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.
1
vote
0answers
25 views
Upper and lower integration inequality
I would like to learn how to prove that the following inequality holds.
Let $F$ be a bounded function on an interval $[a,b]$, so that there exists $B\geq 0$ such that $|f(x)| \leq B$ for every $x\in ...
0
votes
0answers
16 views
Symmetry of Plancherel measure (for $S_n$)
For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$:
$$
\begin{pmatrix}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 ...
2
votes
1answer
49 views
Partitions of $n$: proving $p(n+2)+ p(n) \geq 2p(n+1)$
For $n \geq 2$ give an alternative description of $p(n) - p(n-1)$ as the number of partitions of $n$ which have a certain property.
I have done that part, it is fine. I have not included it here ...
0
votes
1answer
58 views
How to prove $p(n\mathrel{;} \{1, 2, 4\}) = p(n - 4\mathrel{;}\{1, 2, 4\}) + p(n\mathrel{;} \{1, 2\})$?
Let $n_1,...,n_k$ be distinct natural numbers and let $p(n\mathrel{;} \{n_1,...,n_k\})$ denote the number of partitions of $n$ into parts, each of which is equal to one of $n_1,...,n_k$. Show that ...
9
votes
3answers
129 views
Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
Prove
$$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$
I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its complement ...
0
votes
1answer
20 views
Partitions of an interval and convergence of nets
Let $\mathscr{T}$ be the set ob 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
2answers
119 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 ...
1
vote
1answer
19 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
203 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
23 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
31 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
14 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
41 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 ...
-1
votes
2answers
27 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 ...
1
vote
1answer
31 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
4answers
205 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
1
vote
1answer
19 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 ≥ ...
2
votes
1answer
67 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$, ...
3
votes
1answer
38 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
74 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 ...
0
votes
0answers
40 views
General term of this sequence
I wanted to know the General term or the function to generate this sequence I found on OEIS.
It is the number of ways to express 2n+1 as p+2q; where p and q can be odd prime number and even semiprime ...
4
votes
0answers
88 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
59 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 ...
1
vote
1answer
97 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:
$$\sum_{a=1}^n \sum_{b=1}^{\lfloor n/a ...
-2
votes
1answer
123 views
From a Generating Function find $R(x)$ as an infinite product of Quotients
Let $r(n)$ be the number of partitions of $n$ so that no multiple of $3$ appears as a part. For
example, $r(8) = 13$.
Let $R(x) =\sum_0^\infty r(n)x^ n $ be the generating function for $r(n)$.
Find ...
-2
votes
1answer
158 views
Find a form for $Q(x)$ as an infinite product of polynomials
Let $q(n)$ be the number of partitions of $n$ so that no part appears three or more times. For example, $q(8) = 13$
Let $Q(x) = \sum\limits_{n=0}^\infty q(n) x^n$ be the generating function for ...
0
votes
2answers
72 views
Give combinatoric argument for partition counting: $P(n, k) = P(n -1, k -1) + P(n-k,k)$
Suppose you have $n$ identical pieces. You want to split them in $k$ groups. (each group must have $> 0$ pieces)
First, I was ask to answer the basic cases $1 \le k \le n \le 5$
For examble,
...
2
votes
1answer
146 views
bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n
Suppose $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is a partition of $2n$ where $n\in\mathbb N$ satisfying the following conditions:
(1) $\lambda_k=1$.
(2) $\lambda_i−\lambda_{i+1}\leq 1$ for ...
5
votes
3answers
101 views
How to prove it? (one of the Rogers-Ramanujan identities)
Prove the following identity (one of the Rogers-Ramanujan identities) on formal power series by interpreting each side as a generating function for partitions:
...
2
votes
3answers
44 views
How can I find the partitions of this equivalence relation?
I have the following equivalence relation:
$$\{(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 ...
5
votes
1answer
51 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
0answers
23 views
Investigating some statistics of set partitions
I am interested in using some software to investigate a couple of statistics involving set-theoretic partitions of $[n]=\{1,2,3,\cdots,n\}$ for feasibly small $n$. I am simply assuming that there are ...
1
vote
1answer
33 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
39 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 ...
1
vote
1answer
32 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 ...
1
vote
0answers
41 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
...
0
votes
1answer
37 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
46 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
62 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
9 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
0answers
51 views
Number of partitions of a set of n distinct objects
Say I have a set of $n$ distinct objects and I want to divide it into $k$ identical boxes each of which will has exactly $r_i$ objects, $1\leq i \leq k$. How many ways can I do it?
I guess that the ...
4
votes
3answers
90 views
Algorithm to partition sum between buckets in all unique ways
The Problem
I need an algorithm that does this:
Find all the unique ways to partition a given sum across 'buckets' not caring about order
I hope I was clear reasonably coherent in expressing ...
0
votes
2answers
37 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
63 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 ...
2
votes
1answer
52 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
122 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 $x_1$, ...
4
votes
0answers
43 views
Characters of the symmetric group corresponding to partitions into two parts
Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character ...
2
votes
1answer
98 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
42 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 ...
5
votes
2answers
58 views
Partition Bijection
I'm not sure what I'm missing. I think I'm thinking too hard about finding this bijection. Please help!
