number of 1's among all partitions of an integer

I am trying find a recurrence relation for the number of 1's among all partitions of an integer. The OEIS database has an entry mentioning this particular sequence but does not give a recurrence ...
0
votes
2answers
44 views

Introduction to Analysis: Bounded and Defined

This question has been rattling my brain for a while. If $f(x)$ is bounded and defined on the interval $[a,b]$ does this imply $(f(x))^2$ is bounded and defined on $[a,b]$? I would say so. Just not ...
0
votes
0answers
35 views

Refinements and partitions

Suppose $P_1, ... P_n $ are finite partitions of the set $X$. Let $Q_n$ be the smallest common refinenement of the partitions $( P_i ) $. I want to show that $Q_{n+1}$ is a refinement of $Q_n$. My ...
2
votes
1answer
145 views

Counting certain partitions of integers

[Recall that] Young's lattice is a partially ordered set in which all partitions of integers are ordered thus: The elements just one step below any partition are those that you can get by subtracting ...
3
votes
1answer
86 views

A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?

Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why, $$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$ $$\sum_{n=0}^\infty P(n) ...
0
votes
1answer
79 views

Show that there is a sequence $(P_n)$ of partitions of $[a,b]$ such that $||P_n||\to0$ & $\lim_{n\to\infty} S(g,P_n)$ for the $g(x)$ defined.

Let $g(x)= \begin{cases} 0, & \text{if }x\in\mathbb{Q} \\ 1/x, & \text{if }x\not\in\mathbb{Q} \end{cases}$, $x\in[0,1]$. Show that $\exists$ sequence $(P_n)$ of tagged partitions of $[a,b]$ ...
-1
votes
1answer
28 views

Partitions and Equivalence Relations [closed]

I have no idea how to solve this: Suppose that $\{S_i:i \in I\}$ is a partition of a nonempty set $Y$ and $X$ is a nonempty subset of $Y$. Prove that $\{X \cap S_i:i \in I \text{ and } X \cap S_i ...
1
vote
3answers
93 views

Existence of uncountable set of uncountable disjoint subsets of uncountable set

"Can you to any uncountably infinite set $M$ find an uncountably infinite family $F$ consisting of pairwise disjoint uncountably infinite subsets of $M$?" Intuitively, I feel like it should be ...
0
votes
0answers
54 views

A Partition Generated from a Family of Sets

This assertion is so basic that I’d expect it to have been put forward with someone’s name attached 100 years ago. But, I can’t find any reference to it searching the web. Of course, the other ...
2
votes
0answers
53 views

Distributing problem using generating functions

For $r\in\mathbb{Z^*}$, let $a_r$ denote the number of ways to distribute $r$ identical objects into $3$ identical boxes, $b_r$ be the number of distributions so that the boxes are to be non-empty, ...
1
vote
0answers
37 views

explicit random cake cutting

I like to split a given interval, let's say $[0,1]$, randomly to a given number $n$ parts. A random input may be provided, like for example a sequence of random numbers $\omega=(r1, r2, ...
1
vote
3answers
49 views

Partition of an equivalence relation

I am having a hard time with the following problem: In F(R), let f~g iff f(x)=g(x) for all x>c where c is some fixed real number. I proved that it was a equivalence relation by the following: ...
2
votes
1answer
31 views

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4x3 and 5x4 rectangles, What is ...
2
votes
1answer
150 views

Minimum square partitions for 4x3 and 5x4 rectangles

Motivation: Tiling an orthogonal polygon with squares Followup question: What is the first $w$ such that a rectangle, $R_{w\times w−1}$ is minimally-square-partitioned by less than $w$ squares.. Yes, ...
1
vote
1answer
62 views

Help understanding solution to growth of partition function

I'm currently a Combinatorics student trying to parse through this solution. I do not understand the proof currently. Any help understanding it is greatly appreciated. Question Let the number of ...
0
votes
0answers
47 views

The generating function for partitions: an alternate (and false) representation

The combinatorics book I'm going through asked me to find the generating function whose coefficients $p_n$ give the number of integer partitions of $n$. I reasoned the following answer: the generating ...
0
votes
0answers
21 views

Tournament payouts, or more number partitioning/change-making

I was thinking of this when confronted with the problem of tournament payouts... Let's say I have \$100 to be parceled out to 10 different people, each of whom is due $n_1, n_2,..., n_{10}$ dollars. ...
1
vote
1answer
45 views

Name this partition of a set problem.

I have a problem related (presumably) to set theory and I can't find the name of it. I just want you to name it so I can do some more research. Given a set $S$of $n$ elements $S= \{s_{1}, s_{2}, ...
1
vote
1answer
23 views

Prove a graph can be partitioned into two groups where every vertex has half its edges cross?

I'm trying to show that for any graph with more than 2 vertices, the graph can be partitioned into two groups such that for every vertex at least half of the vertices its connected to are in the other ...