# 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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### Partition identity with generating functions

I'd like to show that: The number of partitions of $n$ such that parts appear 2,3 or 5 times is equal to the number of partitions of $n$ into parts congruent to $\pm 2$, $\pm 3$, $6$ mod $12$. The ...
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### The number of partitions of $n$…Subbarao

(Subbarao) The number of partitions of $n$ in which each part appears two, three, or five times equals the number of partitions of $n$ into parts congruent to $2, 3, 6, 9,$ or $10$ modulo $12$. ...
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### Pattern in numbers

I bumped into this mathematical calculation while driving my car. With lot of traffic jams and lot of time to kill, I did some scrambling of the registration numbers of the cars in front of me. I ...
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### Upper and Lower Bound on Partition Function

The partition function $p(n)$ counts the number of ways an integer can be expressed as a sum. For example, $p(4)=5$ as $$4=3+1=2+2=2+1+1=1+1+1+1$$ Hardy and Ramanujan were able to develop a converging ...
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### Number of ways to partition a set with $2n$ elements

In how many ways can I partition $S = \{1,2,\cdots,2n\}$ into $n$ disjoint $2$ element subsets. Suppose if I subsets of $S$ were $S_{1},S_{2},\cdots,S_{n}$, then I can choose $S_{1}$ in $\binom{2n}{2}$...
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### Relation of relative numbers of (restricted) ways to distribute identical / distinct objects into distinct bins

If want to know if the following inequality holds for general values of $s \leq n \ll m$. $$\frac{C_0(n,m,s)}{C_0(n,m)} \leq \frac{p(n,m,s)}{m^n}$$ $C_0(n,m) = \binom{n+m-1}{m-1}$ is the number of ...
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### finite partitions of the square that separate all equipotent sets of points

This question asked whether there exists a finite partition of $[0, 1]^2$ and a finite set of points in $[0, 1]^2$ that can't be affinely transformed to fall into one part of the partition. I would ...
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### Is this right? Showing that if $S\subset \mathbb{R}^n$ arbitrary and $f$ a $C^r$ function then there is $A$ open such that $f$ is $C^r(A)$.

Let $f : S \to \mathbb{R}$ a $C^r$ function, where $S$ is any subset of $\mathbb{R}^n$. We say that $f$ is differentiable on $x_o \in S$ if there is $U_{x_0}$ and $g : U_{x_0} \to \mathbb{R}$ such ...
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### Differentiating between equivalence relations and partitions

What is the purpose of defining equivalence relations and partitions separately when they are the same? Are they used for different purposes?
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### An interesting problem on partitions and odd/even numbers

A number k is expressed as a sum of n non-negative numbers. Let us call the sequence of numbers that are a part of the sum, S. Each element of S is replaced by the remainder it leaves when divided by ...
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### discrete convexity arising in a simple discrete optimization problem

Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
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### Number of Blocks of Size $i$ in Set Partitions

Given a set of $N$ labeled elements $\{1, 2, ..., n\}$, we know that there are $S(N, k)$ ways to partition the elements into $k$ non-empty subsets (where $S(N, k)$ is the Stirling number of the second ...
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### Index set of dyadic partition

Imagine if you have a line from 0 to 1, and you begin partitioning it dyadically. The first point will be at 0, the second at 1 and the third at 0.5, the fourth at 0.25, the fifth at 0.75 etc. Let's ...
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### Determine if given relation is an equivalence relation, and describe the partition?

Can someone walk me through how to do these types of problems in my Discreet Mathematics II Textbook? (ex.): Determine whether the given relation is an equivalence relation on the set. Describe the ...
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### Number of different groups given a list of repeating digits

Suppose that you are given the list[1,1,2,2] . The different groups that can be formed with this list are - ...
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### Equivalence relation on $\mathbb{N}$ with infinitely many equivalence classes

Does there exist an equivalence relation, defined on $\mathbb{N}$, with infinitely many equivalence classes, all of which contain infinitely many elements? I see no reason for such a relation not to ...
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### How to call a partition of $X$ which consists of all singleton subsets of $X$? [duplicate]

In other words, if $X$ is a set, then how do we call $Y=\{\{x\}:x\in X\}$? $\{X\}$ is already named the trivial partition, so that cannot be it. Complete partition and total partition did not yield ...
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### Decomposition of $\mathbb N$ into mutually disjoint infinite subsets

$$\mathbb N =\bigcup_{j\in \mathbb N}\Delta_j$$ where each $\Delta_j$ is an infinite subset of $\mathbb N$ and $\Delta_j\cap \Delta_i=\Phi \ for\ i\neq j.$ Now what I need is a few examples of such ...
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### Formula for Number of possible N element sequences such that their Sum is S?

How many ways I can choose a $N$ element sequence such their cumulative is $S$? Is there any formula for it? Values of $N$ will be greater Than $0$. Here are few examples Let $N=4$ and $S=5$. Their ...
Could someone explain what is the reasoning behind the following equality? Or maybe direct me to a proof of the following equality? $$\sum_{n=0}^{\infty}p(n)x^n = \prod_{k=1}^{\infty}(1-x^k)^{-1}$$ ...