# Tagged Questions

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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### Multiplication partitioning into k distinct elements

Let's say I have a list with the prime factors of a number $n$ and their corresponding exponents. Is there a formula to calculate how many multiplications with $k$ distinct factors of $n$ are ...
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### Partition and equivalence relation

Consider the equivalence relation between non-empty subsets $A , B$ of $\{ 1,2,3, 4,\dots,100\}$ defined by the condition: the greatest element of $A$ is the same as the greatest element of $B .$ ...
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+100

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### Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
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### Determine the number of partition of 20 into at most 5 parts.

Im stuck on these questions, I can kind of compute them on maxima, but I have to figure out why and how to get to a particular generating function or method. Determine the number of partition of 20 ...
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### Notation for the set of all integer partitions

I'm working on a project that involves that set $P = \{\{n_1, \ldots, n_k\} \mid k \in \mathbb{N}, n_i \in \mathbb{N} \text{ and } n_1 + \cdots +n_k = n\}$ of all integer partitions of a number $n$. ...
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### How many solutions of equation

How many solutions of equation $x_1+x_2+x_3+x_4=n$ in $N_0$ such that $x_1\leq x_2\leq x_3 \leq x_4$? I found solutions of $x_1+x_2+x_3=n$ in $N_0$ , $x_1\leq x_2\leq x_3$ in the following way : ...
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### Introduction to proofs: proving a set is a partition.

I've been really trying to understand how some of these proofs work; I've spent a majority of my time studying the material for this class, but I'm still performing poorly in it. It doesn't help that ...
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### Number of ways to build a collection of numbers where $sum = k$, each $0 < n_i <= d_i$ for some corresponding $d_i$, and sum of all $d_i >= k$

I apologize for any (mis|ab)use of notation since I'm not a mathematician. My background is software engineering and computer science. I ran into this problem while trying to figure out the ...
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### The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
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### Partition problem for consecutive $k$th powers with equal sums (another family)

This is the partition problem as applied to a special set, namely the first $n$ $k$th powers. Assume the notation, $$[a_1,a_2,\dots,a_n]^k = a_1^k+a_2^k+\dots+a_n^k$$ I. Family 1 The following ...
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### On t-core partitions

How exactly can one define what is known as a t-core partition? I know (vaguely) that it involves the definition of what is known as "Hook numbers". Anyone cares to provide a link or explain it?
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### Can I partition a non-separable set?

This question flashed in my mind, as I was studying quotient partition of a set. Is there any non-separability condition for which a topological set cannot be partitioned? Particularly, I tried ...
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### Prove that A001399 == A069905

While solving PE and reading SICP I found, that there are two problems, that produce the same OEIS sequence: http://oeis.org/A001399 is a number of partitions of n into at most 3 parts ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...