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The origin of the phase stability of 18 R Mgâ€"Znâ€"Y alloys with a long-period stacking order (LPSO) is studied using first-principles calculations. We calculate the heat of formation as a function of the number of Zn vacancies to discuss the role of Zn atoms. The calculated convex hull indicates that the Zn atoms in the LPSO alloys are stable even if they number about half of the Y atoms. The bonding state with Zn p orbitals leads to the stability of the LPSO structure because the partial density of states of Mg nearest to the solute cluster forms a valley structure. [ABSTRACT FROM AUTHOR]