Sangwook Ree

On k-hypertournament matrices

Abstract A k -hypertournament is a complete k -hypergraph with all k -edges endowed with orientations. The incidence matrix associated with a k -hypertournament is called a k -hypertournament matrix. Some properties of the hypertournament matrices are investigated. The sequences of the numbers of 1’s and −1’s of rows of a k -hypertournament matrix are respectively called the score sequence and the losing score sequence of the matrix and so of the corresponding hypertournament. A necessary and sufficient condition for a sequence to be the score (and losing score) sequence of a k -hypertournament is considered. We also find some conditions for the existence of k -hypertournament matrices with constant score sequence, called regular k -hypertournament matrices.

Lattice Paths and Faber Polynomials

The r-th Faber polynomial of the Laurent series f(t) = t + f 0 + f 1/t + f 2/t 2 + … is the unique polynomial F r (u) of degree r in u such that F r (f) = t r + negative powers of t. We apply Faber polynomials, which were originally used to study univalent functions, to lattice path enumeration.

DIVISORS OF THE PRODUCTS OF CONSECUTIVE INTEGERS

In this Paper, We look at 3 Simple function L assigning to an integer n the smallest positive integer n such that any product of n consecutive numbers is divisible by n. Investigated are the interesting properties of the function. The function L(n) is completely determined by L(pk/), where pk/ is a factor of n, and satisfies L(mn) L(m)+L(n), where the equality holds for infinitely many cases.

A weighted geometric regularity from order restricted statistical inference

In Euclideank-space, the cone of vectors x = (x 1,x 2,...,x k ) satisfyingx 1 ≤x 2 ≤ ... ≤x k and\(\sum\nolimits_{j = 1}^k {x_j } = 0\) is generated by the vectorsv j = (j −k, ...,j −k,j, ...,j) havingj −k’s in its firstj coordinates andj’s for the remainingk −j coordinates, for 1 ≤j <k. In this equal weights case, the average angle between v i and v j over all pairs (i, j) with 1 ≤i <j <k is known to be 60°. This paper generalizes the problem by considering arbitrary weights with permutations.