# 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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### Integer partitions with distinct parts

Let $~~p(n)~~$ denote the number of all partitions of positive integer $~~n~~$ with distinct parts. I would like to find some effective algorithm for calculating $~~p(n)~~$. It seems that dynamic ...
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### How to partition $nk$ objects $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size $k$, so that no combination of $k$ is repeated.

What is an algorithm to partition $nk$ objects a total of $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size exactly $k$, so that no subset of size $k$ is ever repeated? For example, ...
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### # of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers?

How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?
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### Verification of Rogers-Ramanujan identities

In Hardy's book 'Ramanujan', section 6.8 on the Rogers-Ramanujan identities, it states: None of these proofs can be called both "simple" and "straightforward", since the simplest are essentially ...
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### Show that the equivalence classes give a partition of $X$. [duplicate]

"... the following subset of $X$ denoted by $[x]$ $(x\in X)$ is called the equivalence class generated by $x$: $$[x]:= \{y \in X; x\sim y \}$$ Show that the equivalence classes give a partition of ...
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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 ...
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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 ...
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### Equivalence relation and partitions [closed]

Define an equivalence relation on the set R that partitions the real line into subsets of length 1.
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### Time-scale law of Bernoulli-stopped process

This post is quite long, but the problem stated carries no computational burden. Consider the equally spaced partition $t_{i}^n=\frac{i}{n}$ with $i=0,...,n$ of the interval $[0,1]$ into $n$ ...
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### Set partition, partition into X set with Y elements.

How do you partition a set into X number of new sets all that have Y elements? Example: How to partition 18 unique cards are divided to six persons, and each person gets 3 cards each. How many ...
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### Does {$\Bbb Z_0$,$\Bbb Z_1$, $\Bbb Z_2 ,\cdots$, $\Bbb Z_{m-1}$} form a partition of $\Bbb Z$?

"Definition 5. Let X be a nonempty set. By a partition P of X we mean a set of nonempty subsets of X such that (a) If $A, B \in \mathscr P$ and $A \neq B$, then $A \cap B = \emptyset$, (b) ...
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### Set within a partition

Say I have a partition of the set $\{1,2,3,4,5\}$. The partition is $\{\{1,3\},\{2\},\{4\},\{5\}\}$ Is there a word for set within a partition e.g. I want to say, 'one of the sets of the partition ...
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### Number of partitions of $n$ formed by combinations of $2$ and $4$

I'm trying to find the number of partitions of a natural number that are a combination of $2$ and $4$. For example: $$6 = 2+2+2 = 2+4 \Rightarrow p_6 = 2$$ So I start by defining $p_n$ as the ...
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### Use of the “Optimality Principle” in a problem regarding maximization of a product indexed by a partition of a natural number.

I am currently reading a paper (Iterated Binomial Coefficients by S.W. Golomb, The American Mathematical Monthly, 1980, 719-727) that makes use of the "optimality principle" in a couple of proofs. One ...
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### There are n objects and n boxes, how many ways can we place the objects so exactly one box remains empty

A) if both objects and boxes and indistinguishable B) if objects are indistinguishable and boxes are distinguishable My attempt: I know there are n! ways to but n objects into n boxes (both ...
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### Proving Riemann Integrability when $x$ is a function of $n$?

(Context: I have been using this resource from UC Davis to help me out with Riemann integrability.) So, I more or less get how Riemann integrability works when we specify $x$ over an interval, say ...
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### The number of good partitions

Consider a set $S$ of $n$ red balls and $m$ blue balls. It is well known that the number of partitions of this set is the Bell number $B_{n+m}$. We say that a partition $P \subset \mathcal{P}(S)$ of ...
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### name or characterization for the “partition” lattice of integer partitions of some n? (Young lattice row partial order)

Is there a name or characterization for the "partition" lattice of integer partitions of some n? Young's Lattice depicts the integer partitions of numbers. Often Young diagrams are used in displaying ...
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### Prove that $\inf{U(f,P)}\geq 1/2$

Define $f(x)=\begin{cases} x\;\;\;\text{if the point$x\in[0,1]$is rational}\\ 0\;\;\;\text{if the point$x\in[0,1]$is irrational} \end{cases}$ Prove that $\inf{U(f,P)}\geq 1/2$. Let ...
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### Partitioning a Queue

Imaging a situation that we have n people in a queue and each people represent with number 1 and I want to partition the queue in smaller part, there are several ways to partition the queue. For ...
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### Generating functions and integer partitioning [duplicate]

Show that the number of partitions of a positive integer n where no summand appears more than twice is equal to the number of partitions of n where no summand is divisible by 3 So I begin by ...