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Consider a permutation p to be any finite list of distinct positive integers. A statistic is a function St whose domain is all permutations. Let S(p,q) be the set of shuffles of two disjoint permutations p and q. We say that St is shuffle…

This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost…

We study quadratic residue difference sets, GMW difference sets, and difference sets arising from monomial hyperovals, all of which are $(2^d-1, 2^{d-1}-1, 2^{d-2}-1)$ cyclic difference sets in the multiplicative group of the finite field…

A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the…

Our aim in this paper is to deal with the dynamics of following higher order difference equation x_{n+1}=A+B((x_{n-m})/(x_{n}^2)) where A,B>0, and initial values are positive, and m={1,2,...}. Furthermore, we discuss the periodicity,…

We propose an $n$-dimensional analogue of elliptic difference Painlev\'e equation. Some Weyl group acts on a family of rational varieties obtained by successive blow-ups at $m$ points in $\mpp^n(\mc)$, and in many cases they include the…

Let a_1,...,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 solutions of the linear inequality |a_i-a_j| < e_ka_k < a_i+a_j, with e_k=-1 of 1 and k running over the integers 1,...,m with i and j skipped.…

Recently ,mathematicians have been interested in studying the theory of discrete dynamical system, specifically difference equation, such that considerable works about discussing the behavior properties of its solutions (boundedness and…

Several cycle lexicographical orders are found to describe the relative likelihood of elements of the random walks on the symmetric group generated by the conjugacy classes of transpositions, 3-cycles, and n-cycles. Spectral analysis finds…

We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice, and use this cyclic parameterization to count planar well-rounded…

A grid class consists of permutations whose pictorial depiction can be partitioned into increasing and decreasing parts as determined by a given matrix. In this paper, we introduce a method for enumerating cyclic permutations in vector grid…

In our paper we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq \mathbb{Z}/q\mathbb{Z}$ in the case of composite $q$. We obtain a quantitative version of a result of A. Fish about the structure of the…

Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ distinct elements such that the expressions…

More than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive integers $n,k$ with $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose sum is…

Let $(P,\leq)$ be a finite poset (partially ordered set), where $P$ has cardinality $n$. Consider linear extensions of $P$ as permutations $x_1x_2\cdots x_n$ in one-line notation. For distinct elements $x,y\in P$, we define…

A cyclic urn is an urn model for balls of types $0,\ldots,m-1$ where in each draw the ball drawn, say of type $j$, is returned to the urn together with a new ball of type $j+1 \mod m$. The case $m=2$ is the well-known Friedman urn. The…

Classification of differential-difference equation of the form $\ddot{u}_{nm}=F_{nm}\big(t, \{u_{pq}\}|_{(p,q)\in \Gamma}\big)$ are considered according to their Lie point symmetry groups. The set $\Gamma$ represents the point $(n,m)$ and…

For all positive integers $s$ and $t$ exceeding one, a matroid $M$ on $n$ elements is {\em nearly $(s, t)$-cyclic} if there is a cyclic ordering $\sigma$ of its ground set such that every $s-1$ consecutive elements of $\sigma$ are contained…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some…