# 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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### Applications of partitions of natural numbers [closed]

This is a question for applications of partitions in science or technology. I know partitions is an interesting field in combinatorics and in modern algebra because it can be related with symmetric ...
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### Ordered partition of 3 simultaneous integers

I would love help or pointers for the following problem: Given a set of three integers $(i_1,i_2,i_3)$, denote $P_{i_1,i_2,i_3}$ the number which counts how many ordered ways there are of ...
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### Recurrence partition basic

I have some questions regarding (recurrence relation) for partitions, actually I do not know what is the exact term called (it was stated as partitions only in my guide book and upon doing some ...
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### On the inequality $\frac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
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### Does this graph partitioning algorithm achieve anything interesting?

I was musing over graph clustering and partitioning, and isolating clusters, and came up with an algorithm that I think might do some interesting things. I figured I'd run it past here to get some ...
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### Show that there exists a partition $-\infty=t_0<t_1<…<t_k=\infty$ such that $\lim_{t\rightarrow t_j^{-}} F(t)-F(t_{j-1})<\epsilon$

Consider a real-valued random variables $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with cumulative distribution function $F(t):=\mathbb{P}(X\leq t)$. I want to show that ...
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### How many different ways are there to color the faces of a cube with six different colors? [duplicate]

Think about: If the cube is held in a particular orientation, there are 6! ways to paint the six faces. However, if you rotate the cube around, some of these colorings are equivalent. How many ...
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### Partition of number's squares

The problem is to divide $\{k^2\}_{k=1}^{1000}$ into two groups of 500 numbers each, such that they have equal sum. I know that $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6},$$ but it isn't enough ...
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### Set as a union of 3 disjoint sets ,with equal sum

The problem is to find in which value of n the {1,2,3,...n} set can be parted in 3 subsets such that each one has equal sum. I have looked for a lot for this answer, but I can't solve it to the end. ...
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### Restricted Partition Combinatorial Interpretation

Using the definition of $G(N,M;q)$ from 'Theory of Partitions', Andrews, would it be fair to say the denominator provides a partition into at most $N$ parts, and the numerator removes the excess above ...
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### Partition of $\{1,2,3,…n\}$ into $3$ subsets [duplicate]

The problem is to find in which value of n the $\{1,2,3,...n\}$ set can be parted in $3$ subsets such that each one has equal sum. I have looked for a lot for this answer, but I can't find anything....
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### Use a Ferrers diagram to prove that $\sum\limits_{k=1}^m{P_k(n)} = P_m(n+m)$

I can show that $\sum\limits_{k=1}^m{P_k(n)} = P_m(n+m)$ with induction, but I can't figure out how to create an argument based on Ferrers diagrams which proves this equation. Any hints in the right ...
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### Two way partitioning problem's lower bound

In the two-way partitioning problem (as laid out in Slide 7 here), as an example, one possible value for $\nu$ is $-\lambda_{min}(W)1$ which gives the bound as $p^* \ge n\lambda_{min}(W)$. My ...
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### Writing partition function with divisor function

We know this identitiy. $$P(n)=\frac{1}{n}\sum_{k=0}^{n}\sigma _{1}(n-k)P(k)$$ where $P(n)$ is the partition function and $\sigma _{1}(n)$ is the divisor sum function. Can we pull partition function ...
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### delta fine partition on dirichlet function

In many literature, there's a prove of dirichlet functions are henctock integrable, always let P as delta fine partition and then..... My question is how to construct delta fine partition on ...
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### Interpreting the table of classification of the partitions of $n$

I am going through A NON-RECURSIVE EXPRESSION FOR THE NUMBER OF IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUP $S_n$ by AMUNATEGUI. In table I, the classification of the partitions of n according ...
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### Isn't x/E = y/ E ⇔ x E y deduced, not just x/E = y/ E ⇒ x E y from (a) x/ E≠ Ø, (b) x/E ∩ y/ E ≠ Ø ⇔xEy?

"Theorem 3. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. Then (a) Each $x/\mathscr E$ is a nonempty subset of $X$. (b) $x/\mathscr E \cap y/\mathscr E \neq \emptyset$ if and ...
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### The lattice of partial partitions

In perusing A kind of Eulerian numbers connected to Whitney numbers of Dowling lattices I have gotten confused over what seems a very elementary point. The beginning of section 2 of the paper is ...
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### How to create a linear set of proportions summing to 1?

I'm not even really sure what this concept is called. But my objective is to create, for any n, a linear distribution of numbers less than 1 summing up to and plateauing at 1. It would have to work ...
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### How many distinct, non-negative integer solutions are there for $2x_0+\sum_{i=1}^{m}{x_i}=n$?

We are given constants $m$ and $n$. How many non-negative integer solutions are there for $2x_0+\sum_{i=1}^{m}{x_i}=n$ satisfying the condition that$x_i\neq x_j$ if $i\neq j$? I thought a good first ...
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### Interesting orderings open sets of a topology

Let X be a set of points an $\mathcal{Q}$ be a partition on X. The intuition I want to model is that X is a set of worlds considered possible by an agent and $\mathcal{Q}$ is a question whose ...
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### Use generating function to check how many solutions are to get balls from boxes and roll dices

In the box are 4 red balls, 5 blue balls and 2 yellow balls. How many possible solutions there are to get 7 balls from box to have at least 1 red ball and exactly 2 blue balls? I'm not sure I am ...
### Proof that two generating function are equals for the sequence which $n$-th number is:
I am not sure I am doing this exercise good 1) $p_n$ | all parts are pairs different and 2) $p_n$| all parts are not higher than $m$ I found these functions in book, first is:  \prod_{i=1}^\...
Find all $\alpha_i, 1\le i \le n$ and $\beta_j, 1 \le j \le n$, $\alpha_i, \beta_j \in \mathbb{N}$ satisfying the following: (1) $\alpha_i \ge \alpha_{i+1}$ for $1 \le i \le n-1$ and \$\beta_j \...