Goldman, Joichi, and White proved a beautiful theorem showing that the falling factorial generating function for the rook numbers of a Ferrers board factors over the integers. Briggs and Remmel studied an analogue of rook placements where rows are replaced by sets of m rows called levels. They proved a version of the factorization theorem in that setting, but only for certain Ferrers boards. We...

Rook theory has been investigated by many people since its introduction by Kaplansky and Riordan in 1946. Goldman, Joichi and White in 1975 showed that the sum over k of the product of the (n− k)-th rook numbers multiplied by the k-th falling factorial polynomials factorize into a product. In the sequel, different types of generalizations and analogues of this product formula have been derived ...

We investigate the coefficients generated by expressing the falling factorial (xy) as a linear combination of falling factorial products xy for l, m = 1, . . . , k. Algebraic and combinatoric properties are discussed, some in relation to Stirling numbers.

We investigate non-homogeneous linear differential equations of the form x′′(t) + x′(t) − x (t) = p (t) where p (t) is either a polynomial or a factorial polynomial in t. We express the solution of these differential equations in terms of the coefficients of p (t), in the initial conditions, and in the solution of the corresponding homogeneous differential equation y′′(t) + y′(t) − y (t) = 0 wi...

In this article, improved bounds for (kn)! and the factorial polynomial are obtained using Kober’s inequality and Lagrange’s identity.

harkey & Sharkey (2010) have written a most interesting paper, raising interesting ethical and practical issues regarding robots as sole and parttime carers. What competencies can robots achieve in the near term? Can we experience social relatedness with them (and they with us)? Can robots care (in the sense of nurture) for us? Are their risks in deploying robots as carers for the young? Can we...

Let n, p, k be three positive integers. We prove that the numbers n k 3 F 2 (1 − k, −p, p−n ; 1, 1−n ; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application. z k k! , where for an indeterminate a and some positive integer k, the raising factorial is defined by (a) k = a(a + 1). .. (a ...

The conflict between the highland and the plains with the sharing of the highland waters, of late, has assumed an enhanced importance in the context of ecological disputes. While, policy makers and academicians have suggested several methods to resolve the conflict, the entry point in this analysis lies with the “beneficiaries pay” principle. Keeping with this principle, the paper initiates the...

This commentary considers how far one can go in making inferences about functional modularity or segregation, based on the sorts of analyses used by Caplan & Waters in relation to the underlying neuronal infrastructure. Specifically, an attempt is made to relate the “functionalist” approach adopted in the target article to “neuroreductionist” perspectives