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The dihedral groups can be viewed as the set of all functions of the form $x\mapsto\pm x+c$ acting either on $\mathbb{Z}$ or on $\mathbb{Z}/n\mathbb{Z}$. The images of the infinite dihedral group are …

The number $N(i,j)$ of paths from $i$ to $j$ is given by the matrix $B=E+A+A^2+\dots$. The number of paths from $i$ to $j$ passing through $k$ is $N(i,k)N(k,j)$, which is the number of times you have …

In a quite large range of the parameters we can approximate the fraction by a Taylor series to obtain $$ f(x) = \frac{1}{x!}\frac{1}{\frac{-\log c}{\binom{x+n-1}{n-1}} + \mathcal{O}\left(\frac{\log^2 …

The number of necklaces of size $p$ is $\frac{a^p}{p}+\mathcal{O}(a^{p/2})$, hence $$ \prod_{p=1}^nN(p,a)=\frac{a^{n(n+1)/2}}{n!}\prod_{p=1}^n\left(1+\mathcal{O}(a^{-p/2})\right) = \left(c+\mathcal{O} …

I assume that $k$ is fixed, while $n$ tends to $\infty$. I claim that for $p=2$ the sum in question is asymptotically equal to $k\zeta(2)^{k-1}n^{-2}$. First consider those partitions, which contain p …

Put $a=|v_1|$, $b=|v_2|$, $c=|v_1v_2|$. Then we have $$ \sum_{i=0}^a\sum_{j=0}^b\sum_{k=0}^c\binom{a-c}{i-k}\binom{b-c}{j-k}= \sum_{k=0}^c\left(\sum_{i=0}^a\binom{a-c}{i-k}\right)\left(\sum_{j=0}^b\bi …

A similar problem, called the Coupon Collector's younger brothers, has been studied by Foata, Han and Lass (Séminaire Lotharingien de Combinatoire, B47a, 20 pages, 2001, obtainable via http://math.uni …

I assume that $m$ is squarefree, for otherwise the minimum would be equal to 2 no matter what $k$ is. Let $p_1, \ldots, p_k$ be the set of all prime numbers of the form $2^a3^b+1$, where $a, b<t$. Th …

The average score difference does not suffice to predict the probability of the outcome. Suppose all players in team A are of equal strength, while all but one player in team B are somewhat stronger t …

Let $p\neq 3$ be a prime. Then \begin{eqnarray*} \nu_p\left(\frac{(3x)!}{x!^3}\right) & = & \sum_k \left[\frac{3x}{p^k}\right]-3\left[\frac{x}{p^k}\right]\\ & = & \sum_k 3\left\{\frac{x}{p^k}\right\} …

In their article "Stolarsky's conjecture and the sum of digits of polynomial values"( https://www.math.uwaterloo.ca/~kghare/Preprints/PDF/P34_Stolarsky.pdf ), Hare, Laishram and Stoll show in Proposit …

Converging sieves: Let $q_i$ be a sequence of integers with $\sum\frac{1}{q_i}<\infty$, and pick for each $i$ an integer $a_i$ within a finite set $\mathcal{A}$. Then the set of integers $n$ such that …

In higher dimension things become more complicated. For a finite abelian group $G$ define $\mathfrak{s}(G)$ to be the least integer $N$, such that every sequence $x_1, \ldots, x_N$ of elements of $G$ …

The probability that a fixed entry of $v$ is 1 equals $2^{-n^{s+t}}$. Hence the expected Hamming weight of $v$ is $2^{n-n^{s+t}}$. If $s+t\geq 1$, this implies that with high probability the Hamming w …

Let $f$ and $g$ be random mappings. If they commute, then $f(g(1))=g(f(1))$, and this happens with probability $n^{-1}$. Now $f(g(2))=g(f(2))$ also holds with probability $n^{-1}$, but these events ne …