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.

learn more… | top users | synonyms

2
votes
2answers
244 views

Partitions of $n$: proving $p(n+2)+ p(n) \geq 2p(n+1)$

For $n \geq 2$ give an alternative description of $p(n) - p(n-1)$ as the number of partitions of $n$ which have a certain property. I have done that part, it is fine. I have not included it here as ...
5
votes
2answers
334 views

Derivative of Schur function

In his answer to http://mathoverflow.net/questions/129854, R. Stanley says that the partial derivative (over the relevant x[i]) of the Schur function of a partition lambda of n equals the sum the ...
2
votes
1answer
576 views

Count the number of unique equal sized partitions of a set.

Given the integers $[1, ck]$, they will be partitioned into $c$ subsets of size $k$. I want to count the number of unique versions of each subset (where order matters). Clearly, there are ${ck \...
10
votes
6answers
950 views

Can every infinite set be divided into pairwise disjoint subsets of size $n\in\mathbb{N}$?

Let $S$ be an infinite set and $n$ be a natural number. Does there exist partition of $S$ in which each subset has size $n$? This is pretty easy to do for countable sets. Is it true for ...
0
votes
1answer
39 views

Make a partition that contains a set of points??

I am given a set of $M$ points in a segment (the edges are also points in this set) I would like to partition the segment (with equidistant points), in such a way that my partition contains all these ...
1
vote
1answer
352 views

Generating functions of partition numbers

I don't understand at all why: \begin{equation} \sum\limits_{n=0}^\infty p_n x^n = \prod\limits_{k=1}^\infty (1-x^k)^{-1} \end{equation} Where $p_n$ is the number of partitions of $n$. Specifically ...
1
vote
1answer
122 views

Can a p-core of a partition be reached by repeated stripping of p-rimhooks?

in http://mathoverflow.net/questions/42562 I read : "If you strip p-rimhook after p-rimhook off of a partition, this always results in the same p-core, and the choices don't matter." But I must be ...
3
votes
2answers
188 views

Probability distribution of product of integers

I have a scoring system based on 5 factors with integer values from 1 to 5: Score = A * B * C * D * E So the Score can range from 1 to 3125. Each of the factors ...
0
votes
2answers
79 views

Question on combinatorics, partitions. [duplicate]

Let $p$ ($n|$distinct odd parts) be the number of partitions of $n$ into distinct odd parts. Prove that $p(n)$ is odd if and only if $p$($n|$distinct odd parts) is odd by using the theorem on self-...
1
vote
1answer
84 views

Partition parts

Consider the partitions of $n$. For $n = 5,7,9,\ldots$, it appears as if the number of pairwise partitions $\{a,b\}$, where both $a$ and $b$ are composite, equals the total number of individual odd ...
3
votes
1answer
256 views

bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n

Suppose $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is a partition of $2n$ where $n\in\mathbb N$ satisfying the following conditions: (1) $\lambda_k=1$. (2) $\lambda_i−\lambda_{i+1}\leq 1$ for ...
1
vote
1answer
377 views

conjugate partition definition

i would like to understand basic definition of conjugate partition,this is what is said in my book Let $υ = (u_1, u_2, . . . , u_n)$ be a sequence of integers such that $u_1 ≥ u_2 ≥ · · · ≥ u_n ≥ 0....
2
votes
1answer
109 views

Related to the partition of $n$ where $p(n,2)$ denotes the number of partitions $\geq 2$

I was trying the following question from the Number theory book of Zuckerman. If $p(n,2)$ denotes the number of partition of n with parts $\geq2$, prove that $p(n,2)>p(n-1,2)$ for all $n\geq 8$, ...
1
vote
0answers
157 views

Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

I would like to find articles or documentation regarding this process: Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook ...
7
votes
3answers
1k views

Combinatorics: Generating Function related to compositions of a number

My goal is to find the coefficients of the generating function for the following situation: The number $f(n)$ is the sum over all compositions of $n$ into $3$ parts of the product of those parts. Fo ...
3
votes
1answer
77 views

Number of solutions for an equation

I have to find the number of solutions for: $$x_1 + x_2 + x_3 + x_4 = 42$$ when given: $$ (I) 12 <= x_1 <=13 $$ $$ (II) 3 <= x_2 <= 6 $$ $$ (III) 11 <= x_3 <= 18 $$ $$ (IV) 6 <= ...
6
votes
3answers
381 views

How to prove it? (one of the Rogers-Ramanujan identities)

Prove the following identity (one of the Rogers-Ramanujan identities) on formal power series by interpreting each side as a generating function for partitions: $$1+\sum_{k\geq1}\frac{z^k}{(1-z)(1-z^2)...
2
votes
1answer
106 views

Conjugate Ferrers diagrams

Let $\pi=\langle \pi_1,\pi_2,... \rangle , \ \pi_1\ge\pi_2\ge...,$ be a partition of a number and $\pi'=\langle \pi_1',\pi_2',... \rangle$ be a partition conjugated to $\pi$, which means that Ferrers ...
2
votes
1answer
169 views

Total number of parts in the all partitions of $n$

Let's denote $N_k(n)$ as the number of partitions $n$ into at most $k$ parts. Prove that the total number of parts in the all partitions of $n$ is equal to: $$\sum_{a=1}^n \sum_{b=1}^{\lfloor n/a \...
1
vote
1answer
55 views

Balancing two sets while items in one are unmovable

I'm working on a following problem: Given two sets containing jars, each of which is assigned a random weight (weight is a real number), find a way to balance two sets by weight, i.e. the difference ...
0
votes
2answers
299 views

Give combinatoric argument for partition counting: $P(n, k) = P(n -1, k -1) + P(n-k,k)$

Suppose you have $n$ identical pieces. You want to split them in $k$ groups. (each group must have $> 0$ pieces) First, I was ask to answer the basic cases $1 \le k \le n \le 5$ For examble, $$...
3
votes
3answers
1k views

How can I find the partitions of this equivalence relation?

I have the following equivalence relation: $$\{(1,1),(1,4), (2,2), (3,3), (4,1), (4,4)\}$$ On the set: $ A = \{1,2,3,4\}$ How can I find it's partitions? This example will help me understand the ...
6
votes
1answer
150 views

Maximal Zero Sums Partition

You are given $n$ numbers between $-n$ and $n$, the sum of numbers is $0$. Divide the given sequence on disjoint subsequences in such a way that each subsequence has zero sum. Each element should ...
1
vote
1answer
185 views

Balls, Bags, Partitions, and Permutations

We have $n$ distinct colored balls and $m$ similar bags( with the condition $n \geq m$ ). In how many ways can we place these $n$ balls into given $m$ bags? My Attempt: For the moment, if we assume ...
1
vote
1answer
129 views

Determining Stirling number

In the first part of the question I was asked to find the exponential generating function for $s_{n,r}$, the number of ways to distribute $r$ distinct objects into $n$ (a fixed constant) distinct ...
0
votes
1answer
161 views

Generating function: number of partitions that add up to at most $n$

Find a generating function $a_n$, the number of partitions that add up to at most $n$. So I know that if it were asking the number of partitions of the integer $n$, I would have my generating ...
1
vote
1answer
346 views

Combinatorial proof involving partitions and generating functions

Show that any number of partitions of $2r + k$ into $r + k$ parts is the same for any $k$. I've tried this, but I haven't come up with anything; hence why I have nothing written here. But in any ...
7
votes
1answer
107 views

Generating function for $r^\binom{n}{2}$

I'm trying to find a closed form of the generating function $$ G(x) = \sum_{n \ge 0} r^\binom{n}{2} x^n $$ for a real number $0 < r < 1$. I found that $G(x) = 1 + xG(rx)$. Any hints where to ...
2
votes
0answers
22 views

Terminology for breaking partition diagram into “L”'s

When one thinks about partitions, it's quite normal to consider pieces of the partition diagram, such as rows, columns, arms, legs, hooks, etc. One decomposition of particular interest to me is ...
0
votes
2answers
42 views

Relationship Question

Let $\ S\ $ be a non-empty Set, and suppose s$\ \in S $. Assuming $\ S\ $ is finite, what can we deduce about the relationship between $\ |\mathcal P(S\ \setminus \{s\} )| $ and $\ | \mathcal P(S)|?$ ...
3
votes
3answers
679 views

Further clarification needed on proof invovling generating functions and partitions (or alternative proof)

Show with generating functions that every positive integer can be written as a unique sum of distinct powers of $2$. There are 2 parts to the proof that I don't understand. I will point them out as ...
2
votes
1answer
73 views

Question about partition of open sets in $\mathbb{R^n}$

I have to prove that any open set $U \subset \mathbb{R^n}$ is a countable union of disjoint limited rectangles. I proved that it is a countable union of rectangles, the "expected classical" way, I ...
1
vote
4answers
3k views

How many solutions does the equation $x_1 + x_2 + x_3 = 11$ have, where $x_1, x_2, x_3$ are nonnegative integers?

Help me understand problems of this type a bit more intuitively. The solution $C(3+11−1,11)$ seems simple enough, but I got stuck thinking about how many integers you are choosing from within $x_1$, ...
2
votes
1answer
1k views

Finding a generating-function using partitions

Find a generating function for a , the number of partitions of r into (a.) Even integers (b.) Distinct odd integers. I am at a loss of starting this.
0
votes
1answer
92 views

How to find random numbers that can sum up to n?

I have a random integer $n$ and another integer called the summary. I want to know how many ways I can sum a subset of numbers from $1$ to $n$ to produce the value of summary. For example, I have $1,...
5
votes
2answers
106 views

Partition Bijection

I'm not sure what I'm missing. I think I'm thinking too hard about finding this bijection. Please help!
3
votes
2answers
978 views

Number of partitions of $n$ with $k$ parts equals the number of partitions of $n + \binom k {2}$

How do I prove bijectively The number of partitions of $n$ with $k$ parts equals the number of partitions of $n + \binom k {2}$ with $k$ distinct parts
4
votes
2answers
95 views

Partition bijections

How do I prove bijectively that the number of partitions of n with largest part k equals the number of partitions of n with exactly k parts.
5
votes
1answer
847 views

Bijective Proof: Number of Partitions of 2n into n parts

The number of partitions of n is equal to the # of the partitions of 2n divided into n parts. I know that the number of partitions of any integer n into i parts equals the number of partitions of n ...
2
votes
2answers
1k views

common knowledge and concept of coarsening partition

Here is a proof of the equivalence between my definition and Aumann's for "common knowledge". I'm assuming some familiarity with set partitions. Aumann's definition is in terms of the Kolmogorov ...
0
votes
2answers
376 views

Get all possible partitions of number

Is there a way to get all possible partitions of an integer? Possibly specifying max and/or min summand. I'm interested in partitions themselves, not just partition count.
6
votes
1answer
201 views

Formula for evaluation of character on a transposition

Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of $\lambda$...
3
votes
1answer
96 views

Finite Partitions of the Unit Interval

Does the unit interval have a finite partition $P$ such that no element of $P$ contains an open interval? I would think that the answer is no, because each element of $P$ would have Lebesgue measure ...
1
vote
2answers
238 views

Integer solutions

How many positive integer solutions are there to $x_1 + x_2 + x_3 + x_4 < 100$? I haven't seen any problems with "less than", so I'm a bit thrown off. I'm not sure if my answer is correct, but if ...
5
votes
2answers
186 views

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png An equilateral triangle can be divided into 3 congruent shapes. Questions: ...
0
votes
1answer
54 views

Is this a valid partition?

Say we have the $S$ which is the set of all compositions of n >= 0 with an odd num. parts. Define $S_1$ to be the set of all compositions of n >= 0 with an odd num. of parts where at least one part ...
1
vote
1answer
143 views

Question on lower & upper Riemann sums for piecewise.

Partition{$\frac{-\pi}{6},3,2\pi$} $f(x)= 1/2$ if x is rational $sin(x)$ if x is irrational. Im not sure if im doing it correctly: $L(P,f)=(3-\frac{-\pi}{6})(sin(\frac{-\pi}{6}))+(2\pi-3)(sin(2\pi))$...
2
votes
0answers
109 views

What is the correct technical term for this generalization of an integer partition?

Given a vector $v$ with non-negative integer coordinates, is there a technical term for an unordered tuple of vectors $(v_1,\dots, v_k)$ with non-negative integer coordinates such that $v_1+\dots+v_k ...
0
votes
1answer
153 views

Partitions of $n$ into $k$ blocks, without single blocks.

So I'm trying to come up with a recursive formula $f(n)$ which counts the number of all partitions of $[n]$ into $k$ identical blocks, where the number of elements in each box is more than 1. What i'...