# Questions tagged [partitions]

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### Upper bound on decomposition numbers for the symmetric group in a block of weight $w$

The $p$-blocks of the symmetric group $S_n$ are labelled by pairs $(\gamma, w)$ where $\gamma$ is a $p$-core partition, $w \in \mathbb{N}_0$ is the weight of the block, and $|\gamma| + p w = n$. ...
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### Problem concerning cutting of 2n*2n square into 2 equal area connected figures using various cuts without self crossings

We have a square 2n*2n, where n belongs to N. The main problem is to find how many different equal area connected figures could be produced by cuttings without self-crossings. The orientability of the ...
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### Alternating sum of hook lengths: Part I

Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$. Is there a closed formula or a generating function for the ...
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### What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$?

I'm now interested in the modular representation of symmetric groups. It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S_{n}$ over a field ...
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### Hurwitz numbers and $t$-cores

For integers $k \geq 0$ and $d \geq 1$ let $H(k,d)$ be the Hurwitz number which, for the purposes of this posting, will be defined by: H(k,d) \, := \ d! \, \sum_{\lambda \, \vdash 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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### 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 ...
For every set of natural numbers $A$ and for all positive integers $n$, $k$, let $c_k^A(n)$ be the number of compositions of $n$ into $k$ parts from $A$, that is, the number of $(a_1, \dots, a_k) \in ... 0answers 116 views ### Coefficents of these partition-based polyomials are$0, \pm1$This is a follow up on 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$\...
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 ...