Let B be a partially ordered product of three finite chains. For any group G of automorphisms of B, let N,(B, q) denote the rank generating function for G-invariant order ideals of B. If we regard B as a rectangular prism, N,(B, q) can be viewed as a generating function for plane partitions that fit inside B. Similarly, define Nh(B, 4) to be the rank generating function for order ideals of the ...

We introduce the notion of a sturdy frame of abstract algebras which is a common generalization of a sturdy semilattice of semigroups, the sum of lattice ordered systems, the strong distributive lattice of semirings, the sturdy frame of type (2, 2) algebras and the strong b-lattice of semirings. Also, we give some properties and characterizations of the sturdy frame of abstract algebras. As an ...

Two Latin squares of order n are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the 3rst one, we say that the 3rst square is r-self-orthogonal, denoted by r-SOLS(n). It has been proved that the necessary condition for the existence of an r-SOLS(n) is n6 r6 n and r ∈ {n + 1; n − 1}. Zhu and Zhang conjectured that there is a...

Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider for a positive integer s the set Fs(A) ⊂ Zm of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R>0 : Ax = b} contains at least s integer points. We present lower and upper bounds on the so called diagonal s-Frobenius number associated to the set Fs(A). In the case m = 1 we prove an optim...

This paper generalizes the results in Aswal et al. (2003) on dictatorial domains. This is done in two ways. In the first, the notion of connections between pairs of alternatives in Aswal et al. (2003) is weakened to weak connectedness. This notion requires the specification of four preference orderings for every alternative pair. Domains that are linked in the sense of Aswal et al. (2003) with ...

We consider the class Pn of labeled posets on n elements which avoid certain three-element induced subposets. We show that the number of posets in Pn is (n+1) by exploiting a bijection between Pn and the set of regions of the arrangement of hyperplanes in R of the form xi&xj=0 or 1 for 1 i< j n. It also follows that the number of posets in Pn with i pairs (a, b) such that a<b is equal to the nu...

Let A and B be two finite subsets of a field F. In this paper we provide a nontrivial lower bound for |{a + b: a ∈ A, b ∈ B, and P (a, b) 6= 0}| where P (x, y) ∈ F[x, y].

1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ?) where G is a nonempty set and ? is a binary operation on G satisfying: (G1) a ? (b ? c) = (a ? b) ? c, ∀a, b, c ∈ G. A semigroup G is a monoid if it also satisfies: (G2) G has an element e (sometimes denoted by 1G, called the identity of G), such that e ? a = a ? e = a, ∀a ∈ G. A monoid is a group if (G3) below is satisfied...

DOI: 10.1007/s11424-015-2240-3 Received: 8 November 2012 / Revised: 1 July 2014 c ©The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2015 Abstract Let k be a positive integer. For any positive integer x = ∑∞ i=0 xi2 , where xi = 0, 1, we define the weight w(x) of x by w(x) := ∑∞ i=0 xi. For any integer t with 0 < t < 2 k − 1, let St := {(a, b) ∈ Z2|a+ b ≡ t (mod 2 − 1), w(a)+w(b)...

A directed BIBD with parameters (v,b,r,k,λ*) is a BIBD with parameters (v, b, r, k, 2λ*) in which each ordered pair of varieties occurs together in exactly λ* blocks. It is shown that λ*v(v 1) = 0 (mod 3) is a necessary and sufficient condition for the existence of a directed (v, b, r, k, λ*) BIBD with k = 3. Disciplines Physical Sciences and Mathematics Publication Details Seberry, J and Skill...