Maximum Works Performed by Signed Partial Transformations of a Finite Set - Tal, Pokalas P.; Chibueze, Eze; Sanda, Yulari (Mikailalsys Journal of Mathematics and Statistics, 2025)
Let Xn and Xn* be the finite sets {1, 2, 3, ..., n} and {±1, ±2, ±3, ..., ±n} respectively. A map from Xn to Xn is called a transformation on Xn. We call a map a signed transformation if it maps from Xn to Xn*. Let Pn~ be the set of all signed partial transformations on Xn. This set consists of all transformations in Pn~ for which the domain of the transformation is a subset of Xn. The work w(alpha) performed by a transformation alpha is defined as the sum of all distances |i - alpha(i)| for each i in the domain of alpha. In this paper, we characterize all transformations in Pn~ that attain maximum and minimum works, and we deduce formulas for these minimum and maximum values. We further present a range for the values of w(alpha) for all transformations in Pn~.