Tagged Questions

For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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Is it true in the Group of Symmetries that (ab)(ac) = (acb)?

Suppose $1 \le a,b,c \le n$ and $a\ne b$, $a\ne c$, $b\ne c$. Is it true in $S_n$ that $(ab)(ac)=(acb)$? I'm generally new to the subject of Applied/Abstract Algebra and feel as if this is easier ...
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Can you create a formula for the amount of possible permutations of a three digit number, who has a digit sum equal to 4

Can you create a formula for the amount of possible permutations of a three digit number, who has a digit sum equal to n?
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number of permutation in a boolean expression containing only ANDs and ORs

I need to find the number of permutations of some expression which contains only conjunctions and disjunctions e.g.: $$e = x_1x_2 \vee x_3x_4$$ where $x_1x_2$ and $x_3x_4$ are boolean summands, ...
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Number of subsets from an ordered set where adjacent elements may or may not be tied together

Assume we have an ordered set $S$ with a finite number of elements $S=\{1,2,3,\ldots,N\}$. I need to know the number of subsets where adjacent elements from the original set may either be tied ...
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Some properties of finite group of order $p^aq^b$

Let $G$ be a finite group of order $p^aq^b$ ( $p$, $q$ are two distinct primes and $a, b\geq 1$) with $\operatorname{Z}(G)=1$ and $P\in \operatorname{Syl}_p(G)$, $Q\in \operatorname{Syl}_q(G)$. Also ...
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Natural numbers in a circle, combinatorics, existence

I need help with a problem whose solution I'm unaware of. The first $74$ natural numbers are arrange in some manner in a circle. Does there exist an arrangement such that every sum of three ...
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Permutation question (Known answer, don't understand how to get to it)

To save your time, I simplify the question to something like this: There are $18$ people, $4$ of which are teachers. All of them($18$) are going to stand in a row. In how many ways can they be ...
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In how many ways can a positive integer $n$ be expressed as a summation of positive integers less than $n$

For example if I take $n=5$, then I can express it in the following ways: $1+1+1+1+1$ $2+3$ $3+2$ $1+4$ $4+1$ $1+1+3$ $1+3+1$ $3+1+1$ $2+2+1$ $2+1+2$ $1+2+2$ Please note that the order of terms in ...
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Find the string corresponding to a particular lexicographical rank

I came across this question few days back- Given a string - “ thereanswerisyetinsufficientmeaningfulasforadata “ , form all the words with atmost 15 letters and arrange them in lexicographical ...
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Series that converge to every real number via permutation

This great answer at MathOverflow, http://mathoverflow.net/a/29488/8784, shows that the set of permutations of $\mathbb N$ is uncountable. However, I did not grasp the fact that he uses: any ...
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Ordering of basis elements of a Lie-group representations tensor product

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$\textbf{N}\otimes\bar{\textbf{N}} = \bigoplus_J \textbf{r}_J$$ where $\textbf{r}_J$ ...
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How many partitions of $n$ are there?

Considering a partition to be an ordered $n$-tuple $(m_1, m_2, m_3, ..., m_n)$ with all the numbers $m_i$ natural, $m_1 \le m_2 \le m_3 \le ... \le m_n$, and $m_1+m_2+...+m_n=n$: how many of those ...
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Combinatorial proof of an identity between restricted counts of permutations and derangements

In an answer to Counting permutations with given condition, I showed that the number of permutations of $k$ elements that satisfy $\sigma(i+1)\ne\sigma(i)+1$ is $\frac{!(k+1)}k$, which is the number ...
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Circle Permutation w/ Restrictions questions

I was working on two permutation questions, and wasn't sure if I arrived at the correct answer. 1.) In how many ways can a family of four (mother, father, and two children) be seated at a round ...