Charge-orbital ordering near the multicritical point in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>A</mml:mi></mml:math>-site ordered perovskites <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="normal">Sm</mml:mi><mml:mi mathvariant="normal">Ba</mml:mi><mml:msub><mml:mi mathvariant="normal">Mn</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="normal">Nd</mml:mi><mml:mi mathvariant="normal">Ba</mml:mi><mml:msub><mml:mi mathvariant="normal">Mn</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>

Electronic and magnetic structures of $A$-site ordered perovskite-type manganites, $\mathit{Ln}\mathrm{Ba}{\mathrm{Mn}}_{2}{\mathrm{O}}_{6}$ ($\mathit{Ln}=\mathrm{Sm}$ and Nd), located close to the multicritical point of the competing phases, have been investigated by measurements of resonant x-ray scattering, optical spectroscopy, and neutron diffraction. In the $\mathrm{Sm}\mathrm{Ba}{\mathrm{Mn}}_{2}{\mathrm{O}}_{6}$ crystal, two resonant peaks, $(5∕4\phantom{\rule{0.3em}{0ex}}5∕4\phantom{\rule{0.3em}{0ex}}0)$ and $(3∕2\phantom{\rule{0.3em}{0ex}}1∕2\phantom{\rule{0.3em}{0ex}}1∕2)$ in the tetragonal ${a}_{p}\ifmmode\times\else\texttimes\fi{}{a}_{p}\ifmmode\times\else\texttimes\fi{}2{a}_{p}$ setting (${a}_{p}\ensuremath{\approx}3.9\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$), are observed at room temperature and are attributed to the $AABB$-type stacking of $ab$ planes with a checkerboard-type charge and stripe-type orbital orders. The change of the stacking manner from $AABB$- to $ABAB$-type around $190\mathrm{K}$ is evidenced by the disappearance of the resonant peak $(3∕2\phantom{\rule{0.3em}{0ex}}1∕2\phantom{\rule{0.3em}{0ex}}1∕2)$ and a steep increase in Raman scattering intensity of a Jahn-Teller phonon mode below $190\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The powder neutron diffraction study has shown that $\mathrm{Nd}\mathrm{Ba}{\mathrm{Mn}}_{2}{\mathrm{O}}_{6}$ has an A-type antiferromagnetic state below $275\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The abrupt expansion of the $ab$ plane and shrinkage along the $c$ axis upon the antiferromagnetic transition at $275\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ suggest that uniform ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ orbital ordering occurs.