\textbf{Structure-magnetism correlation induced by Mn substitution in bilayered perovskite Sr}$_{\mathbf{3}}$\textbf{(Ru}$_{\mathbf{1-x}}$\textbf{Mn}$_{\mathbf{x}}$\textbf{)}$_{\mathbf{2}}$\textbf{O}$_{\mathbf{7}}$\textbf{ }

Elastic neutron scattering technique was employed to investigate the effect of Mn substitution on the structure, magnetism and their correlation in Sr$_{3}$(Ru$_{1-x}$Mn$_{x})_{2}$O$_{7\, }$(x$=$6{\%}, 12{\%} and 16{\%}) crystals. While parent compound Sr$_{3}$Ru$_{2}$O$_{7\, }$is paramagnetic, a small amount of Mn substitution induces an E-type antiferromagnetic order. With the increase of Mn substitution from 6 {\%} to 16{\%}, the ordered moment at Ru/Mn site increases significantly with an enhanced $T_{N}$ from 20 K for x$=$6 {\%} to 80 K for x$=$ 16{\%}, and the in-plane magnetic correlation lengths increase to achieve the maximum for x$=$ 16{\%} as indicated by the resolution-limited linewidth of the $H $scans through \textbf{Q}$_{\mathbf{AFM}}=$(0.5, 0, 0). Accompanied by the enhancement of $T_{N}$, the (Ru/Mn)O$_{6}$ octahedron rotation is found to be suppressed simultaneously, suggesting a correlation between (Ru/Mn)O$_{6}$ octahedron rotation and magnetism due to Mn substitution. Our findings indicate that Mn substitution on Ru in Sr$_{3}$Ru$_{2}$O$_{7}$ has a significant effect on the microscopic structure and magnetism as well as the correlations between them.