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.

This paper presents a falling‐based optimal foot trajectory planning method (FOFTP) for the effective walking of bipedal robots in three‐dimensional space. Our primary concern is to determine the optimal footstep location for the more balanced walking of bipedal robots based on a measure of falling. A proper strategy for the intermediate trajectory of the swing foot is als...