For a simple drawing D of the complete graph $$K_n$$ , two (plane) subdrawings are compatible if their union is plane. Let $$\mathcal {T}_D$$ be set all plane spanning trees on and {F}(\mathcal {T}_D)$$ compatibility that has vertex for each element in vertices adjacent only corresponding compatible. We show, one hand, connected cylindrical, monotone, or strongly c-monotone drawing. On other we...

A k_n Latin rectangle is a k_n matrix of entries from [1, 2, ..., n] such that no symbol occurs twice in any row or column. An intercalate is a 2_2 Latin subrectangle. Let N(R) be the number of intercalates in R, a randomly chosen k_n Latin rectangle. We obtain a number of results about the distribution of N(R) including its asymptotic expectation and a bound on the probability that N(R)=0. For...

For a complete graph $$K_n$$ of order n, an edge-labeling $$c:E(K_n)\rightarrow \{ -1,1\}$$ satisfying $$c(E(K_n))=0$$ , and spanning forest F we consider the problem to minimize $$|c(E(F'))|$$ over all isomorphic copies $$F'$$ in . In particular, ask under which additional conditions there is zero-sum copy, that is, copy with $$c(E(F'))=0$$ We show always $$|c(E(F'))|\le \Delta (F)+1$$ where $...

let $g$ be a group and $pi(g)$ be the set of primes $p$ such that $g$ contains an element of order $p$. let $nse(g)$ be the set of the number of elements of the same order in $g$. in this paper, we prove that the simple group $l_2(p^2)$ is uniquely determined by $nse(l_2(p^2))$, where $pin{11,13}$‎.‎

Abstract In this paper we study the asymptotic behaviour of a random uniform parking function $\pi_n$ size n . We show that first $k_n$ places $\pi_n(1),\ldots,\pi_n(k_n)$ are asymptotically independent and identically distributed (i.i.d.) on $\{1,2,\ldots,n\}$ , for total variation distance when $k_n = {\rm{o}}(\sqrt{n})$ Kolmogorov $k_n={\rm{o}}(n)$ improving results Diaconis Hicks. Moreover,...

Let $X_{1,n}\le\cdots\le X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ having right heavy tail index $\gamma$. Given known constants $d_{i,n}$, $1\le i\le n$, consider weighted power sums $\sum^{k_n}_{i=1}d_{n+1-i,n}\log^pX_{n+1-i,n}$, where $p>0$ and $k_n$ are positive integers such that $k_n\to\infty$ $k_n/n\to0$ as $n\to\infty$. ...