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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1answer
57 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
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0answers
41 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
34 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
43 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
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0answers
23 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
29 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
119 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
30 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
140 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
383 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
60 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
vote
1answer
53 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
144 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
60 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
40 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
29 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
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0answers
30 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 ...
2
votes
0answers
87 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
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2answers
50 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 ...
2
votes
1answer
165 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
91 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
120 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
vote
3answers
136 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
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0answers
67 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
61 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
38 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
86 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
37 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
187 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
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1answer
97 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 ...
1
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1answer
52 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
28 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 ...
0
votes
1answer
46 views

Number of partitions of integer into parts repeated <= 2 times

The generating function for the number of such partitions is $$ G(q) = \prod_{i=0}^{\infty}(1+q^i+q^{2i}) $$ - that much I understand. Is there any way to transform it into a form ...
1
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0answers
36 views

Partitions that have at most k parts and all parts <= j

Let p¯k(n) be the number of partitions of n with largest part at most k which is equivalent to partition into at most k parts. I do know an expression for that function. ( product of 1/1(1-n) through ...
0
votes
3answers
60 views

Powerset and Partition Creation

Let $A$ be a set with at least three elements If $\mathcal P = \{B_1 ,B_2, B_3\} $ is a partition of $A$, is $\{B_1^c , B_2^c,B_3^c\}$ a partion of A. So I am thinking the answer is no, since there ...
0
votes
1answer
23 views

Equivalence Relations dealing with creating partitions.

Let R be a relation on a set A that is reflexive and transitive but not symmetric. Let R(x) = {y: xRy}. Does the set a = {R(x): x ∈ A} always form a partition of A? I really don't know where to ...
0
votes
1answer
94 views

Partitions on C = {i,-1,-i,1}

Let $ C = \{i, -1, -i, 1\}$ , where $ i^2 = -1 $. The relation $R$ on $C$ given by $xRy$ iff $xy = \pm 1$ is an equivalence relation on $C$. Give the partition of $C$ associated with $R$ I would ...
4
votes
3answers
490 views

Partitions of $n$ into distinct odd and even parts proof

Let $p_\text{odd}(n)$ denote the number of partitions of $n$ into an odd number of parts, and let $p_\text{even}(n)$ denote the number of partitions of $n$ into an even number of parts. How do I ...
0
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0answers
33 views

Need help with a Partitions question

For $m \in \mathbb{N}$, let $C_m = \{x \in R \mid m-1 \leq x^2 < m\}$. Is $\ell=\{C_m \mid m \in \mathbb{N}\}$ a partition of $\mathbb{R}$? If I understand it correctly, $C_m$ will always be a ...
1
vote
0answers
63 views

Running the Greene-Nijenhuis algorithm backwards

Let $Y$ be a Young tableau of shape $\lambda:=(\lambda_1,\ldots,\lambda_n)$, where $\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n$ and $N:=|\lambda|:=\sum_i\lambda_i$. I'll be using the English ...
3
votes
5answers
377 views

In how many ways i can write 12?

In how many ways i can write 12 as an ordered sum of integers where the smallest of that integers is 2? for example 2+10 ; 10+2 ; 2+5+2+3 ; 5+2+2+3; 2+2+2+2+2+2;2+4+6; and many more
2
votes
1answer
368 views

Number of ways to partition a set with $n$ elements to $k$ subsets where at least one subset has $r$ elements

I'm familiar with Stirling numbers of the second kind to compute the number of ways to partition a set with $n$ elements into $k$ non-empty, disjoint subsets. However, there are combinations which I ...
1
vote
0answers
56 views

Teminology: Partitioning a Set Including Empty Partitions

Mostly I see a partition of a set A defined as a collection of non-empty disjoint sets whose union is A. I see one reference that allows empty sets to be included in the partition: ("Potter, M. "Set ...
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2answers
34 views

Partitioning techniques for finding large matrix determinents

I'm in a linear algebra class and we're doing determinants right now. I got this matrix to do: $\begin{matrix} 2 & 1 & 0 & 0 & 0 \\ 3 & -1 & 2 & 0 & 0 \\ 0 & 4 ...
1
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1answer
39 views

About the parity of the partition function.

I am reading this Kolberg's article, where he proofs that the partition function takes both even an odd values infinitely often. http://www.mscand.dk/article.php?id=1555 Although I'm sure it's ...
1
vote
1answer
137 views

Non combinatorial proof of Jacobi triple product?

The jacobi triple product identity in the form given below - $$\prod_{n\geq 1}(1-q^n) (1-xq^{n-1}) (1-\frac{1}{x}q^n) = \sum_{k\geq 0} (-x)^k q^{k \choose 2}$$ can be proved as follows, Let the LHS ...
1
vote
0answers
43 views

Discriminating integer partitions

Given a fixed positive integer, say $n$, letn $P_n$ be the set all partitions of $n$, where each partition itself is a set i.e. order is discarded and each part is less than 5. Can we establish a ...
2
votes
1answer
47 views

Minimum number of questions needed to uniquely determine an integer partition

This came up in an algebra class today, but I'll phrase it a bit differently. Let's say Alice and Bob are playing a game. Alice thinks of an integer partition, and tells Bob the sum of the ...
1
vote
1answer
212 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 ...