# Questions tagged [permutations]

Questions related to permutations, bijections from a finite (or sometimes infinite) set to itself.

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### Component-wise sums of permutations

Given a set $S$ containing all possible permutations of a vector $v = (1, 2, 3, ..., n-1, n)$, find the size of the set $P$, where $P$ is defined as the set of possible component-wise sums obtained by ...
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### Expected sorting time of random permutation using random comparators

In sorting networks, a comparator of positions $i < j$ is an operator which takes a permutation, checks if $p_i > p_j$, and if it is the case, swaps $p_i$ and $p_j$. Using this, we can define ...
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### All the regular $n$-gons are nested tightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?

Let $u_1,u_2,u_3,\dots$ be a permutation of the integers greater than $2$. A unit circle is in a regular $u_1$-gon, which is a regular $u_2$-gon, which is in a regular $u_3$-gon, ad infinitum. Each ...
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### Diagonally dominant matrix via rows permutation

Diagonally dominant matrices are required in many linear algebra algorithms such as the Gauss-Seidel algorithm. Some matrices can be made diagonally dominant by permuting its rows and others cannot. ...
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### Do the columns of the distribution of the DenertMaxDifference statistic on permutations eventually become constant?

The well known Denert statistic on permutations $p$ of $[n]$ can be defined as the number of Denert pairs for $p$, namely, the number of pairs $(i,j)$ with $1\le i<j\le n$ and $j \in [p_i,p_j-1]$ ...
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### Intersecting permutations

Given a subset $P\subset\mathcal{P}_n$ of the permutations of $1,\dots,n$ Question: how can a maximal subset $p\subseteq\lbrace1,\dots,n\rbrace$ be determined, whose elements appear in the same ...
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### Maximal Abelian subgroups of $S_\omega$

Let $S_\omega$ be the group of permutations (bijections) $\varphi:\omega\to\omega$, together with composition as binary operation. Zorn's Lemma implies that every commutative subgroup of $S_\omega$ is ...
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### On nilpotent singular $\mathbb F_2^{n\times n}$ matrices

Let $M$ be a $0/1$ matrix over $\mathbb F_2^{n\times n}$ with determinant $0$. The set of such singular matrices form a semigroup. The set of nilpotent matrices of size $n\times n$ form a semigroup. ...
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### Existence of binary permutations with a given property

Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ Let $$f(n)=n-2^{\ell(n)}$$ Let $a(n)$ be a permutation of the nonnegative integers such that $a(0)=0$, $a(n)=n$ if $n$ is a power of $2$ and ...