An operator formula for the number of halved monotone triangles with prescribed bottom row

Monotone triangles are certain triangular arrays of integers, which correspond to n×n alternating sign matrices when prescribing (1,2,…,n) as bottom row of the monotone triangle. In this article we define halved monotone triangles, a specialization of which correspond to vertically symmetric alternating sign matrices. We derive an operator formula for the number of halved monotone triangles with prescribed bottom row which is analogous to our operator formula for the number of ordinary monotone triangles [I. Fischer, The number of monotone triangles with prescribed bottom row, Adv. in Appl. Math. 37 (2) (2006) 249-267].     

On t-Covering Arrays

This paper concerns construction methods for t-covering arrays. Firstly, a construction method using perfect hash families is discussed by combining with recursion techniques and error-correcting codes. In particular, by using algebraic-geometric codes for this method we obtain infinite families of t-covering arrays which are proved to be better than currently known probabilistic bounds for covering arrays. Secondly, inspired from a result of Roux [16] and also from a recent result of Chateauneuf and Kreher [6] for 3-covering arrays, we present several explicit constructions for t-covering arrays, which can be viewed as generalizations of their results for t-covering arrays.     

Limiting behaviour for arrays of row-wise END random variables under conditions of h-integrability

The authors study L^r-convergence, complete convergence and complete moment convergence for arrays of row-wise extended negatively dependent random variables under some appropriate conditions of h-integrability. The results in this paper extend and improve the results of Sung et al. [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008), pp. 289–300].     

Constructing strength three covering arrays with augmented annealing

A covering arrayCA(N;t,k,v) is an N×k array such that every N×t sub-array contains all t-tuples from v symbols at least once, where t is the strength of the array. One application of these objects is to generate software test suites to cover all t-sets of component interactions. Methods for construction of covering arrays for software testing have focused on two main areas. The first is finding new algebraic and combinatorial constructions that produce smaller covering arrays. The second is refining computational search algorithms to find smaller covering arrays more quickly. In this paper, we examine some new cut-and-paste techniques for strength three covering arrays that combine recursive combinatorial constructions with computational search; when simulated annealing is the base method, this is augmented annealing. This method leverages the computational efficiency and optimality of size obtained through combinatorial constructions while benefiting from the generality of a heuristic search. We present a few examples of specific constructions and provide new bounds for some strength three covering arrays.     

Generalized Riordan arrays

In this paper, we generalize the concept of Riordan array. A generalized Riordan array with respect to c_n is an infinite, lower triangular array determined by the pair (g(t),f(t)) and has the generic element d_n,k=[tⁿ/c_n]g(t)(f(t))^k/c_k, where c_n is a fixed sequence of non-zero constants with c₀=1.We demonstrate that the generalized Riordan arrays have similar properties to those of the classical Riordan arrays. Based on the definition, the iteration matrices related to the Bell polynomials are special cases of the generalized Riordan arrays and the set of iteration matrices is a subgroup of the Riordan group. We also study the relationships between the generalized Riordan arrays and the Sheffer sequences and show that the Riordan group and the group of Sheffer sequences are isomorphic. From the Sheffer sequences, many special Riordan arrays are obtained. Additionally, we investigate the recurrence relations satisfied by the elements of the Riordan arrays. Based on one of the recurrences, some matrix factorizations satisfied by the Riordan arrays are presented. Finally, we give two applications of the Riordan arrays, including the inverse relations problem and the connection constants problem.     

Properties of (θ,s)-continuous functions

Joseph and Kwack [Proc. Amer. Math. Soc. 80 (1980) 341–348] introduced the notion of (θ,s)-continuous functions in order to investigate S-closed spaces due to Thompson [Proc. Amer. Math. Soc. 60 (1976) 335–338]. In this paper, further properties of (θ,s)-continuous functions are obtained and relationships between (θ,s)-continuity, contra-continuity and regular set-connectedness defined by Dontchev et al. [Internat. J. Math. Math. Sci. 19 (1996) 303–310 and elsewhere] are investigated.     

Non-trivial t-designs without repeated blocks exist for all t

We study several classes of arrays generalizing designs and orthogonal arrays. We use these to construct non-trivial t-designs without repeated blocks for all t.     

Sequential binary arrays. I. The square grid

A periodic binary array is said to be sequential if and only if every line of the array is occupied by a given periodic binary sequence or by some cyclic shift or reversal of this sequence. Such arrays are of interest in connection with experimental layouts. This paper, which deals only with arrays built on the square grid, gives some constructions for sequential arrays and some results on the types of arrays possible for certain sequences. The x-step circulant arays are characterized. A technique for exhaustive search to find all possible arrays with a given sequence is outlined and a listing is given for square sequential arrays of period not exceeding seven.     

Sparse anti-magic squares and vertex-magic labelings of bipartite graphs

A sparse anti-magic square is an n×n array whose non-zero entries are the consecutive integers 1,…,m for some m?n² and whose row-sums and column-sums form a set of consecutive integers. We derive some basic properties of these arrays and provide constructions for several infinite families of them. Our main interest in these arrays is their application to constructing vertex-magic labelings for bipartite graphs.     

Limiting behavior for arrays of rowwise ρ *-mixing random variables

We study the limiting behavior of maximal partial sums for arrays of rowwise ?? ^*-mixing random variables and obtain some new results that improve the corresponding theorem of Zhu [M.H. Zhu, Strong laws of large numbers for arrays of rowwise ?? ^*-mixing random variables, Discrete Dyn. Nat. Soc., 2007, Article ID 74296, 6 pp., 2007].