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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Prove; regarding partitions and refinements

If $ f $ is a bounded function, then for every $ \epsilon >0 $ there exists a partition of a real interval $ X $ such that if $ Y $ is a refinement of $ X $. Why is this true?
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2answers
86 views

Prove: Partitions and refinements

Problem: Let $ R $ be the set of partitions of a real interval. Then for all elements in $ R $, every pair of elements has an upper bound. I am having trouble structuring the proof; and intuitively ...
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1answer
42 views

A fast algorithm for a simple multi-objective minimization?

I have a set $S$ consists of n (arbitrary) integer numbers which I want to partition into $k$ subsets $S_i$ each of size $\frac{n}{k}$ (assume $k$ divides $n$). Let $A$ be the arithmetic mean of ...
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42 views

Generalization of Catalan numbers

I am looking for some kind of function describing the number of non-crossing partitions similar to those described by the Catalan numbers. Let's say $C_3$ would be the third Catalan number. $C_3=5$ ...
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1answer
23 views

Counting resticted partitions of a multiset with additional restrictions

Say I have some multiset of integers, for example $M1=\{6,6,4,4,4,2,2\}$. I have a second multiset that consists of some set of valid sums derived from picking without replacement from $M1$, say for ...
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0answers
36 views

Does Euler's recurrence relation for partitions imply that the partition function grows exponentially

Can one, just by manipulating the series, demonstrate that the partition function must be growing exponentially or at least that it is unbounded by any polynomial? If so, then how would it be done. ...
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0answers
21 views

What is wrong with this “inference” about partitions?

Given that: $p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+⋯$ Why can one not state: $p(n)≥p(n-1)+p(n-2)-p(n-5)-p(n-7)$ Here is the logic: the subsequent 2 terms of the relation are additive and the 2 ...
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0answers
35 views

Famous Arithmetic property of lotteries - restricted partition of integer into exactly k distinct parts between a given set

I would like to find a complete explanation regarding a famous arithmetic property of lotteries : Let´s say a friend of us is a regular player of a lottery where 5 numbers are taken from a box ...
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23 views

Determining probabilities of set of independent experiments from probability of different subsets?

I have a population C of candidates C1..Cn An event will occur to Ci with unknown probability Pi (Pi are independent) The population is divided into disjoint sets S1..Sm For each sub set Si, P(Si) is ...
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1answer
82 views

What is the difference between decomposing a number and partitioning a number?

I see that the question has been answered for SETS in "The Decomposition VS. The Partition of a set" but I would like good definitions that distinguish between these terms when used for NUMBERS. I ...
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207 views

Is every integer $\geq5$ the sum of two primes and a power of a prime?

Is every integer $\geq5$ the sum of two primes and a power of a prime (where $1$ is included in the prime powers)? I don't really expect someone to prove this here, but I wonder if the question has ...
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1answer
74 views

Number of Partitioning a deck with m cards in n types into n-element sets.

For exsample, There are 2cards in 3type. AA,BB,CC. Partition 6cards into 2 3-element sets. [AAB,BCC],[AAC,BBC],[ABB,ACC],[ABC,ABC],... 4 ways or Partition 6cards into 3 2-element sets. ...
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1answer
31 views

Partitioning elements into sets

How many ways are there to partition $n$ unique elements into $2$ sets? What about for $k$ sets? I am specifically interested in how to calculate this for varying values of $n$. Moreover, what if ...
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0answers
36 views

the number of ways a planar graph can be partitioned

i have a connected planar graph to cut into k parts and want to know how many possible solutions there are. it clearly depends on the shape of the graph since nodes all in a row cannot be partitioned ...
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1answer
63 views

Solving the equations $x_1= 4 x_2$ and $x_3= 5 x_2$, with the sum of all three being $150$

Here is the problem. A set X is partitioned into subsets x1, x2, and x3. The number of elements in x1 is 4 times the number in x2. And the number in x3 is 5 times the number in x2. If n(x)=150, ...
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2answers
59 views

What is the number $p(n)$ of partitions of an abundant number $n$ into distinct, proper divisors of $n$?

For lack of a better symbol, $p_{\sigma\tau}(n)$ (feel free to suggest something better). For example, $p_{\sigma\tau}(12) \geq 2$ since $12 = 1 + 2 + 3 + 6 = 2 + 4 + 6$. Of course if $n$ is ...
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1answer
92 views

Generating Functions of Partitions?

Show that $2(1-x)^{-3} [(1-x)^{-3} + (1+x)^{-3}]$ is the generating function for the number of ways to toss $r$ identical dice and obtain an even sum. Workings: I'm not too sure on this problem. ...
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2answers
57 views

how interpret this partition identity?

use the symbol $P(N)$ to denote the set of all partitions of a positive integer $N$ and denote by $P_k$ the number of occurrences of $k$ in the partition $P \in P(N)$, so that $$ N = \sum kP_k $$ by ...
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0answers
41 views

how to count possible planar bipartitions?

i want to find out what small fraction of a solution space a metaheuristic search is actually covering. this case comes down to the number of possible bipartitions for a non-bipartite, undirected ...
3
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1answer
154 views

How to find the restricted partition of n into k *distincts* parts between a finite set [1;r]?

It seems to be an opened question. Indeed, it is easy to find: the number of partitions of n into k distinct parts the number of partitions of n into k parts the number of partitions of n into k ...
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0answers
35 views

How to partition into more than two subsets

Given a set $A$ of numbers and the number of desired subsets, $n$, how can I divide the numbers in set $A$ into $n$ subsets where each number in $A$ is used in one and only one subset and the sum of ...
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1answer
154 views

Coin Change Problem - Find the number of ways to make change

I read the solution here http://www.algorithmist.com/index.php/Coin_Change basically to find the number of ways to make change: We are trying to count the number of distinct sets. it says " Since ...
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1answer
53 views

Does and where to does $\lim_{n\to\infty}\sum_{m} \prod_k \frac{1}{\lambda_{k,m}!}$ converge?

Given $n$ you get a number of partitions of $n$ and let's denote $\lambda_{k,m}$ to be the $k$th part of the $m$th partition. Now I built the following sum, that stimulated the following question: $$ ...
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52 views

Partitioning items into fixed size sets

I have the following problem: Given $n$ items, where each item has weight $w_{i}$, $i=1,2\ldots n$, what is the number of ways to partition these items into boxes of fixed size $C$, such that the sum ...
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114 views

Sum over subsets of a multiset

I have a sum that looks like the following for some multiset $S$ and some function $f$ of $n$ variables which does not depend on the ordering of its arguments: $$\sum_{\{k_1,\dots, k_n\}\subset ...
4
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1answer
85 views

How to denote sum over partitions?

How does one denote a sum of partitions of an integer when writing an article? For example, I have a formula regarding an integer-valued variable (say q=5), and I need to write an expression of the ...
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0answers
49 views

partitions of finite set in same-size parts having at most one element in common

Given g ≥ 2, k ≥ 1 and a population of p = kg workers, I'm trying to figure out the longest series of work shifts such that: during each shift, all workers work in k teams of g people; any two ...
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0answers
78 views

Expansion for r-associated Stirling numbers of the second kind

I am looking for a paper or guidance for expanding the r-associated Stirling numbers of the second kind $S_r(n,k)$. $S_r(n,k)$ is the number of ways to partition a set of n objects into k subsets, ...
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0answers
46 views

Expected Max Pseudotree Size

I'm working on a problem where I need to calculate the expected maximum pseudotree size in a randomly-generated pseudoforest with $n$ nodes. Expected maximum value is of course: $$ E(x) = ...
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2answers
112 views

Counting Set Partitions with Constraints

I'm trying to count set partitions under the following constraints: The number of partitions on $n$ elements where the largest cell(s) have exactly $k$ elements All cells have at least two elements ...
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51 views

Truncation of partitions generating function question

$A (x)$ is the generating function for partitions. $B(x)=\sum_{n=0}^{\infty}b_nx^n $ $$b_n =\binom{\text{number of partitions of }n}{\text{into an even number of parts}}-\binom{\text{number of ...
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1answer
34 views

Union of each family is not the whole set

Let $n\geq k>0$, and consider all $\binom{n}{k}$ subsets of $A=\{1,2,\ldots,n\}$ of size $k$. We want to partition it into families so that the union of each family is not equal to $A$. At least ...
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1answer
210 views

Number of Sets of Partitions

I looked at the partitions of numbers, like let's say $n=5$. You get $$ \begin{eqnarray} 5&=&5\\ \hline &=&4+1\\ &=&3+2\\ \hline &=&3+1+1\\ &=&1+2+2\\ \hline ...
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1answer
32 views

What are the number of possible partitions of a set containing n elements?

This question rises immediately if we try to enumerate the number of possible equivalence relations on a set with n elements.
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2answers
124 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
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2answers
73 views

If L(P, f) = U(P, f), prove that f is a constant function on [a, b]

That is what I have for the beginning of my proof, but I'm not sure how to conclude that the function is constant.
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1answer
40 views

Counting the number of possible matchups for teams

A tournament has 16 teams. How many ways are there to match up the teams in 8 pairs? Is it (16 choose 2)(14 choose 2)(12 choose 2)(10 choose 2)(8 choose 2)(6 choose 2)(4 choose 2)(2 choose 2)?
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1answer
41 views

Different flag signal questions

How many different signals can be created by lining up 9 flags in a vertical column in 3 flags are white, 2 are red, and 4 are blue? Is it 9 choose 3 * 6 choose 2 * 4 choose 4?
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1answer
720 views

Divide a set of $n$ elements into $k$ subsets having equal sum

Given $n$ ( $n$ <= 20) non-negative numbers. Is there / Can there be an algorithm with acceptable time complexity that determines whether the n numbers can be divided into $k$ ( $k$ <= 10) ...
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1answer
47 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 $n$.

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 $n$. I do not have any idea to prove this ...
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0answers
54 views

Minimize sum of squared error

I have an array of real numbers, I want to partition them into k sets. In each set, I calculate the sum of squared error. Then, I add up all the sum of squared error for all the set. I want to ...
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2answers
33 views

Prove that the number of partitions of $2010$ into $10$ parts is equal to the number of partitions of $2055$ into $10$ distinct parts.

Prove that the number of partitions of $2010$ into $10$ parts is equal to the number of partitions of $2055$ into $10$ distinct parts. How we can prove this?My idea is to construct a bijection but I ...
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104 views

Partition problem of equal size

I have an array S of size 2n, each element in the array is an integer I want to split it into two arrays of size n, and under this condition, minimize the difference of the sum of integers in the two ...
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45 views

Partition partition with constraint of equal size

I see the problem here Polynomial complexity algorithm of partition problem with sets of equal size This is the well know partition problem but with constraint that the size of both sets must be ...
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1answer
97 views

Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
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40 views

number of pairs that are member of a class of one of the given partitions

Suppose I am given a set of S partition of universal set A, define pair as (a,b) iff elements a,b are members of one of set in at least one partition. I want to know how many pairs in S. Example: $$A ...
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1answer
63 views

Find all Combinations of 1 and 2 which sums up to k.

I have two numbers $1$ and $2$. I have to print all ordered combinations which sums up to $k$. For example: $k=1$ Its only $1$. $k=2$ It's ${1,1},{2}$. $k=3$ Its ${1,1,1},{1,2},{2,1}$ What ...
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1answer
37 views

Proof of Partitions

Let $|n,k|$ denote the number of partitions of $n$ into $k$ distinct parts. Prove $$|n,k| = |n-k,k-1| + |n-k,k|$$ Workings: LHS counts the number of partitions of $n$ into $k$ distinct parts. RHS: ...
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2answers
84 views

Combinatorics] Partition Numbers

Let $R(n,k)$ denote the number of partitions of $n$ into $k$ (non-empty) parts. That is for example $R(7,2) = 3$ because it can be expressed as $1+6, 2+5$ and $3+4$. Prove that: $R(n,1) +R(n,2) + ...
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1answer
37 views

Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...