# 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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### Partitions of $n$: proving $p(n+2)+ p(n) \geq 2p(n+1)$

For $n \geq 2$ give an alternative description of $p(n) - p(n-1)$ as the number of partitions of $n$ which have a certain property. I have done that part, it is fine. I have not included it here as ...
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### Derivative of Schur function

In his answer to http://mathoverflow.net/questions/129854, R. Stanley says that the partial derivative (over the relevant x[i]) of the Schur function of a partition lambda of n equals the sum the ...
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Eg: ...
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### Related to the partition of $n$ where $p(n,2)$ denotes the number of partitions $\geq 2$

I was trying the following question from the Number theory book of Zuckerman. If $p(n,2)$ denotes the number of partition of n with parts $\geq2$, prove that $p(n,2)>p(n-1,2)$ for all $n\geq 8$, ...
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### Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

I would like to find articles or documentation regarding this process: Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook ...
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### Combinatorics: Generating Function related to compositions of a number

My goal is to find the coefficients of the generating function for the following situation: The number $f(n)$ is the sum over all compositions of $n$ into $3$ parts of the product of those parts. Fo ...
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### Conjugate Ferrers diagrams

Let $\pi=\langle \pi_1,\pi_2,... \rangle , \ \pi_1\ge\pi_2\ge...,$ be a partition of a number and $\pi'=\langle \pi_1',\pi_2',... \rangle$ be a partition conjugated to $\pi$, which means that Ferrers ...
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### How can I find the partitions of this equivalence relation?

I have the following equivalence relation: $$\{(1,1),(1,4), (2,2), (3,3), (4,1), (4,4)\}$$ On the set: $A = \{1,2,3,4\}$ How can I find it's partitions? This example will help me understand the ...
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### Maximal Zero Sums Partition

You are given $n$ numbers between $-n$ and $n$, the sum of numbers is $0$. Divide the given sequence on disjoint subsequences in such a way that each subsequence has zero sum. Each element should ...
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### Balls, Bags, Partitions, and Permutations

We have $n$ distinct colored balls and $m$ similar bags( with the condition $n \geq m$ ). In how many ways can we place these $n$ balls into given $m$ bags? My Attempt: For the moment, if we assume ...
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### Determining Stirling number

In the first part of the question I was asked to find the exponential generating function for $s_{n,r}$, the number of ways to distribute $r$ distinct objects into $n$ (a fixed constant) distinct ...
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### Generating function: number of partitions that add up to at most $n$

Find a generating function $a_n$, the number of partitions that add up to at most $n$. So I know that if it were asking the number of partitions of the integer $n$, I would have my generating ...
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### Combinatorial proof involving partitions and generating functions

Show that any number of partitions of $2r + k$ into $r + k$ parts is the same for any $k$. I've tried this, but I haven't come up with anything; hence why I have nothing written here. But in any ...
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### Generating function for $r^\binom{n}{2}$

I'm trying to find a closed form of the generating function $$G(x) = \sum_{n \ge 0} r^\binom{n}{2} x^n$$ for a real number $0 < r < 1$. I found that $G(x) = 1 + xG(rx)$. Any hints where to ...
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### Terminology for breaking partition diagram into “L”'s

When one thinks about partitions, it's quite normal to consider pieces of the partition diagram, such as rows, columns, arms, legs, hooks, etc. One decomposition of particular interest to me is ...
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### Relationship Question

Let $\ S\$ be a non-empty Set, and suppose s$\ \in S$. Assuming $\ S\$ is finite, what can we deduce about the relationship between $\ |\mathcal P(S\ \setminus \{s\} )|$ and $\ | \mathcal P(S)|?$ ...
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### Further clarification needed on proof invovling generating functions and partitions (or alternative proof)

Show with generating functions that every positive integer can be written as a unique sum of distinct powers of $2$. There are 2 parts to the proof that I don't understand. I will point them out as ...
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### Question about partition of open sets in $\mathbb{R^n}$

I have to prove that any open set $U \subset \mathbb{R^n}$ is a countable union of disjoint limited rectangles. I proved that it is a countable union of rectangles, the "expected classical" way, I ...
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### How many solutions does the equation $x_1 + x_2 + x_3 = 11$ have, where $x_1, x_2, x_3$ are nonnegative integers?

Help me understand problems of this type a bit more intuitively. The solution $C(3+11−1,11)$ seems simple enough, but I got stuck thinking about how many integers you are choosing from within $x_1$, ...
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### Finding a generating-function using partitions

Find a generating function for a , the number of partitions of r into (a.) Even integers (b.) Distinct odd integers. I am at a loss of starting this.
### Partitions of $n$ into $k$ blocks, without single blocks.
So I'm trying to come up with a recursive formula $f(n)$ which counts the number of all partitions of $[n]$ into $k$ identical blocks, where the number of elements in each box is more than 1. What i'...