Abstract
In a recent article, Ullman and Velleman studied functions a from an abelian group G to itself that can be expressed as a difference of two bijections b, c from G to itself. In this work we relax the condition that b and c are bijections and instead study functions that can be expressed as the difference of two functions with the same value multiset. We construct all possible functions b, c with same value multiset, such that \(a = b - c\). As a consequence, we obtain a stronger version of Hall’s theorem, which gives a description of b and c in terms of a. We conclude by presenting further directions and questions that relate this new approach to applications of Hall’s theorem.
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Acknowledgements
The authors thank Gary Mullen for calling our attention to the paper that motivated this work [7] and for suggesting possible generalizations of some results on that paper. This work was supported in part by the Puerto Rico Louis Stokes Alliance for Minority Participation (PR-LSAMP) Program at the University of Puerto Rico, NSF grant #1400868.
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Cruz, D., Ramos, A., Rubio, I. (2024). Differences of Functions with the Same Value Multiset. In: Hoffman, F., Holliday, S., Rosen, Z., Shahrokhi, F., Wierman, J. (eds) Combinatorics, Graph Theory and Computing. SEICCGTC 2021. Springer Proceedings in Mathematics & Statistics, vol 448. Springer, Cham. https://doi.org/10.1007/978-3-031-52969-6_3
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DOI: https://doi.org/10.1007/978-3-031-52969-6_3
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