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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1answer
177 views

How many compositions of $n \in N$ are there where each part is greater than $1$?

Can someone help me with this? Let $n \in N$. How many compositions of $n$ are there where each part is greater than $1$? (Number of parts are not restricted)
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1answer
35 views

How does this line work in a problem about restricted compositions of $n$?

I'm trying to follow an example problem of calculating how many $k$-part compositions of $n$ are there, with the restriction that each part is at most 5. During the calculation there is this mess: ...
1
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0answers
61 views

Restricted Partitions of $n$ [duplicate]

The original question was to find the number of ways to split an integer, $n$, into any number of partitions where each of the parts belong to the set $\lbrace 1,3,4,9\rbrace$. Assuming I did this ...
6
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1answer
159 views

Representations of an integer as the sum of other integers

Given a finite set $S$ of (distinct) integers $s_1, \dots, s_n$ and an integer $x$, I'm looking for all representations (where order is important) $$ x=\sum_{i=1}^ks_{t_i} (t_i\in\{1,\dots,n\}) $$ ...
0
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1answer
28 views

Partition set to contain the same number of elements distributing the remainder

Given $|B| = 23$ and number of partitions $P=4$. We want to partition the given set $B = B_1\cup\dots\cup B_P$ so that every partition $B_i$ contains the same amount of elements where the remaining ...
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3answers
261 views

Frobenius coin problem

Suppose that you only have coins worth, say 3 and 5 euros. According to Sylvester result we can find the Frobenius nr $g(3,5)=15-3-5=7$ so 7 is the largest integer that cannot be written as ...
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2answers
103 views

Enumerate partitions of identical objects

I have a problem concerning enumerating partitions of a set of identical objects. I know, now someone is going to talk about Stirling number of second kind, but I'm quite sure this is not the answer. ...
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1answer
60 views

Prove that this set (involving fractional part of any rational number) is a partition of the set of rationals.

For any rational number $x$, we can writte $x=q+\,n/m$ where $q$ is an integer and $0\le n/m<1$. Call $n/m$ the fractional part of $x$. For each rational $r\in \{x : 0\le x<1\}$ ,let $A_r = \{ ...
3
votes
1answer
144 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 ...
2
votes
1answer
42 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
2answers
132 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 ...
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votes
3answers
423 views

Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.

Prove that $$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$ 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 ...
0
votes
1answer
29 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 ...
5
votes
2answers
165 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
249 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
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6answers
482 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
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1answer
28 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
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1answer
89 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
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1answer
37 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
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2answers
78 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
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2answers
49 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 ...
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1answer
56 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
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5answers
307 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
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1answer
78 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
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1answer
99 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
51 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 <= ...
7
votes
3answers
411 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
1answer
64 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 ...
6
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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
86 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 ...
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1answer
128 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 ...
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votes
1answer
151 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 ...
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1answer
200 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 ...
1
vote
1answer
131 views

A partition of an interval to reduce the difference between the upper and lower Darboux sums

Let $$ f(x) = \begin{cases} 2x+1, & x\in [2,4] \\ 7-x, & x\in(4,4.5) \\ 3, & x \in[4.5,6] \end{cases} $$ For $q = 1/4$, find a partition of $[2, 6]$ such that the difference between the ...
0
votes
2answers
160 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
1answer
205 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 ...
3
votes
1answer
181 views

Bell number. Combinatorial proof.

$B_m$ is the Bell's number (the # of all partitions of the set $m$) and $B_m^*$ is the number of partitions of set $m$ with no singleton block We need to prove that $B_m = B_m^* + B_{m+1}^*$. ...
6
votes
3answers
185 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
411 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
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1answer
67 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 ...
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1answer
112 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
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1answer
49 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 ...
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1answer
81 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 ...
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 ...
0
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1answer
97 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 ...
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1answer
166 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
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1answer
87 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
14 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 ...
4
votes
3answers
149 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
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2answers
39 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 $\ | \mathcal P(S)|?$ ...