Maximum Works Performed by Signed Partial Transformations of a Finite Set
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Page Numbers:
500-508
Digital Object Identifier:
10.58578/mjms.v3i3.5962
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Authors:
Pokalas P. Tal1*, Eze Chibueze2, Yulari Sanda3 
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Abstract
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~.
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References
East, J., & McNamara, P. J. (2011). On the Work Performed by a Transformation Semigroup. Australasian Journal of Combinatorics, 49, 95-109. http://ajc.maths.uq.edu.au/pdf/49/ajc_v49_p095.pdf
Knuth, D. E. (1973). The Art of Computer Programming-3, Sorting and Searching. Addison-Wesley Publishing Company. https://seriouscomputerist.atariverse.com/media/pdf/book/Art%20of%20Computer%20Programming%20-%20Volume%203%20(Sorting%20&%20Searching).pdf
Tal, P. P., Mahmud, M. S., Mbah, M. A. & Ndubuisi, R. U. (2022). Maximum and Minimum Works Performed by . Modern Applied Science,16(2). Canadian Centre of Science. https://ccsenet.org/journal/index.php/mas/article/download/0/0/47106/50454
Diaconis, P., & Graham, R. L. (1977). Spearman's Footrule as a Measure of Disarray. Journal of the Royal Statistical Society: Series B (Methodological), 39(2), 262-268. https://doi.org/10.1111/j.2517-6161.1977.tb01624.x
Aitken, W. (1999). Total Relative Displacement of Permutations. Journal of Combinatorial Theory, 87(1), 1-21.https://doi.org/10.1006/jcta.1998.2943.
Gallero, R., Montorsi, G., Benedetto, S., & Cancellieri, G. (2001). Interleaver Properties and their Applications to the Trellis Complexity Analysis of Turbo Codes. Institute of Electrical and Electronics Engineers Transactions on Communications, 49, 793-807. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=e96d59c1a43a0429950150cc39360ccf0bdb3175
Ravichandran, V., & Srinivasan, N. (2003). Measures for Displacement of Permutations Used for Speech Scrambling. Journal of Indian Acad. Mathematics, 25(2), 251-259.
Imam, A. T., & Tal, P. P. (2019). On Maximum Works and Stretches Performed by Transformations of a Finite Set. Journal of the Nigerian Association of Mathematical Physics, 53, 21-28.