# Questions tagged [partitions]

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### Number of distinct sum of two squares under sum condition [closed]

Given integer $0\leq n$ and $x_1+x_2=n$ holds with $x_1,x_2$ non-negative what is the distribution of and maximum number of distinct $d$'s achievable in the sum of squares function $$x_1^2+x_2^2=d?$$...
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### Optimal partition search

Given an integer $n$, and 2 real sequences $\{a_1, \dots, a_n\}$ and $\{b_1, \dots, b_n\}$, with $a_i$, $b_i$ > 0, for all $i$. For a fixed $m < n$ let $\{P_1, \dots, P_m\}$ be a partition of the ...
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### Write large $n$ as $n_1+\ldots+n_k\ (n_1<\ldots<n_k)$ with $\varphi(n_1),\ldots,\varphi(n_k)\in\{x^k:\ x\in\mathbb Z\}$

Let $\varphi$ denote Euler's totient function. QUESTION. Is it true that for each positive integer $k$ large integers $n$ can be written as $n_1+\ldots+n_k$ with $n_1,\ldots,n_k$ distinct positive ...
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### Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero [closed]

Edit: I got rid of my old definitions. Everything should be clear now Since no one has answered my question on MSE, I’m hoping to get an answer here. I apologize if you dislike my writing. I am an ...
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### How many partitions are there regarding Tverberg's theorem?

"Specifically, for any set of $(d+1)(r-1)+1$ points there exists a point $x$ (not necessarily one of the given points) and a partition of the given points into $r$ subsets, such that $x$ belongs to ...
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### Meinardus theorem at use: problems with conditions

I am working on an enumerative problem related to knot theory, and I have found the following generating function $$F(z)=\prod_{n\geq 1} \frac{1}{(1-z^{2n+1})^2}.$$ I am interested on getting ...
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### Divisibility of polynomials over partitions

This is a follow up from my earlier MO question. Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its ...
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### Size of parities in counting partitions into odd parts

Let $p_{odd}(n)$ be the number of partitions of $n$ into odd parts (see here). For instance, one has the generating function $$\prod_{k\geq1}\frac1{1-q^{2k-1}}.$$ QUESTION. What is the size of this ...
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### Possible oversight in paper of Greene and Kleitman on chains in dominance order on partitions?

This question is about a possible lacuna in a paper of Greene and Kleitman which Zarathustra Brady made me aware of. The paper in question is "Longest Chains in the Lattice of Integer Partitions ...
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### Asymptotics of the Steenrod algebra / $s$-partitions?

Recall that an $s$-partition is a partition of a natural number $n$ such that each part is of the form $2^r-1$. By a fundamental theorem of Milnor, the number $p_s(n)$ of $s$-partitions of $n$ counts ...
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### Iterated Inverse structures: polynomial representation of integer partitioning of preimages in Sigma Matrices (reference request)

I am studying iterated preimage structures of functions on a finite set. The main structure of interest to me, the Sigma Matrix, is derived from a matrix listing the element-wise preimage sets at ...
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### Every possible number of partitions by restricting parts?

Write $p(n)$ for the number of integer partitions of $n$. For $S \subseteq \{1, \ldots, n\}$, let $p_S(n)$ be the number of partitions of $n$ with all parts in $S$. So $p(n) = p_{\{1,\ldots,n\}}(n)$....
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### enumerate line partitions of points in the plane

Let $S$ be a nonempty subset of $\mathbb{R}^2$ and $l$ a line in $\mathbb{R}^2$ disjoint from $S$. Then $l$ partitions $S$ into two disjoint sets $S = S_1 \cup S_2$ in the obvious way. Note that, at ...
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### Given a set of $n$ points in $[0,1)^d$, how do I partition the space into hyperrectangles such that each hyperrectangle contains exactly one point?

I'm new to this forum so I apologize if my question is ill-posed or too general. I have the following problem. Given a set of $n$ points in the unit hypercube, $[0,1)^d$, how can I partition the unit ...
Suppose I have a chocolate bar of integer length $L$, and there are $m\leq L$ people that are going to share it. We do not know ahead of time how much each person should receive, all we know is that ...